Generalization of the Fuzzy Fejér–Hadamard Inequalities for Non-Convex Functions over a Rectangle Plane

Abstract

Integral inequalities with generalized convexity play a vital role in both theoretical and applied mathematics. The theory of integral inequalities is one of the branches of mathematics that is now developing at the quickest rate due to its wide range of applications. We define a new Hermite–Hadamard inequality for the novel class of coordinated ƛ-pre-invex fuzzy number-valued mappings (C-ƛ-pre-invex F N V Ms) and examine the idea of C-ƛ-pre-invex F N V Ms in this paper. Furthermore, using C-ƛ-pre-invex F N V Ms, we construct several new integral inequalities for fuzzy double Riemann integrals. Several well-known results, as well as recently discovered results, are included in these findings as special circumstances. We think that the findings in this work are new and will help to stimulate more research in this area in the future. Additionally, unique choices lead to new outcomes.

1. Introduction

Convexity is a key component of mathematical sciences and is useful in many domains, including optimization theory, economics, engineering, variational inequalities, management science, and Riemannian manifolds. Complex issues are simplified by convex sets and functions, which allow for effective computing solutions. Convex analysis-derived notions continue to be useful in a wide range of scientific and engineering fields. A strong mathematical idea known as convexity can be utilized to provide a theoretical framework for the development of efficient algorithms in a range of fields, as well as to simplify challenging mathematical issues. Convexity-derived integral inequalities provide a comprehensive understanding of the behavior of complex systems. Mathematical rigor is provided by these inequalities. They are essential tools for engineers and physicists because of their capacity to simulate, understand, and predict a broad variety of natural processes. The identification of new applications and connections made possible by this field of study will probably contribute to our comprehension of the physical universe. In conclusion, Jensen’s research and subsequent developments in convex analysis have made it clearer how useful convex functions are for comprehending optimization issues. This provides helpful methods and theoretical foundations for figuring out the optimal solutions in a range of scenarios. Mathematical convexity continues to be an important area of study and application in many different fields.

Generalized convexity ideas are used in approximation theory and probability distributions to approximate non-convex functions with convex functions. This approximation is useful for a wide range of computational and numerical techniques. To summarize, the mathematical framework for establishing and analyzing integral inequalities is shared by the closely related fields of study of generalized convexity and integral inequalities. The applications of these concepts in many domains, including physics, functional analysis, and optimization, highlight the importance of understanding the relationship between generalized convexity and integral inequality in theoretical and practical contexts. There are many different kinds of inequality in the literature. Hermite–Hadamard inequality, sometimes known as double inequality, is the most important component in optimization issues. In this context, we take into consideration the well-known inequality for convex functions due to Hadamard and Hermite separately; refer to Ref. [1].

Theorem 1.

Let $\mathcal{P}:\left[\mathcal{p},\mathcal{q}\right]\to \mathfrak{R}$ be a convex mapping, where $\mathfrak{R}$ is a set of real number. Then, the following outcome holds:

$$\mathcal{P}\left({\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)\le {\displaystyle \frac{1}{\mathcal{q}-\mathcal{p}}}{\int }_{\mathcal{p}}^{\mathcal{q}}\mathcal{P}\left(\mathrm{ҩ}\right)d\mathrm{ҩ}\le {\displaystyle \frac{\mathcal{P}\left(\mathcal{q}\right)+\mathcal{P}\left(\mathcal{p}\right)}{2}}.$$

(1)

One can check the concavity of the mappings by replacing the symbol “$\le ”$ with $“\ge ”$ in (1).

The continuous investigation into improved versions of the double inequality demonstrates the mathematical significance of this discovery and its broad applicability in other fields; refer to Refs. [2,3]. Invex functions are now significant extensions of convex functions in mathematical optimization and related fields. The first introduction of invex functions, which generalized classical convex mappings, was made by the authors in [4], who also went over some of their intriguing characteristics. A modification and generalization of conventional convex mappings, the modified forms of invex sets and pre-invex functions, were introduced by Ben and Mond together in [5]. One characteristic that sets this type of invexity apart is that the differentiable pre-invex mappings are invex, but not the other way around. For further research related to pre-invexity and its applications, see [6,7,8] and the references therein.

With interval-valued analysis, one may successfully handle errors and uncertainties in a variety of computational tasks. This method is especially helpful for applications that demand precise forecasts and dependable outcomes, since it represents numerical values as intervals, ensuring that results are based on uncertainties in input data. It provides a practical and cautious method of computing by portraying numerical values as intervals. Interval analysis has several applications in various domains, including mathematics, computer science, engineering, and natural science, thanks to Moore’s [9] contributions (see [10]). Mathematicians are inspired to extend integral inequalities to interval-valued mappings due to exact results in a range of areas. In the beginning, writers in [11] linked Jensen-type and Hadamard-type results in the setup of set-valued functions using h-convex mappings. Afzal et al. [12] created three well-known inequalities that provide insight into the characteristics and behavior of stochastic processes inside a probability space by fusing the ideas of set-valued analysis and h-convex mappings. The concept of pre-invex functions was utilized by the authors in [13] to produce double inequality for set-valued mappings. The concept of pre-invex functions on coordinates is used by the authors in [14] to produce a number of double inequality conclusions on the rectangular plane. Sharma et al. [15] used fractional integral operators to build various novel product forms of the double inequality by utilizing the concept of (h1, h2)-pre-invex functions in the creation of set-valued mappings. In the setting of set-valued mappings, Zhou et al. [16] created an improved version of the double inequalities by utilizing pre-invex exponential type functions via fractional integrals. Up-down pre-invex mappings in a fuzzy setup were employed by Khan et al. [17] to obtain Fejér- and Hermite-type results. Noor et al. [18] produced several Hermite–Hadamard-type theorems linked to special functions using power mean integral inequalities, utilizing the idea of (h1, h2)-pre-invex mappings. Sun et al. [19] developed a number of novel double inequalities for h-pre-invex functions with applications by utilizing local fractional integrals. Using the idea of (s,m,φ)-type functions, the writers of [20] created a number of innovative forms of double inequalities for generalized pre-invex mappings that had some intriguing characteristics. Based on Godunova–Levin pre-invex mappings, Ali et al. [21] created several novel variations of Hermite-Hadamard-type findings using partial-order relations. Various novel Hermite–Hadamard- and Fejér-type findings were developed by Tariq et al. [22] by merging fractional operators and generalized pre-invex mappings. Sitho et al. [23] proved mid-point and trapezoidal inequality for differentiable pre-invex functions using the concept of quantum integrals. Applications of Trapezium-type inequalities for h-pre-invex functions to specific means were studied by Latif et al. [24]. The use by Delavar et al [25] of fractional integrals resulted in new limits for the mid-point type and Hermite–Hadamard’s trapezoid inequality. Further details on these findings and some other fascinating recent advances can be found in the references [26,27,28,29,30,31,32].

Additionally, we describe various applications for random variables within the context of error limits that also generalize other conclusions. This is the first time in the literature that we have established error bounds for quadrature-type formulas using this class of generalized fuzzy convexity. Since we cannot compare two intervals, the majority of the research is based on fuzzy partial-order or pseudo-order relationships, which show serious problems in several of the inequality results. The benefit of this order relationship is that intervals can be easily compared. More significantly, though, the interval difference’s endpoints are substantially closer together, allowing for a more accurate outcome. In 2023, different authors used distinct classes of convexities to derive different solutions utilizing Bhunias–Samanata-order relation; see [33,34,35].

Since the fuzzy set notion was introduced [36], fuzzy set theory has grown to be an effective tool for processing ambiguous or subjective data in mathematical models and for modeling uncertainty. The primary research areas have focused on a variety of applications in pressing issues, such as population dynamics [37,38], medicine [39,40,41], renewable and sustainable energy [42,43,44,45,46], engineering issues [47], and image processing [48,49]. In the same vein, fuzzy mathematical analysis is a crucial subject from both a theoretical and practical application standpoint [50,51]. For further study related to basic operations, Aumann’s integrations, and fractional integrals, where integrable mappings comprise interval-valued and fuzzy-number-valued mappings, see [52,53,54,55,56,57,58] and the references therein. Moreover, Khan et al. [59,60,61,62], and Matloka [63] presented the new versions of integral inequalities for fuzzy-number-valued mappings, real-valued mappings, and interval-value mappings with the help of coordinated convexity. Recently, Khan et al. [64] gave the fractional versions of inequalities via coordinated convexity and non-convexity, where integrable functions are fuzzy-number-valued mappings. For more information, see [65] and the reference therein.

This work is innovative and important because it uses various choices of bifunction “$\mathrm{\hslash }$” to introduce a more generalized class of pre-invex functions called coordinated $\mathrm{ƛ}$-pre-invex $\mathsf{F}$·$\mathsf{N}$·$\mathsf{V}$·$\mathsf{M}$s, which unify diverse previously reported findings. A more generalized version of inequality is deduced with this class, since pre-invexity offers more good features than conventional convex maps, and convexity and pre-invexity are two independent concepts.

We are defining a new class of pre-invexity for the first time thanks to the literature on developed inequalities, and, in particular, these articles [13,14,63]. Using these ideas, we are developing a number of novel variants of the well-known double- and trapezoid-type inequalities and their relationship to Fejér ’s work. This essay is organized as follows: building on the preparatory work in Section 2, In Section 3, we introduce a new class of pre-invexity and discuss some of its fascinating aspects. The primary findings of this work are reported in Section 3, where we share modified Hermite–Hadamard–Fejér-type results, as well as establishing various versions of well-known double-type inequalities. Section 4 concludes with a review of some closing observations and recommendations for further research.

2. Preliminaries

In this section, we go over a few recent definitions and findings that might bolster the study’s main conclusions. Moreover, several concepts are utilized in articles without being clarified; refer to Refs. [50,52,53,54,55,56,57,58]. In the following results, to avoid confusion, we will not include the symbols $\left(R\right)$, $\left(IR\right)$, $\left(FR\right)$, $\left(ID\right)$, or $\left(FD\right)$ before the integral sign. Moreover, the notions ${\mathbb{R}}_I$, ${\mathbb{R}}_I^{+}$, ${\mathbb{F}}_0$ and ${\mathbb{F}}_0^{+}$ and ${\mathbb{R}}_I$ will be used for a set of fuzzy number, set of positive fuzzy number, set of intervals, and collection of positive intervals, respectively.

Definition 1

([52,54]). Given $\widetilde{\mathrm{Ɋ}}\in {\mathbb{F}}_0$, the level sets or cut sets are given by

$${\left[\widetilde{\mathrm{Ɋ}}\right]}^{\mathrm{ɤ}}=\left\{\mathrm{ҩ}\in \mathfrak{R}|\widetilde{\mathrm{Ɋ}}\left(\mathrm{ҩ}\right)>\mathrm{ɤ}\right\},$$

(2)

$\forall$ $\mathrm{ɤ}\in (0,1]$ and by

$${\left[\widetilde{\mathrm{Ɋ}}\right]}^0=\left\{\mathfrak{ҩ}\in \mathfrak{R}|\widetilde{\mathrm{Ɋ}}\left(\mathrm{ҩ}\right)>0\right\}.$$

(3)

These sets are known as $\mathrm{ɤ}$-level sets or $\mathrm{ɤ}$-cut sets of $\widetilde{\mathrm{Ɋ}}$.

Proposition 1

([40]). Let $\widetilde{\mathrm{Ɋ}},\widetilde{\mathrm{Ŋ}}\in {\mathbb{F}}_0$. Then, relation $“{\le }_{\mathbb{F}}”$ is given on ${\mathbb{F}}_0$ by $\widetilde{\mathrm{Ɋ}}{\le }_{\mathbb{F}}\widetilde{\mathrm{Ŋ}}$ when and only when ${{\left[\widetilde{\mathrm{Ɋ}}\right]}^{\mathrm{ɤ}}\le }_I{\left[\widetilde{\mathrm{Ŋ}}\right]}^{\mathrm{ɤ}}$, for every $\mathrm{ɤ}\in [0,1],$ which are left- and right-order relations.

Proposition 2

([37]). Let $\widetilde{\mathrm{Ɋ}},\widetilde{\mathrm{Ŋ}}\in {\mathbb{F}}_0$. Then, relation $“{\supseteq }_{\mathbb{F}}”$ is given on ${\mathbb{F}}_0$ by $\widetilde{\mathrm{Ɋ}}{\supseteq }_{\mathbb{F}}\widetilde{\mathrm{Ŋ}}$ when and only when ${\left[\widetilde{\mathrm{Ɋ}}\right]}^{\mathrm{ɤ}}{\supseteq }_I{\left[\widetilde{\mathrm{Ŋ}}\right]}^{\mathrm{ɤ}}$ for every $\mathrm{ɤ}\in [0,1],$ which is the $UD$-order relation on ${\mathbb{F}}_0$.

Remember the approaching notions, which are given in the literature. If $\widetilde{\mathrm{Ɋ}},\widetilde{\mathrm{Ŋ}}\in {\mathbb{F}}_0$ and $\mathcal{t}\in \mathfrak{R}$, then, for every $\mathrm{ɤ}\in \left[0,1\right],$ the arithmetic operation addition $“\oplus ”$, multiplication $“\otimes ”$, and scaler multiplication $“\odot ”$ are defined by

$${\left[\widetilde{\mathrm{Ɋ}}\oplus \widetilde{\mathrm{Ŋ}}\right]}^{\mathrm{ɤ}}={\left[\widetilde{\mathrm{Ɋ}}\right]}^{\mathrm{ɤ}}+{\left[\widetilde{\mathrm{Ŋ}}\right]}^{\mathrm{ɤ}},$$

(4)

$${\left[\widetilde{\mathrm{Ɋ}}\otimes \widetilde{\mathrm{Ŋ}}\right]}^{\mathrm{ɤ}}={\left[\widetilde{\mathrm{Ɋ}}\right]}^{\mathrm{ɤ}}\times {\left[\widetilde{\mathrm{Ŋ}}\right]}^{\mathrm{ɤ}},$$

(5)

$${\left[\mathcal{t}\odot \widetilde{\mathrm{Ɋ}}\right]}^{\mathrm{ɤ}}=\mathcal{t}{\left[\widetilde{\mathrm{Ɋ}}\right]}^{\mathrm{ɤ}}.$$

(6)

The equations numbered from (4) to (6) directly influence these results.

Theorem 2

([55]). Let $\tilde{\mathcal{P}}:[\mathcal{p},\mathcal{q}]\subset \mathfrak{R}\to {\mathbb{F}}_0$ be an $\mathsf{F}$·$\mathsf{N}$·$\mathsf{V}$·$\mathsf{M}$, its $\mathsf{I}$·$\mathsf{V}$·$\mathsf{M}$s are classified according to their $\mathrm{ɤ}$-levels ${\mathcal{P}}_{\mathrm{ɤ}}:[\mathcal{p},\mathcal{q}]\subset \mathfrak{R}\to {\mathbb{R}}_I$ are given by ${\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ҩ}\right)=\left[{\mathcal{P}}_{*}\left(\mathrm{ҩ},\mathrm{ɤ}\right),{\mathcal{P}}^{*}\left(\mathrm{ҩ},\mathrm{ɤ}\right)\right]$ $\forall$ $\mathrm{ҩ}\in [\mathcal{p},\mathcal{q}]$ and $\forall$ $\mathrm{ɤ}\in (0,1].$ Then, $\tilde{\mathcal{P}}$ is $FA$-integrable over $[\mathcal{p},\mathcal{q}]$ if and only if ${\mathcal{P}}_{*}\left(\mathrm{ҩ},\mathrm{ɤ}\right)$ and ${\mathcal{P}}^{*}\left(\mathrm{ҩ},\mathrm{ɤ}\right)$ are both $A$-integrable over $[\mathcal{p},\mathcal{q}]$. Moreover, if $\tilde{\mathcal{P}}$ is $FA$-integrable over $\left[\mathcal{p},\mathcal{q}\right],$ then

$${\left[\left(FA\right){\int }_{\mathcal{p}}^{\mathcal{q}}\tilde{\mathcal{P}}\left(\mathrm{ҩ}\right)d\mathrm{ҩ}\right]}^{\mathrm{ɤ}}=\left[\left(A\right){\int }_{\mathcal{p}}^{\mathcal{q}}{\mathcal{P}}_{*}\left(\mathrm{ҩ},\mathrm{ɤ}\right)d\mathrm{ҩ}, \left(A\right){\int }_{\mathcal{p}}^{\mathcal{q}}{\mathcal{P}}^{*}\left(\mathrm{ҩ},\mathrm{ɤ}\right)d\mathrm{ҩ}\right]$$

(7)

$$=\left(IA\right){\int }_{\mathcal{p}}^{\mathcal{q}}{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ҩ}\right)d\mathrm{ҩ}$$

(8)

$\forall$ $\mathrm{ɤ}\in (0,1].$ For all $\mathrm{ɤ}\in \left(0,1\right],$ ${\mathcal{F}\mathcal{A}}_{\left(\left[\mathcal{p},\mathcal{q}\right],\mathrm{ɤ}\right)}$ denotes the collection of all $FA$-integrable $\mathsf{F}$·$\mathsf{N}$·$\mathsf{V}$·$\mathsf{M}$s over $[\mathcal{p},\mathcal{q}]$.

Here, we will give some definitions of Aumann’s fractional integrals over a rectangle plane.

Definition 2

([57]). Let $\mathcal{P}:\left[\mathrm{ⱱ},\mathrm{ⱳ}\right]\to {\mathbb{R}}_I^{+}$ be $\mathsf{I}$·$\mathsf{V}$·$\mathsf{M}$ and $\mathcal{P}\in {\mathcal{I}\mathcal{R}}_{\left[\mathrm{ⱱ},\mathrm{ⱳ}\right]}$. Subsequently, the interval integrals of $\mathcal{P}$ using the Riemann–Liouville approach are delineated as follows:

$${\mathcal{I}}_{{\mathrm{ⱱ}}^{+}}^{\alpha }\mathcal{P}\left(\mathrm{ʑ}\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_{\mathrm{ⱱ}}^{\mathrm{ʑ}}{\left(\mathtt{ʑ}-\mathtt{t}\right)}^{\alpha -1}\mathcal{P}\left(\mathtt{t}\right)d\mathtt{t}\left(\mathrm{ʑ}>\mathrm{ⱱ}\right),$$

(9)

$${\mathcal{I}}_{{\mathrm{ⱳ}}^{-}}^{\alpha }\mathcal{P}\left(\mathrm{ʑ}\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_{\mathrm{ʑ}}^{\mathrm{ⱳ}}{\left(\mathtt{t}-\mathtt{ʑ}\right)}^{\alpha -1}\mathcal{P}\left(\mathtt{t}\right)d\mathtt{t}\left(\mathrm{ʑ}<\mathrm{ⱳ}\right),$$

(10)

where $\alpha >0$ and $\Gamma$ is the gamma function.

In a recent study by Allahviranloo et al. [58], they proposed a fuzzy adaptation of fractional integral formulations, which can be expressed as follows:

Definition 3.

Let $\alpha >0$ and $L\left(\left[\mathrm{ⱱ},\mathrm{ⱳ}\right],{\mathbb{F}}_0\right)$ be the collection of all Lebesgue measurable $\mathsf{F}$·$\mathsf{N}$·$\mathsf{V}$·$\mathsf{M}$s on $[\mathrm{ⱱ},\mathrm{ⱳ}]$. Then, the fuzzy left and right Riemann–Liouville fractional integral of $\tilde{\mathcal{P}}\in L\left(\left[\mathrm{ⱱ},\mathrm{ⱳ}\right],{\mathbb{F}}_0\right)$ with order $\alpha >0$ are defined by

$${\mathcal{I}}_{{\mathrm{ⱱ}}^{+}}^{\alpha }\tilde{\mathcal{P}}\left(\mathrm{ʑ}\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_{\mathrm{ⱱ}}^{\mathrm{ʑ}}{\left(\mathtt{ʑ}-\mathtt{t}\right)}^{\alpha -1}\tilde{\mathcal{P}}\left(\mathtt{t}\right)d\mathtt{t},\hspace{1em}\hspace{1em}\left(\mathrm{ʑ}>\mathrm{ⱱ}\right),$$

(11)

and

$${\mathcal{I}}_{{\mathrm{ⱳ}}^{-}}^{\alpha }\tilde{\mathcal{P}}\left(\mathrm{ʑ}\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_{\mathrm{ʑ}}^{\mathrm{ⱳ}}{\left(\mathtt{t}-\mathtt{ʑ}\right)}^{\alpha -1}\tilde{\mathcal{P}}\left(\mathtt{t}\right)d\mathtt{t},\hspace{1em}\hspace{1em}\left(\mathrm{ʑ}<\mathrm{ⱳ}\right),$$

(12)

respectively, where $\Gamma \left(\mathrm{ʑ}\right)={\int }_0^{\infty }{\mathtt{t}}^{\mathrm{ʑ}-1}{e}^{-\mathtt{t}}d\mathtt{t}$ is the Euler gamma function.

The fuzzy fractional $\mathrm{ʑ}$-based integrals, both left and right, in the Riemann–Liouville framework, which rely on endpoint functions, can be characterized as follows:

$$\begin{gathered} {\left[{\mathcal{I}}_{{\mathrm{ⱱ}}^{+}}^{\alpha }\tilde{\mathcal{P}}\left(\mathrm{ʑ}\right)\right]}^{\mathrm{ɤ}}={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_{\mathrm{ⱱ}}^{\mathrm{ʑ}}{\left(\mathtt{ʑ}-\mathtt{t}\right)}^{\alpha -1}{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathtt{t}\right)d\mathtt{t} \\ ={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_{\mathrm{ⱱ}}^{\mathrm{ʑ}}{\left(\mathtt{ʑ}-\mathtt{t}\right)}^{\alpha -1}\left[{\mathcal{P}}_{*}\left(\mathtt{t},\mathrm{ɤ}\right),{\mathcal{P}}^{*}\left(\mathtt{t},\mathrm{ɤ}\right)\right]d\mathtt{t}, \left(\mathrm{ʑ}>\mathrm{ⱱ}\right), \end{gathered}$$

(13)

where

$${\mathcal{I}}_{{\mathrm{ⱱ}}^{+}}^{\alpha }{\mathcal{P}}_{*}\left(\mathrm{ʑ},\mathrm{ɤ}\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_{\mathrm{ⱱ}}^{\mathrm{ʑ}}{\left(\mathtt{ʑ}-\mathtt{t}\right)}^{\alpha -1}{\mathcal{P}}_{*}\left(\mathtt{t},\mathrm{ɤ}\right)d\mathtt{t}, \left(\mathrm{ʑ}>\mathrm{ⱱ}\right),$$

(14)

and

$${\mathcal{I}}_{{\mathrm{ⱱ}}^{+}}^{\alpha }{\mathcal{P}}^{*}\left(\mathrm{ʑ},\mathrm{ɤ}\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_{\mathrm{ⱱ}}^{\mathrm{ʑ}}{\left(\mathtt{ʑ}-\mathtt{t}\right)}^{\alpha -1}{\mathcal{P}}^{*}\left(\mathtt{t},\mathrm{ɤ}\right)d\mathtt{t}, \left(\mathrm{ʑ}>\mathrm{ⱱ}\right).$$

(15)

The fuzzy Riemann–Liouville fractional integral on the right side, symbolized as ${\left[{\mathcal{I}}_{{\mathrm{ⱳ}}^{-}}^{\alpha }\tilde{\mathcal{P}}\left(\mathrm{ʑ}\right)\right]}^{\mathrm{ɤ}}$, can likewise be formulated by incorporating both left and right endpoint functions.

Theorem 3

([59]). Let $\mathrm{ƛ}:\left[0,1\right]\to {\mathbb{R}}^{+}$ and $\tilde{\mathcal{P}}:\left[\mathcal{p},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]\to {\mathbb{F}}_0^{+}$ be a pre-invex $\mathsf{F}$·$\mathsf{N}$·$\mathsf{V}$·$\mathsf{M}$ on $\left[\mathcal{p},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right],$ whose $\mathrm{ɤ}$-cuts set up the sequence of $\mathsf{I}$·$\mathsf{V}$·$\mathsf{M}$s ${\mathcal{P}}_{\mathrm{ɤ}}:\left[\mathcal{p},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]\subset \mathbb{R}\to {\mathbb{R}}_I^{+}$ are given by ${\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ʑ}\right)=\left[{\mathcal{P}}_{*}\left(\mathrm{ʑ},\mathrm{ɤ}\right), {\mathcal{P}}^{*}\left(\mathrm{ʑ},\mathrm{ɤ}\right)\right]$ for all $\mathrm{ʑ}\in \left[\mathcal{p},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]$ and for all $\mathrm{ɤ}\in \left[0,1\right]$. If $\tilde{\mathcal{P}}\in L\left(\left[\mathcal{p},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right],{\mathbb{F}}_0\right)$, then

$$\begin{array}{cc}{\displaystyle \frac{1}{\alpha \mathrm{ƛ}\left(\frac{1}{2}\right)}}\tilde{\mathcal{P}}\left({\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)\hfill & {\le }_{\mathbb{F}}{\displaystyle \frac{\Gamma \left(\alpha \right)}{{\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\alpha }}}\left[{\mathcal{I}}_{{\mathcal{p}}^{+}}^{\alpha }\tilde{\mathcal{P}}\left(\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\oplus {\mathcal{I}}_{{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\alpha }\tilde{\mathcal{P}}\left(\mathcal{p}\right)\right]\hfill \\ & {\le }_{\mathbb{F}}\left[\tilde{\mathcal{P}}\left(\mathcal{p}\right)\oplus \tilde{\mathcal{P}}\left(\mathcal{q}\right)\right]{\int }_0^1{\mathtt{t}}^{\beta -1}\left[\mathrm{ƛ}\left(\mathtt{t}\right)+\mathrm{ƛ}\left(1-\mathtt{t}\right)\right]d\mathtt{t}.\hfill \end{array}$$

(16)

Interval and fuzzy integrals in the style of Aumann are described as follows for $\mathsf{C}$-$\mathsf{I}$·$\mathsf{V}$·$\mathsf{M}$ $\mathcal{P}\left(\mathrm{ҩ},\mathrm{ʑ}\right)$ and $\mathsf{C}$-$\mathsf{F}$·$\mathsf{N}$·$\mathsf{V}$·$\mathsf{M}$ $\tilde{\mathcal{P}}\left(\mathrm{ҩ},\mathrm{ʑ}\right)$.

Theorem 4

([61]). Let $\tilde{\mathcal{P}}:▽=\left[\mathrm{ⱱ},\mathrm{ⱳ}\right]\times \left[\mathcal{p},\mathcal{q}\right]\subset {\mathbb{R}}^2\to {\mathbb{F}}_0$ be an $\mathsf{F}$·$\mathsf{N}$·$\mathsf{V}$·$\mathsf{M}$ on coordinates, whose $\mathrm{ɤ}$-cuts set up the sequence of $\mathsf{I}$·$\mathsf{V}$·$\mathsf{M}s$ ${\mathcal{P}}_{\mathrm{ɤ}}:▽\subset {\mathbb{R}}^2\to {\mathbb{R}}_I$ are given by ${\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ҩ},\mathrm{ʑ}\right)=\left[{\mathcal{P}}_{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right), {\mathcal{P}}^{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)\right]$ for all $\left(\mathrm{ҩ},\mathrm{ʑ}\right)\in ▽=\left[\mathrm{ⱱ},\mathrm{ⱳ}\right]\times \left[\mathcal{p},\mathcal{q}\right]$ and for all $\mathrm{ɤ}\in \left[0,1\right].$ Then, $\tilde{\mathcal{P}}$ is fuzzy double-integrable ($FD$-integrable) over $▽$ if and only if ${\mathcal{P}}_{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)$ and ${\mathcal{P}}^{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)$ both are $D$-integrable over $▽.$ Moreover, if $\tilde{\mathcal{P}}$ is $FD$-integrable over $▽,$ then

$$\begin{array}{cc}{\left[\left(FD\right){\int }_{\mathrm{ⱱ}}^{\mathrm{ⱳ}}{\int }_{\mathcal{p}}^{\mathcal{q}}\tilde{\mathcal{P}}\left(\mathrm{ҩ},\mathrm{ʑ}\right)d\mathrm{ʑ}d\mathrm{ҩ}\right]}^{\mathrm{ɤ}}\hfill & =\left[\left(D\right){\int }_{\mathrm{ⱱ}}^{\mathrm{ⱳ}}{\int }_{\mathcal{p}}^{\mathcal{q}}{\mathcal{P}}_{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)d\mathrm{ʑ}d\mathrm{ҩ}, \left(D\right){\int }_{\mathrm{ⱱ}}^{\mathrm{ⱳ}}{\int }_{\mathcal{p}}^{\mathcal{q}}{\mathcal{P}}^{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)d\mathrm{ʑ}d\mathrm{ҩ}\right]\hfill \\ & =\left(ID\right){\int }_{\mathrm{ⱱ}}^{\mathrm{ⱳ}}{\int }_{\mathcal{p}}^{\mathcal{q}}{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ҩ},\mathrm{ʑ}\right)d\mathrm{ʑ}d\mathrm{ҩ},\hfill \end{array}$$

(17)

for all $\mathrm{ɤ}\in \left[0,1\right].$

The family of all $FD$-integrable functions of $\mathsf{F}$·$\mathsf{N}$·$\mathsf{V}$·$\mathsf{M}$s over coordinates and $D$-integrable functions over coordinates are denoted by ${\mathcal{F}\mathfrak{O}}_{▽}$ and ${\mathfrak{O}}_{\left(▽,\mathrm{ɤ}\right)},$ for all $\mathrm{ɤ}\in \left[0,1\right].$

The primary specification for the fuzzy Riemann–Liouville fractional integral concerning the coordinates of the function $\tilde{\mathcal{P}}\left(\mathrm{ҩ},\mathrm{ʑ}\right)$ is presented as follows.

Definition 4

([60]). Let $\tilde{\mathcal{P}}:▽\to {\mathbb{F}}_0$ and $\tilde{\mathcal{P}}\in {\mathcal{F}\mathfrak{O}}_{▽}$. The double fuzzy interval Riemann–Liouville-type integrals ${\mathcal{I}}_{{\mathrm{ⱱ}}^{+},{\mathcal{p}}^{+}}^{\alpha ,\beta }, {\mathcal{I}}_{{\mathrm{ⱱ}}^{+},{\mathcal{q}}^{-}}^{\alpha ,\beta },{\mathcal{I}}_{{\mathrm{ⱳ}}^{-},{\mathcal{p}}^{+}}^{\alpha ,\beta },{\mathcal{I}}_{{\mathrm{ⱳ}}^{-},{\mathcal{q}}^{-}}^{\alpha ,\beta }$ of $\tilde{\mathcal{P}}$ order $\alpha ,\beta >0$ are defined by

$${\mathcal{I}}_{{\mathrm{ⱱ}}^{+},{\mathcal{p}}^{+}}^{\alpha ,\beta }\tilde{\mathcal{P}}\left(\mathrm{ҩ},\mathrm{ʑ}\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}}{\int }_{\mathrm{ⱱ}}^{\mathrm{ҩ}}{\int }_{\mathcal{p}}^{\mathrm{ʑ}}{{\left(\mathtt{ҩ}-\mathtt{t}\right)}^{\alpha -1}\left(\mathtt{ʑ}-\mathtt{s}\right)}^{\beta -1}\tilde{\mathcal{P}}\left(\mathtt{t},\mathtt{s}\right)d\mathtt{s}d\mathtt{t}, \left(\mathrm{ҩ}>\mathrm{ⱱ},\mathcal{ʑ}>\mathcal{p}\right),$$

(18)

$${\mathcal{I}}_{{\mathrm{ⱱ}}^{+},{\mathcal{q}}^{-}}^{\alpha ,\beta }\tilde{\mathcal{P}}\left(\mathrm{ҩ},\mathrm{ʑ}\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}}{\int }_{\mathrm{ⱱ}}^{\mathrm{ҩ}}{\int }_{\mathrm{ʑ}}^{\mathcal{q}}{{\left(\mathtt{ҩ}-\mathtt{t}\right)}^{\alpha -1}\left(\mathtt{s}-\mathtt{ʑ}\right)}^{\beta -1}\tilde{\mathcal{P}}\left(\mathtt{t},\mathtt{s}\right)d\mathtt{s}d\mathtt{t}, \left(\mathrm{ҩ}>\mathrm{ⱱ},\mathcal{ʑ}<\mathcal{q}\right),$$

(19)

$${\mathcal{I}}_{{\mathrm{ⱳ}}^{-},{\mathcal{p}}^{+}}^{\alpha ,\beta }\tilde{\mathcal{P}}\left(\mathrm{ҩ},\mathrm{ʑ}\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}}{\int }_{\mathrm{ҩ}}^{\mathrm{ⱳ}}{\int }_{\mathcal{p}}^{\mathrm{ʑ}}{{\left(\mathtt{t}-\mathtt{ҩ}\right)}^{\alpha -1}\left(\mathtt{ʑ}-\mathtt{s}\right)}^{\beta -1}\tilde{\mathcal{P}}\left(\mathtt{t},\mathtt{s}\right)d\mathtt{s}d\mathtt{t}, \left(\mathrm{ҩ}<\mathrm{ⱳ},\mathcal{ʑ}>\mathcal{p}\right),$$

(20)

$${\mathcal{I}}_{{\mathrm{ⱳ}}^{-},{\mathcal{q}}^{-}}^{\alpha , \beta }\tilde{\mathcal{P}}\left(\mathrm{ҩ},\mathrm{ʑ}\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}}{\int }_{\mathrm{ҩ}}^{\mathrm{ⱳ}}{\int }_{\mathrm{ʑ}}^{\mathcal{q}}{{\left(\mathtt{t}-\mathtt{ҩ}\right)}^{\alpha -1}\left(\mathtt{s}-\mathtt{ʑ}\right)}^{\beta -1}\tilde{\mathcal{P}}\left(\mathtt{t},\mathtt{s}\right)d\mathtt{s}d\mathtt{t}, (\mathrm{ҩ}<\mathrm{ⱳ},\mathcal{ʑ}<\mathcal{q})$$

(21)

Presented below is the recently formulated notion of $\mathsf{C}$-$\mathrm{ƛ}$-pre-invexity across fuzzy number space within the codomain through the fuzzy relation denoted by the following.

Definition 5.

The $\mathsf{F}$·$\mathsf{N}$·$\mathsf{V}$·$\mathsf{M}$ $\tilde{\mathcal{P}}:▽\to {\mathbb{F}}_0$ is referred to as $\mathsf{C}$-$\mathrm{ƛ}$-pre-invex $\mathsf{F}$·$\mathsf{N}$·$\mathsf{V}$·$\mathsf{M}$ on an invex set $▽$ if

$$\begin{gathered} \tilde{\mathcal{P}}\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right) \\ {\le }_{\mathbb{F}}\mathrm{ƛ}\left(\mathtt{t}\right)\mathrm{ƛ}\left(\mathrm{ԟ}\right)\tilde{\mathcal{P}}\left(\mathcal{ⱱ},\mathcal{p}\right)\oplus \mathrm{ƛ}\left(\mathtt{t}\right)\mathrm{ƛ}\left(1-\mathrm{ԟ}\right)\tilde{\mathcal{P}}\left(\mathcal{ⱱ},\mathcal{q}\right)\oplus \mathrm{ƛ}\left(1-\mathtt{t}\right)\mathrm{ƛ}\left(\mathrm{ԟ}\right)\tilde{\mathcal{P}}\left(\mathcal{ⱳ},\mathcal{p}\right)\oplus \mathrm{ƛ}\left(1-\mathtt{t}\right)\mathrm{ƛ}\left(1-\mathrm{ԟ}\right)\tilde{\mathcal{P}}\left(\mathcal{ⱳ},\mathcal{q}\right), \end{gathered}$$

(22)

for all $\left(\mathrm{ⱱ}, \mathrm{ⱳ}\right), \left(\mathcal{p},\mathcal{q}\right)\in ▽,$ and $\mathtt{t},\mathrm{ԟ}\in \left[0,1\right],$ where $\tilde{\mathcal{P}}\left(\mathrm{ҩ}\right){\ge }_{\mathbb{F}}\tilde{0}.$ If inequality (22) is reversed, then $\tilde{\mathcal{P}}$ is referred to as coordinate $\mathrm{ƛ}$-pre-concave $\mathsf{F}$·$\mathsf{N}$·$\mathsf{V}$·$\mathsf{M}$ on $▽$.

Lemma 1.

Let $\tilde{\mathcal{P}}:▽\to {\mathbb{F}}_0$ be a $\mathsf{C}$-$\mathsf{F}$·$\mathsf{N}$·$\mathsf{V}$·$\mathsf{M}$ on $▽$. Then, $\tilde{\mathcal{P}}$ is $\mathsf{C}$-$\mathrm{ƛ}$-pre-invex $\mathsf{F}$·$\mathsf{N}$·$\mathsf{V}$·$\mathsf{M}$ on $▽$ if and only if there exist two $\mathsf{C}$-$\mathrm{ƛ}$-pre-invex $\mathsf{F}$·$\mathsf{N}$·$\mathsf{V}$·$\mathsf{M}$s ${\tilde{\mathcal{P}}}_{\mathrm{ҩ}}:\left[\mathcal{p},\mathcal{q}\right]\to {\mathbb{F}}_0$, ${\tilde{\mathcal{P}}}_{\mathrm{ҩ}}\left(w\right)=\tilde{\mathcal{P}}\left(\mathrm{ҩ},w\right)$ and ${\tilde{\mathcal{P}}}_{\mathrm{ʑ}}:\left[\mathrm{ⱱ},\mathrm{ⱳ}\right]\to {\mathbb{F}}_0$, ${\tilde{\mathcal{P}}}_{\mathrm{ʑ}}\left(z\right)=\tilde{\mathcal{P}}\left(z,\mathrm{ʑ}\right)$.

Theorem 5.

Let $\tilde{\mathcal{P}}:▽\to {\mathbb{F}}_0$ be an $\mathsf{F}$·$\mathsf{N}$·$\mathsf{V}$·$\mathsf{M}$ on $▽$. Subsequently, derived from ɤ-levels, we acquire the set of $\mathsf{I}$·$\mathsf{V}$·$\mathsf{M}s$ ${\mathcal{P}}_{\mathrm{ɤ}}:▽\to {\mathbb{R}}_I^{+}\subset {\mathbb{R}}_I$, which are expressed as

$${\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ҩ},\mathrm{ʑ}\right)=\left[{\mathcal{P}}_{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right), {\mathcal{P}}^{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)\right],$$

(23)

for all $\left(\mathrm{ҩ},\mathrm{ʑ}\right)\in ▽$ and for all $\mathrm{ɤ}\in \left[0,1\right]$. Then, $\tilde{\mathcal{P}}$ is $\mathsf{C}$-$\mathrm{ƛ}$-pre-invex $\mathsf{F}$·$\mathsf{N}$·$\mathsf{V}$·$\mathsf{M}$ on $▽,$ if and only if, for all $\mathrm{ɤ}\in \left[0,1\right],$ ${\mathcal{P}}_{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)$ and ${\mathcal{P}}^{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)$ both are $\mathsf{C}$-$\mathrm{ƛ}$-pre-invex.

Proof. 

Assume that for each $\mathrm{ɤ}\in \left[0,1\right],$ ${\mathcal{P}}_{*}\left(\mathcal{x},\mathrm{ɤ}\right)$ and ${\mathcal{P}}^{*}\left(\mathcal{x},\mathrm{ɤ}\right)$ are $\mathsf{C}$-$\mathrm{ƛ}$-pre-invex on $▽$. Then, from Equation (22), for all $\left(\mathrm{ⱱ},\mathrm{ⱳ}\right), \left(\mathcal{p},\mathcal{q}\right)\in ▽, \mathtt{t}$ and $\mathrm{ԟ}\in \left[0,1\right]$, we have

$$\begin{gathered} {\mathcal{P}}_{*}\left(\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right),\mathrm{ɤ}\right) \\ \le \mathrm{ƛ}\left(\mathtt{t}\right){\mathrm{ƛ}\left(\mathrm{ԟ}\right)\mathcal{P}}_{*}\left(\left(\mathcal{ⱱ},\mathcal{p}\right),\mathrm{ɤ}\right)+\mathrm{ƛ}\left(\mathtt{t}\right)\mathrm{ƛ}\left(1-\mathrm{ԟ}\right){\mathcal{P}}_{*}\left(\left(\mathcal{ⱱ},\mathcal{q}\right),\mathrm{ɤ}\right)+{\mathrm{ƛ}\left(\mathrm{ԟ}\right)\mathrm{ƛ}\left(1-\mathtt{t}\right)\mathcal{P}}_{*}\left(\left(\mathcal{ⱱ},\mathcal{p}\right),\mathrm{ɤ}\right)+\mathrm{ƛ}\left(1-\mathtt{t}\right)\mathrm{ƛ}\left(1-\mathrm{ԟ}\right){\mathcal{P}}_{*}\left(\left(\mathcal{ⱱ},\mathcal{q}\right),\mathrm{ɤ}\right), \end{gathered}$$

and

$$\begin{gathered} {\mathcal{P}}^{*}\left(\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right),\mathrm{ɤ}\right) \\ \le \mathrm{ƛ}\left(\mathtt{t}\right){\mathrm{ƛ}\left(\mathrm{ԟ}\right)\mathcal{P}}_{*}\left(\left(\mathcal{ⱱ},\mathcal{p}\right),\mathrm{ɤ}\right)+\mathrm{ƛ}\left(\mathtt{t}\right)\mathrm{ƛ}\left(1-\mathrm{ԟ}\right){\mathcal{P}}^{*}\left(\left(\mathcal{ⱱ},\mathcal{q}\right),\mathrm{ɤ}\right)+\mathrm{ƛ}\left(\mathrm{ԟ}\right)\mathrm{ƛ}\left(1-\mathtt{t}\right){\mathcal{P}}^{*}\left(\left(\mathcal{ⱱ},\mathcal{p}\right),\mathrm{ɤ}\right)+\mathrm{ƛ}\left(1-\mathtt{t}\right)\mathrm{ƛ}\left(1-\mathrm{ԟ}\right){\mathcal{P}}^{*}\left(\left(\mathcal{ⱱ},\mathcal{q}\right),\mathrm{ɤ}\right), \end{gathered}$$

Then, by Equations (4), (6), and (23), we obtain

$${\mathcal{P}}_{\mathrm{ɤ}}\left(\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\right)$$

$$=\left[{\mathcal{P}}_{*}\left(\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right),\mathrm{ɤ}\right), {\mathcal{P}}^{*}\left(\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right),\mathrm{ɤ}\right)\right]$$

$${\le }_I\mathrm{ƛ}\left(\mathtt{t}\right)\mathrm{ƛ}\left(\mathrm{ԟ}\right)\left[{\mathcal{P}}_{*}\left(\left(\mathcal{ⱱ},\mathcal{p}\right),\mathrm{ɤ}\right), {\mathcal{P}}^{*}\left(\left(\mathcal{ⱱ},\mathcal{p}\right),\mathrm{ɤ}\right)\right]+\mathrm{ƛ}\left(\mathtt{t}\right)\mathrm{ƛ}\left(1-\mathrm{ԟ}\right)\left[{\mathcal{P}}_{*}\left(\left(\mathcal{ⱱ},\mathcal{q}\right),\mathrm{ɤ}\right), {\mathcal{P}}^{\begin{array}{c} \\ \\ \begin{array}{c} \\ *\end{array}\end{array}}\left(\left(\mathcal{ⱱ},\mathcal{q}\right),\mathrm{ɤ}\right)\right]$$

$$+\mathrm{ƛ}\left(\mathrm{ԟ}\right)\mathrm{ƛ}\left(1-\mathtt{t}\right)\left[{\mathcal{P}}_{*}\left(\left(\mathcal{ⱱ},\mathcal{p}\right),\mathrm{ɤ}\right), {\mathcal{P}}^{*}\left(\left(\mathcal{ⱱ},\mathcal{p}\right),\mathrm{ɤ}\right)\right]+\mathrm{ƛ}\left(1-\mathtt{t}\right)\mathrm{ƛ}\left(1-\mathrm{ԟ}\right)\left[{\mathcal{P}}_{*}\left(\left(\mathcal{ⱱ},\mathcal{q}\right),\mathrm{ɤ}\right), {\mathcal{P}}^{*}\left(\left(\mathcal{ⱱ},\mathcal{q}\right),\mathrm{ɤ}\right)\right]$$

That is,

$$\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)$$

$${\le }_{\mathbb{F}}\mathrm{ƛ}\left(\mathtt{t}\right)\mathrm{ƛ}\left(\mathrm{ԟ}\right)\tilde{\mathcal{P}}\left(\mathcal{ⱱ},\mathcal{p}\right)\oplus \mathrm{ƛ}\left(\mathtt{t}\right)\mathrm{ƛ}\left(1-\mathrm{ԟ}\right)\tilde{\mathcal{P}}\left(\mathcal{ⱱ},\mathcal{q}\right)\oplus \mathrm{ƛ}\left(1-\mathtt{t}\right)\mathrm{ƛ}\left(1-\mathrm{ԟ}\right)\tilde{\mathcal{P}}\left(\mathcal{ⱳ},\mathcal{p}\right)\oplus \mathrm{ƛ}\left(1-\mathtt{t}\right)\mathrm{ƛ}\left(1-\mathrm{ԟ}\right)\tilde{\mathcal{P}}\left(\mathcal{ⱳ},\mathcal{q}\right),$$

hence, $\tilde{\mathcal{P}}$ is $\mathsf{C}$-$\mathrm{ƛ}$-pre-invex $\mathsf{F}$·$\mathsf{N}$·$\mathsf{V}$·$\mathsf{M}$ on $▽$.

Conversely, let $\tilde{\mathcal{P}}$ be $\mathsf{C}$-$\mathrm{ƛ}$-pre-invex $\mathsf{F}$·$\mathsf{N}$·$\mathsf{V}$·$\mathsf{M}$ on $▽.$ Then, for all $\left(\mathrm{ⱱ},\mathrm{ⱳ}\right), \left(\mathcal{p},\mathcal{q}\right)\in ▽,\mathtt{t}$ and $\mathrm{ԟ}\in \left[0,1\right],$ we have

$$\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)$$

$${\le }_{\mathbb{F}}\mathrm{ƛ}\left(\mathtt{t}\right)\mathrm{ƛ}\left(\mathrm{ԟ}\right)\tilde{\mathcal{P}}\left(\mathcal{ⱱ},\mathcal{p}\right)\oplus \mathrm{ƛ}\left(\mathtt{t}\right)\mathrm{ƛ}\left(1-\mathrm{ԟ}\right)\tilde{\mathcal{P}}\left(\mathcal{ⱱ},\mathcal{q}\right)\oplus \mathrm{ƛ}\left(1-\mathtt{t}\right)\mathrm{ƛ}\left(\mathrm{ԟ}\right)\tilde{\mathcal{P}}\left(\mathcal{ⱳ},\mathcal{p}\right)\oplus \mathrm{ƛ}\left(1-\mathtt{t}\right)\mathrm{ƛ}\left(1-\mathrm{ԟ}\right)\tilde{\mathcal{P}}\left(\mathcal{ⱳ},\mathcal{q}\right)$$

Therefore, again from Equation (23), for each $\mathrm{ɤ}\in \left[0,1\right]$, we have

$${\mathcal{P}}_{\mathrm{ɤ}}\left(\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\right)$$

$$=\left[{\mathcal{P}}_{*}\left(\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right),\mathrm{ɤ}\right), {\mathcal{P}}^{*}\left(\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right),\mathrm{ɤ}\right)\right]$$

Again, using Equations (4) and (6), we obtain

$$\begin{gathered} \mathrm{ƛ}\left(\mathtt{t}\right)\mathrm{ƛ}\left(\mathrm{ԟ}\right){\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathcal{p}\right)+\mathrm{ƛ}\left(\mathtt{t}\right)\mathrm{ƛ}\left(1-\mathrm{ԟ}\right){\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathcal{q}\right)+\mathrm{ƛ}\left(1-\mathtt{t}\right)\mathrm{ƛ}\left(\mathrm{ԟ}\right){\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱳ},\mathcal{p}\right)+\mathrm{ƛ}\left(1-\mathtt{t}\right)\mathrm{ƛ}\left(1-\mathrm{ԟ}\right){\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱳ},\mathcal{q}\right) \\ =\mathrm{ƛ}\left(\mathtt{t}\right)\mathrm{ƛ}\left(\mathrm{ԟ}\right)\left[{\mathcal{P}}_{*}\left(\left(\mathrm{ⱱ},\mathcal{p}\right),\mathrm{ɤ}\right), {\mathcal{P}}^{*}\left(\left(\mathrm{ⱱ},\mathcal{p}\right),\mathrm{ɤ}\right)\right]+\mathrm{ƛ}\left(\mathtt{t}\right)\mathrm{ƛ}\left(1-\mathrm{ԟ}\right)\left[{\mathcal{P}}_{*}\left(\left(\mathrm{ⱱ},\mathcal{q}\right),\mathrm{ɤ}\right), {\mathcal{P}}^{*}\left(\left(\mathrm{ⱱ},\mathcal{q}\right),\mathrm{ɤ}\right)\right] \\ +\mathrm{ƛ}\left(\mathrm{ԟ}\right)\mathrm{ƛ}\left(1-\mathtt{t}\right)\left[{\mathcal{P}}_{*}\left(\left(\mathrm{ⱱ},\mathcal{p}\right),\mathrm{ɤ}\right), {\mathcal{P}}^{*}\left(\left(\mathrm{ⱱ},\mathcal{p}\right),\mathrm{ɤ}\right)\right]+\mathrm{ƛ}\left(1-\mathtt{t}\right)\mathrm{ƛ}\left(1-\mathrm{ԟ}\right)\left[{\mathcal{P}}_{*}\left(\left(\mathrm{ⱱ},\mathcal{q}\right),\mathrm{ɤ}\right), {\mathcal{P}}^{*}\left(\left(\mathrm{ⱱ},\mathcal{q}\right),\mathrm{ɤ}\right)\right], \end{gathered}$$

for all $\mathcal{x},\omega \in ▽$ and $\mathtt{t}\in \left[0,1\right].$ Then, by the $\mathsf{C}$-$\mathrm{ƛ}$-pre-invexity of $\tilde{\mathcal{P}}$, we have for all $\mathcal{x},\omega \in ▽$ and $\mathtt{t}\in \left[0,1\right]$ the case that

$${\mathcal{P}}_{*}\left(\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right),\mathrm{ɤ}\right)$$

$$\le \mathrm{ƛ}\left(\mathtt{t}\right)\mathrm{ƛ}\left(\mathrm{ԟ}\right){\mathcal{P}}_{*}\left(\mathrm{ⱱ},\mathcal{p}\right)+\mathrm{ƛ}\left(\mathtt{t}\right)\mathrm{ƛ}\left(1-\mathrm{ԟ}\right){\mathcal{P}}_{*}\left(\mathrm{ⱱ},\mathcal{q}\right)+\mathrm{ƛ}\left(1-\mathtt{t}\right)\mathrm{ƛ}\left(\mathrm{ԟ}\right){\mathcal{P}}_{*}\left(\mathrm{ⱳ},\mathcal{p}\right)+\mathrm{ƛ}\left(1-\mathtt{t}\right)\mathrm{ƛ}\left(1-\mathrm{ԟ}\right){\mathcal{P}}_{*}\left(\mathrm{ⱳ},\mathcal{q}\right),$$

and

$${\mathcal{P}}^{*}\left(\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right),\mathrm{ɤ}\right)$$

$$\le \mathrm{ƛ}\left(\mathtt{t}\right)\mathrm{ƛ}\left(\mathrm{ԟ}\right){\mathcal{P}}^{*}\left(\mathrm{ⱱ},\mathcal{p}\right)+\mathrm{ƛ}\left(\mathtt{t}\right)\mathrm{ƛ}\left(1-\mathrm{ԟ}\right){\mathcal{P}}^{*}\left(\mathrm{ⱱ},\mathcal{q}\right)+\mathrm{ƛ}\left(1-\mathtt{t}\right)\mathrm{ƛ}\left(\mathrm{ԟ}\right){\mathcal{P}}^{*}\left(\mathrm{ⱳ},\mathcal{p}\right)+\mathrm{ƛ}\left(1-\mathtt{t}\right)\mathrm{ƛ}\left(1-\mathrm{ԟ}\right){\mathcal{P}}^{*}\left(\mathrm{ⱳ},\mathcal{q}\right),$$

for each $\mathrm{ɤ}\in \left[0,1\right].$ Hence, the result follows. □

Example 1.

We consider the $\mathsf{F}$·$\mathsf{N}$·$\mathsf{V}$·$\mathsf{M}$ $\tilde{\mathcal{P}}:\left[0,1\right]\times \left[0,1\right]\to {\mathbb{F}}_0$ defined by

$$\mathcal{P}\left(\mathrm{ҩ},\mathrm{ʑ}\right)\left(\sigma \right)=\left\{\begin{array}{c}{\displaystyle \frac{\sigma -\mathrm{ҩ}\mathrm{ʑ}}{5-\mathrm{ҩ}\mathrm{ʑ}}}, \sigma \in \left[\mathrm{ҩ}\mathrm{ʑ}, 5\right]\\ {\displaystyle \frac{\left(6+e^{\mathrm{ҩ}}\right)\left(6+e^{\mathrm{ʑ}}\right)-\sigma }{\left(6+e^{\mathrm{ҩ}}\right)\left(6+e^{\mathrm{ʑ}}\right)-5}}, \sigma \in \left(5,\left(6+e^{\mathrm{ҩ}}\right)\left(6+e^{\mathrm{ʑ}}\right)\right]\\ 0 , otherwise,\end{array}\right.$$

(24)

Then, for each $\mathrm{ɤ}\in \left[0,1\right],$ we have ${\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ҩ}\right)=\left[\left(1-\mathrm{ɤ}\right)\mathrm{ҩ}\mathrm{ʑ}+5\mathrm{ɤ},\left(1-\mathrm{ɤ}\right)\left(6+e^{\mathrm{ҩ}}\right)\left(6+e^{\mathrm{ʑ}}\right)+5\mathrm{ɤ}\right]$. Since endpoint functions ${\mathcal{P}}_{\mathrm{*}}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right),$ ${\mathcal{P}}^{\mathrm{*}}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)$ are $\mathsf{C}$-$\mathrm{ƛ}$-pre-invex functions for each $\mathrm{ɤ}\in \left[0,1\right]$, $\tilde{\mathcal{P}}\left(\mathrm{ҩ},\mathrm{ʑ}\right)$ is a $\mathsf{C}$-$\mathrm{ƛ}$-pre-invex $\mathsf{F}$·$\mathsf{N}$·$\mathsf{V}$·$\mathsf{M}$.

Upon examination of Lemma 1 and Example 1, it becomes apparent that every $\mathrm{ƛ}$-pre-invex $\mathsf{F}$·$\mathsf{N}$·$\mathsf{V}$·$\mathsf{M}$ satisfies $\mathsf{C}$-$\mathrm{ƛ}$-pre-invex $\mathsf{F}$·$\mathsf{N}$·$\mathsf{V}$·$\mathsf{M}$ conditions, utilizing ${\mathrm{\hslash }}_1\left(\mathrm{ҩ},\mathrm{ʑ}\right)=\mathrm{ʑ}-\mathrm{ҩ}$ and ${\mathrm{\hslash }}_2\left(\mathrm{ҩ},\mathrm{ʑ}\right)=\mathrm{ʑ}-\mathrm{ҩ}$. However, the reverse assertion does not hold.

Remark 1.

When setting ${\mathcal{P}}_{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)={\mathcal{P}}^{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)$ with $\mathrm{ɤ}=1$, $\mathcal{P}$ is deemed a classical $\mathsf{C}$-$\mathrm{ƛ}$-pre-invex function if it satisfies the specified inequality outlined here:

$$\mathcal{P}\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)$$

$$\le \mathrm{ƛ}\left(\mathtt{t}\right){\mathrm{ƛ}\left(\mathrm{ԟ}\right)\mathcal{P}}_{*}\left(\mathrm{ⱱ},\mathcal{p}\right)+\mathrm{ƛ}\left(\mathtt{t}\right)\mathrm{ƛ}\left(1-\mathrm{ԟ}\right){\mathcal{P}}_{*}\left(\mathrm{ⱱ},\mathcal{q}\right)+{\mathrm{ƛ}\left(\mathrm{ԟ}\right)\mathrm{ƛ}\left(1-\mathtt{t}\right)\mathcal{P}}_{*}\left(\mathrm{ⱱ},\mathcal{p}\right)+\mathrm{ƛ}\left(1-\mathtt{t}\right)\mathrm{ƛ}\left(1-\mathrm{ԟ}\right){\mathcal{P}}_{*}\left(\mathrm{ⱱ},\mathcal{q}\right).$$

(25)

We define $\mathrm{ƛ}\left(\mathtt{t}\right)=\mathtt{t}, \mathrm{ƛ}\left(\mathrm{ԟ}\right)=\mathrm{ԟ}$, along with ${\mathcal{P}}_{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)={\mathcal{P}}^{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)$ with $\mathrm{ɤ}=1$, leading to the classification of $\mathcal{P}$ as a classical $\mathsf{C}$-pre-invex function, provided that $\mathcal{P}$ satisfies the following inequality:

$$\mathcal{P}\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)$$

$$\le \mathtt{t}\mathtt{ԟ}\mathcal{P}\left(\mathrm{ⱱ},\mathcal{p}\right)+\mathtt{t}\left(1-\mathrm{ԟ}\right)\mathcal{P}\left(\mathrm{ⱱ},\mathcal{q}\right)+\left(1-\mathtt{t}\right)\mathrm{ԟ}\mathcal{P}\left(\mathrm{ⱳ},\mathcal{p}\right)+\left(1-\mathtt{t}\right)\left(1-\mathrm{ԟ}\right)\mathcal{P}\left(\mathrm{ⱳ},\mathcal{q}\right)$$

(26)

By setting $\mathrm{ƛ}\left(\mathtt{t}\right)=\mathtt{t}, \mathrm{ƛ}\left(\mathrm{ԟ}\right)=\mathrm{ԟ}$ and ${\mathcal{P}}_{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)\ne {\mathcal{P}}^{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)$ with $\mathrm{ɤ}=1$, and ${\mathcal{P}}_{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)$ is an affine function and ${\mathcal{P}}^{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)$ is a pre-concave function. If the stated inequality is true here, see [62]:

$$\mathcal{P}\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)$$

$$\supseteq \mathtt{t}\mathtt{ԟ}\mathcal{P}\left(\mathrm{ⱱ},\mathcal{p}\right)+\mathtt{t}\left(1-\mathrm{ԟ}\right)\mathcal{P}\left(\mathrm{ⱱ},\mathcal{q}\right)+\left(1-\mathtt{t}\right)\mathrm{ԟ}\mathcal{P}\left(\mathrm{ⱳ},\mathcal{p}\right)+\left(1-\mathtt{t}\right)\left(1-\mathrm{ԟ}\right)\mathcal{P}\left(\mathrm{ⱳ},\mathcal{q}\right)$$

(27)

Definition 6.

Let $\tilde{\mathcal{P}}:▽\to {\mathbb{F}}_0$ be an $\mathsf{F}$·$\mathsf{N}$·$\mathsf{V}$·$\mathsf{M}$ on $▽$, given by

$${\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ҩ},\mathrm{ʑ}\right)=\left[{\mathcal{P}}_{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right), {\mathcal{P}}^{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)\right],$$

(28)

for all $\left(\mathrm{ҩ},\mathrm{ʑ}\right)\in ▽$ and for all $\mathrm{ɤ}\in \left[0,1\right]$. If ${\mathcal{P}}_{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)$ and ${\mathcal{P}}^{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)$ are $\mathsf{C}$-$\mathrm{ƛ}$-pre-invex (pre-concave) and affine functions on $▽$, for all $\mathrm{ɤ}\in \left[0,1\right],$ respectively, then $\tilde{\mathcal{P}}$ is a $\mathsf{C}$-left-$\mathrm{ƛ}$-pre-invex (pre-concave) $\mathsf{F}$·$\mathsf{N}$·$\mathsf{M}$ on $▽$.

Definition 7.

Let $\tilde{\mathcal{P}}:▽\to {\mathbb{F}}_0$ be an $\mathsf{F}$·$\mathsf{N}$·$\mathsf{V}$·$\mathsf{M}$ on $▽$, defined by

$${\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ҩ},\mathrm{ʑ}\right)=\left[{\mathcal{P}}_{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right), {\mathcal{P}}^{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)\right],$$

for all $\left(\mathrm{ҩ},\mathrm{ʑ}\right)\in ▽$ and for all $\mathrm{ɤ}\in \left[0,1\right]$. If ${\mathcal{P}}_{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)$ and ${\mathcal{P}}^{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)$ are $\mathsf{C}$-$\mathrm{ƛ}$-affine and $\mathrm{ƛ}$-pre-invex (pre-concave) functions on $▽$, respectively, then $\tilde{\mathcal{P}}$ is a $\mathsf{C}$-right-$\mathrm{ƛ}$-pre-invex (pre-concave) $\mathsf{F}$·$\mathsf{N}$·$\mathsf{V}$·$\mathsf{M}$ on $▽$.

Theorem 6.

Let $▽$ be a coordinated invex set, and let $\tilde{\mathcal{P}}:▽\to {\mathbb{F}}_0$ be an $\mathsf{F}$·$\mathsf{N}$·$\mathsf{V}$·$\mathsf{M}$. Subsequently, derived from ɤ-levels, we acquire the set of $\mathsf{I}$·$\mathsf{V}$·$\mathsf{M}s$ ${\mathcal{P}}_{\mathrm{ɤ}}:▽\to {\mathbb{R}}_I^{+}\subset {\mathbb{R}}_I$, which are expressed as

$${\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ҩ},\mathrm{ʑ}\right)=\left[{\mathcal{P}}_{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right), {\mathcal{P}}^{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)\right],$$

for all $\left(\mathrm{ҩ},\mathrm{ʑ}\right)\in ▽$ and for all $\mathrm{ɤ}\in \left[0,1\right]$. Then, $\tilde{\mathcal{P}}$ is $\mathsf{C}$-$\mathrm{ƛ}$-pre-concave $\mathsf{F}$·$\mathsf{N}$·$\mathsf{V}$·$\mathsf{M}$ on $▽,$ if and only if, for all $\mathrm{ɤ}\in \left[0,1\right],$ ${\mathcal{P}}_{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)$ and ${\mathcal{P}}^{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)$ are $\mathsf{C}$-$\mathrm{ƛ}$-pre-concave and $\mathrm{ƛ}$-pre-invex functions, respectively.

Proof. 

The proof for Theorem 6 follows a methodology akin to that of Theorem 5. □

Example 2.

We consider the $\mathsf{F}$·$\mathsf{N}$·$\mathsf{V}$·$\mathsf{M}$s $\tilde{\mathcal{P}}:\left[0,1\right]\times \left[0,1\right]\to {\mathbb{F}}_0$ defined by

$$\tilde{\mathcal{P}}\left(\mathrm{ҩ},\mathrm{ʑ}\right)\left(\sigma \right)=\left\{\begin{array}{c}{\displaystyle \frac{\sigma -\left(6-e^{\mathrm{ҩ}}\right)\left(6-e^{\mathrm{ʑ}}\right)}{\left(6-e^{\mathrm{ҩ}}\right)\left(6-e^{\mathrm{ʑ}}\right)-25}}, \sigma \in \left[\left(6-e^{\mathrm{ҩ}}\right)\left(6-e^{\mathrm{ʑ}}\right), 25\right]\\ {\displaystyle \frac{35\mathrm{ҩ}\mathrm{ʑ}-\sigma }{35\mathrm{ҩ}\mathrm{ʑ}-25}}, \sigma \in \left(25, 35\mathrm{ҩ}\mathrm{ʑ}\right]\\ 0, otherwise.\end{array}\right.$$

(29)

Then, for each $\mathrm{ɤ}\in \left[0,1\right],$ we have ${\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ҩ},\mathrm{ʑ}\right)=\left[\left(1-\mathrm{ɤ}\right)\left(6-e^{\mathrm{ҩ}}\right)\left(6-e^{\mathrm{ʑ}}\right)+25\mathrm{ɤ},35\left(1-\mathrm{ɤ}\right)\mathrm{ҩ}\mathrm{ʑ}+25\mathrm{ɤ}\right]$. Since endpoint functions ${\mathcal{P}}_{\mathrm{*}}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right),$ ${\mathcal{P}}^{\mathrm{*}}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)$ are coordinate $\mathrm{ƛ}$-pre-concave and $\mathrm{ƛ}$-affine functions with $\mathrm{ƛ}\left(\mathtt{t}\right)=\mathtt{t},\mathrm{ƛ}\left(\mathrm{ԟ}\right)=\mathrm{ԟ}$ and by means of ${\mathrm{\hslash }}_1\left(\mathrm{ҩ},\mathrm{ʑ}\right)=\mathrm{ʑ}-\mathrm{ҩ}$ and ${\mathrm{\hslash }}_2\left(\mathrm{ҩ},\mathrm{ʑ}\right)=\mathrm{ʑ}-\mathrm{ҩ}$, for each $\mathrm{ɤ}\in \left[0,1\right]$, $\tilde{\mathcal{P}}\left(\mathrm{ҩ},\mathrm{ʑ}\right)$ is $\mathsf{C}$-$\mathrm{ƛ}$-pre-concave $\mathsf{F}$·$\mathsf{N}$·$\mathsf{V}$·$\mathsf{M}$.

3. Main Results

Below, we present the primary findings for $\mathsf{C}$-integral inequalities applicable to the Hermite–Hadamard type, employing fuzzy fractional operators within $\mathsf{C}$-$\mathrm{ƛ}$-pre-invex $\mathsf{F}$·$\mathsf{N}$·$\mathsf{V}$·$\mathsf{M}$s.

Theorem 7.

Let $\tilde{\mathcal{P}}:▽=\left[\mathrm{ⱱ},\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]\times \left[\mathcal{p},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]\to {\mathbb{F}}_0^{+}$ be a coordinate $\mathrm{ƛ}$-pre-invex $\mathsf{F}$·$\mathsf{N}$·$\mathsf{V}$·$\mathsf{M}$ on $▽$, where $\mathrm{ƛ}:\left[0,1\right]\to {\mathbb{R}}^{+}$. Subsequently, derived fromɤ-levels, we acquire the set of $\mathsf{I}$·$\mathsf{V}$·$\mathsf{M}s$ ${\mathcal{P}}_{\mathrm{ɤ}}:▽\to {\mathbb{R}}_I^{+}\subset {\mathbb{R}}_I$, which are expressed as ${\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ҩ},\mathrm{ʑ}\right)=\left[{\mathcal{P}}_{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right), {\mathcal{P}}^{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)\right]$ for all $\left(\mathrm{ҩ},\mathrm{ʑ}\right)\in ▽$ and for all $\mathrm{ɤ}\in \left[0,1\right]$. If ${\tilde{\mathcal{P}}\in \mathcal{F}\mathfrak{O}}_{▽}$, then the following inequalities hold:

$${\displaystyle \frac{1}{{\mathrm{ƛ}}^2\left(\frac{1}{2}\right)}}\tilde{\mathcal{P}}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}}, {\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)$$

$$\begin{gathered} {\le }_{\mathbb{F}}{\displaystyle \frac{\Gamma \left(\alpha +1\right)}{{2\mathrm{ƛ}\left(\frac{1}{2}\right)\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }}}\left[{\mathcal{I}}_{{\mathrm{ⱱ}}^{+}}^{\alpha }\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)\oplus {\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-}}^{\alpha }\tilde{\mathcal{P}}\left(\mathrm{ⱱ},{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)\right] \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\oplus {\displaystyle \frac{\Gamma \left(\beta +1\right)}{{2\mathrm{ƛ}\left(\frac{1}{2}\right)\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}\left[{\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta }\tilde{\mathcal{P}}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\oplus {\mathcal{I}}_{{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\beta }\tilde{\mathcal{P}}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}},\mathcal{p}\right)\right] \end{gathered}$$

$$\begin{gathered} {\le }_{\mathbb{F}}{\displaystyle \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }{\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}\left[{\mathcal{I}}_{{\mathrm{ⱱ}}^{+},{\mathcal{p}}^{+}}^{\alpha , \beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\oplus {\mathcal{I}}_{{\mathrm{ⱱ}}^{+},{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\alpha , \beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right) \\ \oplus {\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-},{\mathcal{p}}^{+}}^{\alpha , \beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\oplus {\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-},{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\alpha , \beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}\right)\right] \end{gathered}$$

$$\begin{gathered} {\le }_{\mathbb{F}}{\displaystyle \frac{\beta \Gamma \left(\alpha +1\right)}{{\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }}}\left[{\mathcal{I}}_{{\mathrm{ⱱ}}^{+}}^{\alpha }\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right)\oplus {\mathcal{I}}_{{\mathrm{ⱱ}}^{+}}^{\alpha }\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\oplus {\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-}}^{\alpha }\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}\right) \\ \oplus {\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-}}^{\alpha }\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\right]\times {\int }_0^1{\mathrm{ԟ}}^{\beta -1}\left[\mathrm{ƛ}\left(\mathrm{ԟ}\right)+\mathrm{ƛ}\left(1-\mathrm{ԟ}\right)\right]d\mathrm{ԟ} \end{gathered}$$

$$\begin{gathered} \oplus {\displaystyle \frac{\alpha \Gamma \left(\beta +1\right)}{{\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}\left[{\mathcal{I}}_{{\mathcal{p}}^{+}}^{ \beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\oplus {\mathcal{I}}_{{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right)\oplus {\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right) \\ \oplus {\mathcal{I}}_{{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right)\right]\times {\int }_0^1{\mathtt{t}}^{\alpha -1}\mathrm{ƛ}\left(\mathtt{t}\right)+\mathrm{ƛ}\left(1-\mathtt{t}\right)d\mathtt{t} \end{gathered}$$

$${\le }_{\mathbb{F}}\alpha \beta \left[\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}\right)\oplus \tilde{\mathcal{P}}\left(\mathrm{ⱳ},\mathcal{p}\right)\oplus \tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{q}\right)\oplus \tilde{\mathcal{P}}\left(\mathrm{ⱳ},\mathcal{q}\right)\right]\times {\int }_0^1{\mathrm{ԟ}}^{\beta -1}\left[\mathrm{ƛ}\left(\mathrm{ԟ}\right)+\mathrm{ƛ}\left(1-\mathrm{ԟ}\right)\right]d\mathrm{ԟ}{\int }_0^1{\mathtt{t}}^{\alpha -1}\left[\mathrm{ƛ}\left(\mathtt{t}\right)+\mathrm{ƛ}\left(1-\mathtt{t}\right)\right]d\mathtt{t}.$$

(30)

If $\tilde{\mathcal{P}}\left(\mathrm{ҩ},\mathrm{ʑ}\right)$ $\mathsf{C}$-$\mathrm{ƛ}$-pre-concave $\mathsf{F}$·$\mathsf{N}$·$\mathsf{V}$·$\mathsf{M}$, then,

$${\displaystyle \frac{1}{{\mathrm{ƛ}}^2\left(\frac{1}{2}\right)}}\tilde{\mathcal{P}}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}}, {\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)$$

$$\begin{gathered} {\ge }_{\mathbb{F}}{\displaystyle \frac{\Gamma \left(\alpha +1\right)}{{2\mathrm{ƛ}\left(\frac{1}{2}\right)\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }}}\left[{\mathcal{I}}_{{\mathrm{ⱱ}}^{+}}^{\alpha }\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)\oplus {\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-}}^{\alpha }\tilde{\mathcal{P}}\left(\mathrm{ⱱ},{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)\right] \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\oplus {\displaystyle \frac{\Gamma \left(\beta +1\right)}{{2\mathrm{ƛ}\left(\frac{1}{2}\right)\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}\left[{\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta }\tilde{\mathcal{P}}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\oplus {\mathcal{I}}_{{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\beta }\tilde{\mathcal{P}}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}},\mathcal{p}\right)\right] \end{gathered}$$

$$\begin{gathered} {\ge }_{\mathbb{F}}{\displaystyle \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }{\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}\left[{\mathcal{I}}_{{\mathrm{ⱱ}}^{+},{\mathcal{p}}^{+}}^{\alpha , \beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\oplus {\mathcal{I}}_{{\mathrm{ⱱ}}^{+},{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\alpha , \beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right) \\ \oplus {\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-},{\mathcal{p}}^{+}}^{\alpha , \beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\oplus {\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-},{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\alpha , \beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}\right)\right] \end{gathered}$$

$$\begin{gathered} {\ge }_{\mathbb{F}}{\displaystyle \frac{\beta \Gamma \left(\alpha +1\right)}{{\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }}}\left[{\mathcal{I}}_{{\mathrm{ⱱ}}^{+}}^{\alpha }\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right)\oplus {\mathcal{I}}_{{\mathrm{ⱱ}}^{+}}^{\alpha }\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\oplus {\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-}}^{\alpha }\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}\right) \\ \oplus {\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-}}^{\alpha }\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\right]\times {\int }_0^1{\mathrm{ԟ}}^{\beta -1}\left[\mathrm{ƛ}\left(\mathrm{ԟ}\right)+\mathrm{ƛ}\left(1-\mathrm{ԟ}\right)\right]d\mathrm{ԟ} \end{gathered}$$

$$\begin{gathered} \oplus {\displaystyle \frac{\alpha \Gamma \left(\beta +1\right)}{{\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}\left[{\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\oplus {\mathcal{I}}_{{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right)\oplus {\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right) \\ \oplus {\mathcal{I}}_{{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right)\right]\times {\int }_0^1{\mathtt{t}}^{\alpha -1}\left[\mathrm{ƛ}\left(\mathtt{t}\right)+\mathrm{ƛ}\left(1-\mathtt{t}\right)\right]d\mathtt{t} \end{gathered}$$

$${\ge }_{\mathbb{F}}\alpha \beta \left[\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}\right)\oplus \tilde{\mathcal{P}}\left(\mathrm{ⱳ},\mathcal{p}\right)\oplus \tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{q}\right)\oplus \tilde{\mathcal{P}}\left(\mathrm{ⱳ},\mathcal{q}\right)\right]\times {\int }_0^1{\mathrm{ԟ}}^{\beta -1}\left[\mathrm{ƛ}\left(\mathrm{ԟ}\right)+\mathrm{ƛ}\left(1-\mathrm{ԟ}\right)\right]d\mathrm{ԟ}{\int }_0^1{\mathtt{t}}^{\alpha -1}\left[\mathrm{ƛ}\left(\mathtt{t}\right)+\mathrm{ƛ}\left(1-\mathtt{t}\right)\right]d\mathtt{t}.$$

(31)

Proof. 

Let $\tilde{\mathcal{P}}:\left[\mathrm{ⱱ},\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]\times \left[\mathcal{p},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]\to {\mathbb{F}}_0$ be a $\mathsf{C}$-$\mathrm{ƛ}$-pre-invex $\mathsf{F}$·$\mathsf{N}$·$\mathsf{V}$·$\mathsf{M}$. Then, by hypothesis, we have

$$\begin{gathered} {\displaystyle \frac{1}{{\mathrm{ƛ}}^2\left(\frac{1}{2}\right)}}\tilde{\mathcal{P}}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}},{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right){\le }_{\mathbb{F}}\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\hspace{1em}\hspace{1em}\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right) \\ \oplus \tilde{\mathcal{P}}\left(\mathrm{ⱱ}+\mathtt{t}{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\mathrm{ԟ}{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right). \end{gathered}$$

By using Theorem 5, for every $\mathrm{ɤ}\in \left[0,1\right]$, we have

$$\begin{array}{c}{\displaystyle \frac{1}{{\mathrm{ƛ}}^2\left(\frac{1}{2}\right)}}{\mathcal{P}}_{*}\left(\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}},{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right),\mathrm{ɤ}\right) \\ \le {\mathcal{P}}_{*}\left(\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\hspace{1em}\hspace{1em}\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right),\mathrm{ɤ}\right)+{\mathcal{P}}_{*}\left(\left(\mathrm{ⱱ}+\mathtt{t}{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\mathrm{ԟ}{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right),\mathrm{ɤ}\right), \\ {\displaystyle \frac{1}{{\mathrm{ƛ}}^2\left(\frac{1}{2}\right)}}{\mathcal{P}}^{*}\left(\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}},{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right),\mathrm{ɤ}\right) \\ \le {\mathcal{P}}^{*}\left(\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\hspace{1em}\hspace{1em}\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right),\mathrm{ɤ}\right)+{\mathcal{P}}^{*}\left(\left(\mathrm{ⱱ}+\mathtt{t}{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\mathrm{ԟ}{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right),\mathrm{ɤ}\right).\end{array}$$

By using Lemma 1, we have

$$\begin{array}{c}{\displaystyle \frac{1}{\mathrm{ƛ}\left(\frac{1}{2}\right)}}{\mathcal{P}}_{*}\left(\left(\mathrm{ҩ},{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right),\mathrm{ɤ}\right)\le {\mathcal{P}}_{*}\left(\left(\mathrm{ҩ},\hspace{1em}\hspace{1em}\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right),\mathrm{ɤ}\right)+{\mathcal{P}}_{*}\left(\left(\mathrm{ҩ},\mathcal{p}+\mathrm{ԟ}{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right),\mathrm{ɤ}\right),\\ {\displaystyle \frac{1}{\mathrm{ƛ}\left(\frac{1}{2}\right)}}{\mathcal{P}}^{*}\left(\left(\mathrm{ҩ},{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right),\mathrm{ɤ}\right)\le {\mathcal{P}}^{*}\left(\left(\mathrm{ҩ},\hspace{1em}\hspace{1em}\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right),\mathrm{ɤ}\right)+{\mathcal{P}}^{*}\left(\left(\mathrm{ҩ},\mathcal{p}+\mathrm{ԟ}{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right),\mathrm{ɤ}\right),\end{array}$$

(32)

and

$$\begin{array}{c}{\displaystyle \frac{1}{\mathrm{ƛ}\left(\frac{1}{2}\right)}}{\mathcal{P}}_{*}\left(\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}},\mathrm{ʑ}\right),\mathrm{ɤ}\right)\le {\mathcal{P}}_{*}\left(\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right), \mathrm{ʑ}\right),\mathrm{ɤ}\right)+{\mathcal{P}}_{*}\left(\left(\mathrm{ⱱ}+\mathtt{t}{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right), \mathrm{ʑ}\right),\mathrm{ɤ}\right), \\ {\displaystyle \frac{1}{\mathrm{ƛ}\left(\frac{1}{2}\right)}}{\mathcal{P}}^{*}\left(\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}},\mathrm{ʑ}\right),\mathrm{ɤ}\right)\le {\mathcal{P}}^{*}\left(\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right), \mathrm{ʑ}\right),\mathrm{ɤ}\right)+{\mathcal{P}}^{*}\left(\left(\mathrm{ⱱ}+\mathtt{t}{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right), \mathrm{ʑ}\right),\mathrm{ɤ}\right).\end{array}$$

(33)

From (32) and (33), we have

$${\displaystyle \frac{1}{\mathrm{ƛ}\left(\frac{1}{2}\right)}}\left[{\mathcal{P}}_{*}\left(\left(\mathrm{ҩ},{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right),\mathrm{ɤ}\right),{\mathcal{P}}^{*}\left(\left(\mathrm{ҩ},{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right),\mathrm{ɤ}\right)\right]$$

$${\le }_I\left[{\mathcal{P}}_{*}\left(\left(\mathrm{ҩ},\hspace{1em}\hspace{1em}\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right),\mathrm{ɤ}\right),{\mathcal{P}}^{*}\left(\left(\mathrm{ҩ},\hspace{1em}\hspace{1em}\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right),\mathrm{ɤ}\right)\right]$$

$$+\left[{\mathcal{P}}_{*}\left(\left(\mathrm{ҩ},\mathcal{p}+\mathrm{ԟ}{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right),\mathrm{ɤ}\right),{\mathcal{P}}^{*}\left(\left(\mathrm{ҩ},\mathcal{p}+\mathrm{ԟ}{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right),\mathrm{ɤ}\right)\right],$$

and

$${\displaystyle \frac{1}{\mathrm{ƛ}\left(\frac{1}{2}\right)}}\left[{\mathcal{P}}_{*}\left(\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}},\mathrm{ʑ}\right),\mathrm{ɤ}\right),{\mathcal{P}}^{*}\left(\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}},\mathrm{ʑ}\right),\mathrm{ɤ}\right)\right]$$

$${\le }_I\left[{\mathcal{P}}_{*}\left(\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathrm{ʑ}\right),\mathrm{ɤ}\right),{\mathcal{P}}^{*}\left(\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathrm{ʑ}\right),\mathrm{ɤ}\right)\right]$$

$$+\left[{\mathcal{P}}_{*}\left(\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathrm{ʑ}\right),\mathrm{ɤ}\right),{\mathcal{P}}^{*}\left(\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathrm{ʑ}\right),\mathrm{ɤ}\right)\right],$$

It follows that

$${\displaystyle \frac{1}{\mathrm{ƛ}\left(\frac{1}{2}\right)}}{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ҩ},{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right){\le }_I{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ҩ},\hspace{1em}\hspace{1em}\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)+{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ҩ},\mathcal{p}+\mathrm{ԟ}{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right),$$

(34)

and

$${\displaystyle \frac{1}{\mathrm{ƛ}\left(\frac{1}{2}\right)}}{\mathcal{P}}_{\mathrm{ɤ}}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}},\mathrm{ʑ}\right){\le }_I{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathrm{ʑ}\right)+{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathrm{ʑ}\right).$$

(35)

Since ${\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ҩ},.\right)$ and ${\mathcal{P}}_{\mathrm{ɤ}}\left(.,\mathrm{ʑ}\right)$, are both $\mathsf{C}$-$\mathrm{ƛ}$-pre-invex-$\mathsf{I}$·$\mathsf{V}$·$\mathsf{M}$s, from inequality (16), for every $\mathrm{ɤ}\in \left[0,1\right]$, for inequalities (34) and (35), we have

$$\begin{gathered} {\displaystyle \frac{1}{\beta \mathrm{ƛ}\left(\frac{1}{2}\right)}}{{\mathcal{P}}_{\mathrm{ɤ}}}_{\mathrm{ҩ}}\left({\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right){\le }_I{\displaystyle \frac{\Gamma \left(\beta \right)}{{\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}\left[{\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta }{{\mathcal{P}}_{\mathrm{ɤ}}}_{\mathrm{ҩ}}\left(\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)+{\mathcal{I}}_{{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\beta }{{\mathcal{P}}_{\mathrm{ɤ}}}_{\mathrm{ҩ}}\left(\mathcal{p}\right)\right]{\le }_I[{{\mathcal{P}}_{\mathrm{ɤ}}}_{\mathrm{ҩ}}\left(\mathcal{p}\right) \\ +{{\mathcal{P}}_{\mathrm{ɤ}}}_{\mathrm{ҩ}}\left(\mathcal{q}\right)]{\int }_0^1{\mathrm{ԟ}}^{\beta -1}\left[\mathrm{ƛ}\left(\mathrm{ԟ}\right)+\mathrm{ƛ}\left(1-\mathrm{ԟ}\right)\right]d\mathrm{ԟ} \end{gathered}$$

(36)

and

$$\begin{gathered} {\displaystyle \frac{1}{\alpha \mathrm{ƛ}\left(\frac{1}{2}\right)}}{{\mathcal{P}}_{\mathrm{ɤ}}}_{\mathrm{ʑ}}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}}\right){\le }_I{\displaystyle \frac{\Gamma \left(\alpha \right)}{{\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }}}\left[{\mathcal{I}}_{{\mathrm{ⱱ}}^{+}}^{\alpha }{{\mathcal{P}}_{\mathrm{ɤ}}}_{\mathrm{ʑ}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right)+{\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-}}^{\alpha }{{\mathcal{P}}_{\mathrm{ɤ}}}_{\mathrm{ʑ}}\left(\mathrm{ⱱ}\right)\right]{\le }_I[{{\mathcal{P}}_{\mathrm{ɤ}}}_{\mathrm{ʑ}}\left(\mathrm{ⱱ}\right) \\ +{{\mathcal{P}}_{\mathrm{ɤ}}}_{\mathrm{ʑ}}\left(\mathrm{ⱳ}\right)]{\int }_0^1{\mathtt{t}}^{\alpha -1}\mathrm{ƛ}\left(\mathtt{t}\right)+\mathrm{ƛ}\left(1-\mathtt{t}\right)d\mathtt{t} \end{gathered}$$

(37)

Since ${{\mathcal{P}}_{\mathrm{ɤ}}}_{\mathrm{ҩ}}\left(w\right)={\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ҩ},w\right)$, (36) can be written as

$$\begin{gathered} {\displaystyle \frac{1}{\beta \mathrm{ƛ}\left(\frac{1}{2}\right)}}{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ҩ},{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right){\le }_I{\displaystyle \frac{\Gamma \left(\beta \right)}{{\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}\left[{\mathcal{I}}_{{\mathcal{p}}^{+}}^{\alpha }{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ҩ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)+{\mathcal{I}}_{{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\alpha }{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ҩ},\mathcal{p}\right)\right]{\le }_I[{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ҩ},\mathcal{p}\right) \\ +{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ҩ},\mathcal{q}\right)]{\int }_0^1{\mathrm{ԟ}}^{\beta -1}\left[\mathrm{ƛ}\left(\mathrm{ԟ}\right)+\mathrm{ƛ}\left(1-\mathrm{ԟ}\right)\right]d\mathrm{ԟ}. \end{gathered}$$

(38)

That is,

$$\begin{gathered} {\displaystyle \frac{1}{\beta \mathrm{ƛ}\left(\frac{1}{2}\right)}}{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ҩ},{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right){\le }_I{\displaystyle \frac{1}{{\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}[{\int }_{\mathcal{p}}^{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{\left(\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)-\mathrm{ԟ}\right)}^{\beta -1}{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ҩ},\mathrm{ԟ}\right)d\mathrm{ԟ} \\ +{\int }_{\mathcal{p}}^{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{\left(\mathrm{ԟ}-\mathcal{p}\right)}^{\beta -1}{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ҩ},\mathrm{ԟ}\right)d\mathrm{ԟ}] \end{gathered}$$

$${\le }_I\left[{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ҩ},\mathcal{p}\right)+{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ҩ},\mathcal{q}\right)\right]{\int }_0^1{\mathrm{ԟ}}^{\beta -1}\left[\mathrm{ƛ}\left(\mathrm{ԟ}\right)+\mathrm{ƛ}\left(1-\mathrm{ԟ}\right)\right]d\mathrm{ԟ}.$$

Multiplying double inequality (38) by ${\displaystyle \frac{{\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)-\mathrm{ҩ}\right)}^{\alpha -1}}{{\left({\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right)}^{\alpha }}}$ and integrating with respect to $\mathrm{ҩ}$ over $\left[\mathrm{ⱱ},\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right],$ we have

$${\displaystyle \frac{1}{{\beta \left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }\mathrm{ƛ}\left(\frac{1}{2}\right)}}{\int }_{\mathrm{ⱱ}}^{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ҩ},{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right){\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)-\mathrm{ҩ}\right)}^{\alpha -1}d\mathrm{ҩ}$$

$$\begin{gathered} {\le }_I{\displaystyle \frac{1}{{{\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}{\int }_{\mathrm{ⱱ}}^{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{\int }_{\mathcal{p}}^{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)-\mathrm{ҩ}\right)}^{\alpha -1}{\left(\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)-\mathrm{ԟ}\right)}^{\beta -1}{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ҩ},\mathrm{ԟ}\right)d\mathrm{ԟ}d\mathrm{ҩ} \\ +{\int }_{\mathrm{ⱱ}}^{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{\int }_{\mathcal{p}}^{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)-\mathrm{ҩ}\right)}^{\alpha -1}{\left(\mathrm{ԟ}-\mathcal{p}\right)}^{\beta -1}{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ҩ},\mathrm{ԟ}\right)d\mathrm{ԟ}d\mathrm{ҩ} \end{gathered}$$

$$\begin{array}{c}{\le }_I{\displaystyle \frac{1}{{\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }}}[{\int }_{\mathrm{ⱱ}}^{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)-\mathrm{ҩ}\right)}^{\alpha -1}{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ҩ},\mathcal{p}\right)d\mathrm{ҩ}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\int }_{\mathrm{ⱱ}}^{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)-\mathrm{ҩ}\right)}^{\alpha -1}{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ҩ},\mathcal{q}\right)d\mathrm{ҩ}]{\int }_0^1{\mathrm{ԟ}}^{\beta -1}\left[\mathrm{ƛ}\left(\mathrm{ԟ}\right)+\mathrm{ƛ}\left(1-\mathrm{ԟ}\right)\right]d\mathrm{ԟ}\hfill \end{array}$$

(39)

Again, multiplying double inequality (38) by ${\displaystyle \frac{{\left(\mathrm{ҩ}-\mathrm{ⱱ}\right)}^{\alpha -1}}{{\left({\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right)}^{\alpha }}}$ and integrating with respect to $\mathrm{ҩ}$ over $\left[\mathrm{ⱱ},\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right],$ we have

$${\displaystyle \frac{1}{{\beta \left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }\mathrm{ƛ}\left(\frac{1}{2}\right)}}{\int }_{\mathrm{ⱱ}}^{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ҩ},{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right){\left(\mathrm{ҩ}-\mathrm{ⱱ}\right)}^{\alpha -1}d\mathrm{ҩ}$$

$${\le }_I{\displaystyle \frac{1}{{\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }{\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}{\int }_{\mathrm{ⱱ}}^{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{\int }_{\mathcal{p}}^{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{\left(\mathrm{ҩ}-\mathrm{ⱱ}\right)}^{\alpha -1}{\left(\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)-\mathrm{ԟ}\right)}^{\beta -1}{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ҩ},\mathrm{ԟ}\right)d\mathrm{ԟ}d\mathrm{ҩ}$$

$$+{\displaystyle \frac{1}{{\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }{\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}{\int }_{\mathrm{ⱱ}}^{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{\int }_{\mathcal{p}}^{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{\left(\mathrm{ҩ}-\mathrm{ⱱ}\right)}^{\alpha -1}{\left(\mathrm{ԟ}-\mathcal{p}\right)}^{\beta -1}{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ҩ},\mathrm{ԟ}\right)d\mathrm{ԟ}d\mathrm{ҩ}$$

$$\begin{array}{c}{\le }_I{\displaystyle \frac{1}{{\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }}}[{\int }_{\mathrm{ⱱ}}^{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{\left(\mathrm{ҩ}-\mathrm{ⱱ}\right)}^{\alpha -1}{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ҩ},\mathcal{p}\right)d\mathrm{ҩ}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\int }_{\mathrm{ⱱ}}^{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{\left(\mathrm{ҩ}-\mathrm{ⱱ}\right)}^{\alpha -1}{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ҩ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)d\mathrm{ҩ}]{\int }_0^1{\mathrm{ԟ}}^{\beta -1}\left[\mathrm{ƛ}\left(\mathrm{ԟ}\right)+\mathrm{ƛ}\left(1-\mathrm{ԟ}\right)\right]d\mathrm{ԟ}\hfill \end{array}$$

(40)

From (39), we have

$${\displaystyle \frac{\Gamma \left(\alpha +1\right)}{{2\mathrm{ƛ}\left(\frac{1}{2}\right)\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }}}\left[{\mathcal{I}}_{{\mathrm{ⱱ}}^{+}}^{\alpha }{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)\right],$$

$${\le }_I{\displaystyle \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }{\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}\left[{\mathcal{I}}_{{\mathrm{ⱱ}}^{+},{\mathcal{p}}^{+}}^{\alpha , \beta }{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)+{\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-},{\mathcal{p}}^{+}}^{\alpha , \beta }{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right)\right]$$

$${\le }_I{\displaystyle \frac{\beta \Gamma \left(\alpha +1\right)}{{\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }}}\left[{\mathcal{I}}_{{\mathrm{ⱱ}}^{+}}^{\alpha }{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right)+{\mathcal{I}}_{{\mathrm{ⱱ}}^{+}}^{\alpha }{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\right]{\int }_0^1{\mathrm{ԟ}}^{\beta -1}\left[\mathrm{ƛ}\left(\mathrm{ԟ}\right)+\mathrm{ƛ}\left(1-\mathrm{ԟ}\right)\right]d\mathrm{ԟ}$$

(41)

From (40), we have

$${\displaystyle \frac{\Gamma \left(\alpha +1\right)}{{2\mathrm{ƛ}\left(\frac{1}{2}\right)\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }}}\left[{\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-}}^{\alpha }{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)\right]$$

$${\le }_I{\displaystyle \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }{\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}\left[{\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-},{\mathcal{p}}^{+}}^{\alpha , \beta }{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)+{\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-},{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\alpha , \beta }{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathcal{p}\right)\right]$$

$${\le }_I{\displaystyle \frac{\beta \Gamma \left(\alpha +1\right)}{{\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }}}\left[{\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-}}^{\alpha }{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathcal{p}\right)+{\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-}}^{\alpha }{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\right]{\int }_0^1{\mathrm{ԟ}}^{\beta -1}\left[\mathrm{ƛ}\left(\mathrm{ԟ}\right)+\mathrm{ƛ}\left(1-\mathrm{ԟ}\right)\right]d\mathrm{ԟ}$$

(42)

This is because, derived from ɤ-levels, we acquire the set of $\mathsf{I}$·$\mathsf{V}$·$\mathsf{M}s$ ${\mathcal{P}}_{\mathrm{ɤ}}:▽\to {\mathbb{R}}_I^{+}\subset {\mathbb{R}}_I$, which are expressed as

$${\displaystyle \frac{\Gamma \left(\alpha +1\right)}{{2\mathrm{ƛ}\left(\frac{1}{2}\right)\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }}}\left[{\mathcal{I}}_{{\mathrm{ⱱ}}^{+}}^{\alpha }\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)\right]$$

$${\le }_{\mathbb{F}}{\displaystyle \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }{\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}\left[{\mathcal{I}}_{{\mathrm{ⱱ}}^{+},{\mathcal{p}}^{+}}^{\alpha , \beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\oplus {\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-},{\mathcal{p}}^{+}}^{\alpha , \beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right)\right]$$

$${\le }_{\mathbb{F}}{\displaystyle \frac{\beta \Gamma \left(\alpha +1\right)}{{\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }}}\left[{\mathcal{I}}_{{\mathrm{ⱱ}}^{+}}^{\alpha }\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right)\oplus {\mathcal{I}}_{{\mathrm{ⱱ}}^{+}}^{\alpha }\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\right]{\int }_0^1{\mathrm{ԟ}}^{\beta -1}\left[\mathrm{ƛ}\left(\mathrm{ԟ}\right)+\mathrm{ƛ}\left(1-\mathrm{ԟ}\right)\right]d\mathrm{ԟ}.$$

(43)

and

$${\displaystyle \frac{\Gamma \left(\alpha +1\right)}{{2\mathrm{ƛ}\left(\frac{1}{2}\right)\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }}}\left[{\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-}}^{\alpha }\tilde{\mathcal{P}}\left(\mathrm{ⱱ},{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)\right]$$

$${\le }_{\mathbb{F}}{\displaystyle \frac{\beta \Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }{\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}\left[{\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-},{\mathcal{p}}^{+}}^{\alpha , \beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\oplus {\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-},{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\alpha , \beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}\right)\right]$$

$${\le }_{\mathbb{F}}{\displaystyle \frac{\beta \Gamma \left(\alpha +1\right)}{{\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }}}\left[{\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-}}^{\alpha }\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}\right)\oplus {\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-}}^{\alpha }\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\right]{\int }_0^1{\mathrm{ԟ}}^{\beta -1}\left[\mathrm{ƛ}\left(\mathrm{ԟ}\right)+\mathrm{ƛ}\left(1-\mathrm{ԟ}\right)\right]d\mathrm{ԟ}.$$

(44)

Similarly, since ${\tilde{\mathcal{P}}}_{\mathrm{ʑ}}\left(z\right)=\tilde{\mathcal{P}}\left(z,\mathrm{ʑ}\right)$ then, from (37), (43) and (44), we have

$${\displaystyle \frac{\Gamma \left(\beta +1\right)}{{2\mathrm{ƛ}\left(\frac{1}{2}\right)\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}\left[{\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta }\tilde{\mathcal{P}}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\right]$$

$${\le }_{\mathbb{F}}{\displaystyle \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }{\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}\left[{\mathcal{I}}_{{\mathrm{ⱱ}}^{+},{\mathcal{p}}^{+}}^{\alpha , \beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\oplus {\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-},{\mathcal{p}}^{+}}^{\alpha , \beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\right]$$

$${\le }_{\mathbb{F}}{\displaystyle \frac{\alpha \Gamma \left(\beta +1\right)}{{\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}\left[{\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\oplus {\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\right].$$

(45)

and

$${\displaystyle \frac{\Gamma \left(\beta +1\right)}{{2\mathrm{ƛ}\left(\frac{1}{2}\right)\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\alpha }}}\left[{\mathcal{I}}_{{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\beta }\tilde{\mathcal{P}}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}},\mathcal{p}\right)\right]$$

$${\le }_{\mathbb{F}}{\displaystyle \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }{\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}\left[{\mathcal{I}}_{{\mathrm{ⱱ}}^{+},{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\alpha , \beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right)\oplus {\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-},{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\alpha , \beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}\right)\right]$$

$${\le }_{\mathbb{F}}{\displaystyle \frac{\alpha \Gamma \left(\beta +1\right)}{{\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}\left[{\mathcal{I}}_{{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}\right)\oplus {\mathcal{I}}_{{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right)\right].$$

(46)

The second, third, and fourth inequalities of (30) will be the consequence of adding the inequalities (43)–(46).

The second, third, and fourth inequalities within Equation (30) arise from the incorporation of the inequalities (43)–(46).

Now, considering any $\mathrm{ɤ}\in \left[0,1\right]$, we encounter the left segment of inequality (16).

$$\begin{gathered} {\displaystyle \frac{1}{{\mathrm{ƛ}}^2\left(\frac{1}{2}\right)}}{\mathcal{P}}_{\mathrm{ɤ}}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}},{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right){\le }_I{\displaystyle \frac{\Gamma \left(\beta +1\right)}{{\mathrm{ƛ}\left(\frac{1}{2}\right)\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}[{\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta }{\mathcal{P}}_{\mathrm{ɤ}}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right) \\ +{\mathcal{I}}_{{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\beta }{\mathcal{P}}_{\mathrm{ɤ}}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}},\mathcal{p}\right)] \end{gathered}$$

(47)

and

$$\begin{gathered} {\displaystyle \frac{1}{{\mathrm{ƛ}}^2\left(\frac{1}{2}\right)}}{\mathcal{P}}_{\mathrm{ɤ}}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}},{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right){\le }_I{\displaystyle \frac{\Gamma \left(\alpha +1\right)}{{\mathrm{ƛ}\left(\frac{1}{2}\right)\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }}}[{\mathcal{I}}_{{\mathrm{ⱱ}}^{+}}^{\alpha }{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right) \\ +{\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-}}^{\alpha }{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)] \end{gathered}$$

(48)

The subsequent inequality emerges from the combination of the two inequalities (47) and (48):

$$\begin{gathered} {\displaystyle \frac{1}{{\mathrm{ƛ}}^2\left(\frac{1}{2}\right)}}{\mathcal{P}}_{\mathrm{ɤ}}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}},{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right){\le }_I{\displaystyle \frac{\Gamma \left(\alpha +1\right)}{{\mathrm{ƛ}\left(\frac{1}{2}\right)\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }}}[{\mathcal{I}}_{{\mathrm{ⱱ}}^{+}}^{\alpha }{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right) \\ +{\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-}}^{\alpha }{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)] \end{gathered}$$

$$+{\displaystyle \frac{\Gamma \left(\beta +1\right)}{{\mathrm{ƛ}\left(\frac{1}{2}\right)\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}\left[{\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta }{\mathcal{P}}_{\mathrm{ɤ}}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)+{\mathcal{I}}_{{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\beta }{\mathcal{P}}_{\mathrm{ɤ}}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}},\mathcal{p}\right)\right]$$

Likewise, as we acquire the collection of $\mathsf{I}$·$\mathsf{V}$·$\mathsf{M}$s ${\mathcal{P}}_{\mathrm{ɤ}}:▽\to {\mathbb{R}}_I^{+}$ for $\mathrm{ɤ}\in \left[0,1\right]$, the inequality can be articulated in the following manner:

$${\displaystyle \frac{1}{{\mathrm{ƛ}}^2\left(\frac{1}{2}\right)}}\tilde{\mathcal{P}}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}},{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)$$

$$\begin{gathered} {\le }_{\mathbb{F}}{\displaystyle \frac{\Gamma \left(\alpha +1\right)}{{\mathrm{ƛ}\left(\frac{1}{2}\right)\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }}}\left[{\mathcal{I}}_{{\mathrm{ⱱ}}^{+}}^{\alpha }\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)\oplus {\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-}}^{\alpha }\tilde{\mathcal{P}}\left(\mathrm{ⱱ},{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)\right] \\ \oplus {\displaystyle \frac{\Gamma \left(\beta +1\right)}{{\mathrm{ƛ}\left(\frac{1}{2}\right)\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}[{\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta }\tilde{\mathcal{P}}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right) \\ \oplus {\mathcal{I}}_{{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\beta }\tilde{\mathcal{P}}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}},\mathcal{p}\right)]. \end{gathered}$$

(49)

This is the primary inequality in Equation (30).

Now, considering any $\mathrm{ɤ}\in \left[0,1\right]$, we observe the right section of inequality (16):

$$\begin{gathered} {\displaystyle \frac{\Gamma \left(\beta \right)}{{\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}\left[{\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta }{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)+{\mathcal{I}}_{{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\beta }{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathcal{p}\right)\right]{\le }_I\left[{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathcal{p}\right)+{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\right] \\ \times {\int }_0^1{\mathrm{ԟ}}^{\beta -1}\left[\mathrm{ƛ}\left(\mathrm{ԟ}\right)+\mathrm{ƛ}\left(1-\mathrm{ԟ}\right)\right]d\mathrm{ԟ} \end{gathered}$$

(50)

$$\begin{array}{c}{\displaystyle \frac{\Gamma \left(\beta \right)}{{\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}\left[{\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta }{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)+{\mathcal{I}}_{{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\beta }{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right)\right]{\le }_I\left[{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱳ},\mathcal{p}\right)+{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱳ},\mathcal{q}\right)\right]\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times {\int }_0^1{\mathrm{ԟ}}^{\beta -1}\left[\mathrm{ƛ}\left(\mathrm{ԟ}\right)+\mathrm{ƛ}\left(1-\mathrm{ԟ}\right)\right]d\mathrm{ԟ}\hfill \end{array}$$

(51)

$${\displaystyle \frac{\Gamma \left(\alpha \right)}{{\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }}}\left[{\mathcal{I}}_{{\mathrm{ⱱ}}^{+}}^{\alpha }{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right)+{\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-}}^{\alpha }{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathcal{p}\right)\right]{\le }_I\left[{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathcal{p}\right)+{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱳ},\mathcal{p}\right)\right]\times {\int }_0^1{\mathtt{t}}^{\alpha -1}\mathrm{ƛ}\left(\mathtt{t}\right)+\mathrm{ƛ}\left(1-\mathtt{t}\right)d\mathtt{t}$$

(52)

$$\begin{array}{c}{\displaystyle \frac{\Gamma \left(\alpha \right)}{{\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }}}\left[{\mathcal{I}}_{{\mathrm{ⱱ}}^{+}}^{\alpha }{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)+{\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-}}^{\alpha }{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\right]{\le }_I\left[{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathcal{q}\right)+{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱳ},\mathcal{q}\right)\right]\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times {\int }_0^1{\mathtt{t}}^{\alpha -1}\mathrm{ƛ}\left(\mathtt{t}\right)+\mathrm{ƛ}\left(1-\mathtt{t}\right)d\mathtt{t}\hfill \end{array}$$

(53)

Summing inequalities (50)–(53), and then multiplying the resultant with $\alpha \beta$, we have

$$\begin{gathered} {\displaystyle \frac{\beta \Gamma \left(\alpha +1\right)}{{\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }}}[{\mathcal{I}}_{{\mathrm{ⱱ}}^{+}}^{\alpha }{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right)+{\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-}}^{\alpha }{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathcal{p}\right)+{\mathcal{I}}_{{\mathrm{ⱱ}}^{+}}^{\alpha }{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right) \\ +{\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-}}^{\alpha }{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)] \end{gathered}$$

$$\begin{gathered} +{\displaystyle \frac{\alpha \Gamma \left(\beta +1\right)}{{\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}[{\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta }{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)+{\mathcal{I}}_{{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\beta }{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathcal{p}\right)+{\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta }{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right) \\ +{\mathcal{I}}_{{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\beta }{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right)] \end{gathered}$$

$${\le }_I\left[{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathcal{p}\right)+{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathcal{q}\right)+{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱳ},\mathcal{p}\right)+{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱳ},\mathcal{q}\right)\right]\times {\int }_0^1{\mathrm{ԟ}}^{\beta -1}\left[\mathrm{ƛ}\left(\mathrm{ԟ}\right)+\mathrm{ƛ}\left(1-\mathrm{ԟ}\right)\right]d\mathrm{ԟ}{\int }_0^1{\mathtt{t}}^{\alpha -1}\mathrm{ƛ}\left(\mathtt{t}\right)+\mathrm{ƛ}\left(1-\mathtt{t}\right)d\mathtt{t}.$$

This is because, derived from ɤ-levels, we acquire the set of $\mathsf{I}$·$\mathsf{V}$·$\mathsf{M}s$ ${\mathcal{P}}_{\mathrm{ɤ}}:▽\to {\mathbb{R}}_I^{+}\subset {\mathbb{R}}_I$, which is expressed as

$$\begin{gathered} {\displaystyle \frac{\beta \Gamma \left(\alpha +1\right)}{{\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }}}[{\mathcal{I}}_{{\mathrm{ⱱ}}^{+}}^{\alpha }\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right)\oplus {\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-}}^{\alpha }\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}\right)\oplus {\mathcal{I}}_{{\mathrm{ⱱ}}^{+}}^{\alpha }\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right) \\ \oplus {\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-}}^{\alpha }\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)] \end{gathered}$$

$$\begin{gathered} \oplus {\displaystyle \frac{\alpha \Gamma \left(\beta +1\right)}{{\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}[{\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\oplus {\mathcal{I}}_{{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}\right)\oplus {\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right) \\ \oplus {\mathcal{I}}_{{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right)] \end{gathered}$$

$${\le }_{\mathbb{F}}\left[\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}\right)\oplus \tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{q}\right)\oplus \tilde{\mathcal{P}}\left(\mathrm{ⱳ},\mathcal{p}\right)\oplus \tilde{\mathcal{P}}\left(\mathrm{ⱳ},\mathcal{q}\right)\right]\times {\int }_0^1{\mathrm{ԟ}}^{\beta -1}\left[\mathrm{ƛ}\left(\mathrm{ԟ}\right)+\mathrm{ƛ}\left(1-\mathrm{ԟ}\right)\right]d\mathrm{ԟ}{\int }_0^1{\mathtt{t}}^{\alpha -1}\mathrm{ƛ}\left(\mathtt{t}\right)+\mathrm{ƛ}\left(1-\mathtt{t}\right)d\mathtt{t}.$$

(54)

Here lies the ultimate inequality within (30), marking the establishment of the conclusion. □

Example 3.

We consider the $\mathsf{F}$·$\mathsf{N}$·$\mathsf{V}$·$\mathsf{M}$s $\tilde{\mathcal{P}}:\left[0, 2\right]\times \left[0, 2\right]\to {\mathbb{F}}_0$, characterized by

$$\mathcal{P}\left(\mathrm{ҩ},\mathrm{ʑ}\right)\left(\sigma \right)=\left\{\begin{array}{c}{\displaystyle \frac{\sigma }{\left(2-\sqrt{\mathrm{ҩ}}\right)\left(2-\sqrt{\mathrm{ʑ}}\right)}}, \sigma \in \left[0, \left(2-\sqrt{\mathrm{ҩ}}\right)\left(2-\sqrt{\mathrm{ʑ}}\right)\right]\\ {\displaystyle \frac{2\left(2-\sqrt{\mathrm{ҩ}}\right)\left(2-\sqrt{\mathrm{ʑ}}\right)-\sigma }{\left(2-\sqrt{\mathrm{ҩ}}\right)\left(2-\sqrt{\mathrm{ʑ}}\right)}}, \sigma \in \left(\left(2-\sqrt{\mathrm{ҩ}}\right)\left(2-\sqrt{\mathrm{ʑ}}\right), 2\left(2-\sqrt{\mathrm{ҩ}}\right)\left(2-\sqrt{\mathrm{ʑ}}\right)\right]\\ 0, otherwise,\end{array}\right.$$

(55)

then, for each $\mathrm{ɤ}\in \left[0,1\right],$ we have ${\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ҩ},\mathrm{ʑ}\right)=\left[\mathrm{ɤ}\left(2-\sqrt{\mathrm{ҩ}}\right)\left(2-\sqrt{\mathrm{ʑ}}\right),\left(2-\mathrm{ɤ}\right)\left(2-\sqrt{\mathrm{ҩ}}\right)\left(2-\sqrt{\mathrm{ʑ}}\right)\right]$. Consequently, the endpoint functions ${\mathcal{P}}_{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right),$ ${\mathcal{P}}^{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)$ are $\mathsf{C}$-$\mathrm{ƛ}$-pre-invex functions for each $\mathrm{ɤ}\in \left[0,1\right]$ with respect to ${\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)=\mathrm{ⱳ}-\mathrm{ⱱ},$ and ${\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)=\mathcal{q}-\mathcal{p}$. Hence, $\tilde{\mathcal{P}}\left(\mathrm{ҩ},\mathrm{ʑ}\right)$ is a $\mathsf{C}$-$\mathrm{ƛ}$-pre-invex $\mathsf{F}$·$\mathsf{N}$·$\mathsf{V}$·$\mathsf{M}$ with respect to ${\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)=\mathrm{ⱳ}-\mathrm{ⱱ},$ and ${\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)=\mathcal{q}-\mathcal{p}$.

$${\mathcal{P}}_{\mathrm{ɤ}}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}}, {\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)=\left[\mathrm{ɤ},\left(2-\mathrm{ɤ}\right)\right]$$

$$\begin{array}{c}{\displaystyle \frac{\Gamma \left(\alpha +1\right)}{{4\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }}}\left[{\mathcal{I}}_{{\mathrm{ⱱ}}^{+}}^{\alpha }\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)\oplus {\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-}}^{\alpha }\tilde{\mathcal{P}}\left(\mathrm{ⱱ},{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)\right]\hfill \\ \oplus {\displaystyle \frac{\Gamma \left(\beta +1\right)}{{4\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}\left[{\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta }\tilde{\mathcal{P}}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\oplus {\mathcal{I}}_{{\mathcal{q}}^{-}}^{\beta }\tilde{\mathcal{P}}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}},\mathcal{p}\right)\right]\hfill \end{array}$$

$$=\left[\mathrm{ɤ}\left(2-{\displaystyle \frac{\sqrt{2}}{4}}-{\displaystyle \frac{\sqrt{2}}{8}}\mathrm{\hslash }\right),\left(2-\mathrm{ɤ}\right)\left(2-{\displaystyle \frac{\sqrt{2}}{4}}-{\displaystyle \frac{\sqrt{2}}{8}}\mathrm{\hslash }\right)\right]$$

$$\begin{gathered} {\displaystyle \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{4\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }{\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}[{\mathcal{I}}_{{\mathrm{ⱱ}}^{+},{\mathcal{p}}^{+}}^{\alpha , \beta }{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\oplus {\mathcal{I}}_{{\mathrm{ⱱ}}^{+},{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\alpha , \beta }{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right) \\ \oplus {\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-},{\mathcal{p}}^{+}}^{\alpha , \beta }{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\oplus {\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-},{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\alpha , \beta }{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathcal{p}\right)] \end{gathered}$$

$$=\left[\mathrm{ɤ}\left({\displaystyle \frac{33}{8}}-\sqrt{2}-{\displaystyle \frac{\sqrt{2}}{2}}\mathrm{\hslash }+{\displaystyle \frac{\mathrm{\hslash }}{8}}+{\displaystyle \frac{{\mathrm{\hslash }}^2}{32}}\right),\left(2-\mathrm{ɤ}\right)\left({\displaystyle \frac{33}{8}}-\sqrt{2}-{\displaystyle \frac{\sqrt{2}}{2}}\mathrm{\hslash }+{\displaystyle \frac{\mathrm{\hslash }}{8}}+{\displaystyle \frac{{\mathrm{\hslash }}^2}{32}}\right)\right]$$

$$\begin{gathered} {\displaystyle \frac{\Gamma \left(\alpha +1\right)}{{8\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }}}[{\mathcal{I}}_{{\mathrm{ⱱ}}^{+}}^{\alpha }\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right)\oplus {\mathcal{I}}_{{\mathrm{ⱱ}}^{+}}^{\alpha }\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\oplus {\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-}}^{\alpha }\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}\right) \\ \oplus {\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-}}^{\alpha }\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)] \end{gathered}$$

$$\begin{gathered} \oplus {\displaystyle \frac{\Gamma \left(\beta +1\right)}{{8\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}[{\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\oplus {\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\oplus {\mathcal{I}}_{{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}\right) \\ \oplus {\mathcal{I}}_{{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right)] \end{gathered}$$

$$=\left[{\displaystyle \frac{34\sqrt{2}+\left(\sqrt{2}-4\right)\mathrm{\hslash }-24}{8\sqrt{2}}}\mathrm{ɤ},{\displaystyle \frac{34\sqrt{2}+\left(\sqrt{2}-4\right)\mathrm{\hslash }-24}{8\sqrt{2}}}\left(2-\mathrm{ɤ}\right)\right]$$

$${\displaystyle \frac{{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathcal{p},\mathrm{ⱳ}\right)+{\mathcal{P}}_{\mathrm{ɤ}}\left(\sigma ,\mathrm{ⱳ}\right)+{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathcal{p},\mathcal{q}\right)+{\mathcal{P}}_{\mathrm{ɤ}}\left(\sigma ,\mathcal{q}\right)}{4}}=\left[\mathrm{ɤ}\left({\displaystyle \frac{9}{2}}-2\sqrt{2}\right),\left(2-\mathrm{ɤ}\right)\left({\displaystyle \frac{9}{2}}-2\sqrt{2}\right)\right].$$

That is,

$$\left[\mathrm{ɤ},\left(2-\mathrm{ɤ}\right)\right]{\le }_I\left[\mathrm{ɤ}\left(2-{\displaystyle \frac{\sqrt{2}}{4}}-{\displaystyle \frac{\sqrt{2}}{8}}\mathrm{\hslash }\right),\left(2-\mathrm{ɤ}\right)\left(2-{\displaystyle \frac{\sqrt{2}}{4}}-{\displaystyle \frac{\sqrt{2}}{8}}\mathrm{\hslash }\right)\right]$$

$${\le }_I\left[\mathrm{ɤ}\left({\displaystyle \frac{33}{8}}-\sqrt{2}-{\displaystyle \frac{\sqrt{2}}{2}}\mathrm{\hslash }+{\displaystyle \frac{\mathrm{\hslash }}{8}}+{\displaystyle \frac{{\mathrm{\hslash }}^2}{32}}\right),\left(2-\mathrm{ɤ}\right)\left({\displaystyle \frac{33}{8}}-\sqrt{2}-{\displaystyle \frac{\sqrt{2}}{2}}\mathrm{\hslash }+{\displaystyle \frac{\mathrm{\hslash }}{8}}+{\displaystyle \frac{{\mathrm{\hslash }}^2}{32}}\right)\right]$$

$${\le }_I\left[{\displaystyle \frac{34\sqrt{2}+\left(\sqrt{2}-4\right)\mathrm{\hslash }-24}{8\sqrt{2}}}\mathrm{ɤ},{\displaystyle \frac{34\sqrt{2}+\left(\sqrt{2}-4\right)\mathrm{\hslash }-24}{8\sqrt{2}}}\left(2-\mathrm{ɤ}\right)\right]$$

$${\le }_I\left[\mathrm{ɤ}\left({\displaystyle \frac{9}{2}}-2\sqrt{2}\right),\left(2-\mathrm{ɤ}\right)\left({\displaystyle \frac{9}{2}}-2\sqrt{2}\right)\right]$$

Hence, Theorem 7 has been verified.

Remark 2.

Setting $\alpha =1$ and $\beta =1$, and $\mathrm{ƛ}\left(\mathtt{t}\right)=\mathtt{t},\mathrm{ƛ}\left(\mathrm{ԟ}\right)=\mathrm{ԟ}$, from (30), as a result, there will be inequity; see [62]:

$$\tilde{\mathcal{P}}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}}, {\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)$$

$$\begin{gathered} {\le }_{\mathbb{F}}{\displaystyle \frac{\begin{array}{c} \\ 1\end{array}}{2}}[{\displaystyle \frac{1}{{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}}{\int }_{\mathrm{ⱱ}}^{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}\tilde{\mathcal{P}}\left(\mathrm{ҩ},{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)d\mathrm{ҩ} \\ \oplus {\displaystyle \frac{1}{{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}}{\int }_{\mathcal{p}}^{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}\tilde{\mathcal{P}}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}},\mathrm{ʑ}\right)d\mathrm{ʑ}] \end{gathered}$$

$${\le }_{\mathbb{F}}{\displaystyle \frac{1}{{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}}{\int }_{\mathrm{ⱱ}}^{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{\int }_{\mathcal{p}}^{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}\tilde{\mathcal{P}}\left(\mathrm{ҩ},\mathrm{ʑ}\right)d\mathrm{ʑ}d\mathrm{ҩ}$$

$${\le }_{\mathbb{F}}{\displaystyle \frac{\begin{array}{c} \\ 1\end{array}}{4{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}}\left[{\int }_{\mathrm{ⱱ}}^{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}\tilde{\mathcal{P}}\left(\mathrm{ҩ},\mathcal{p}\right)d\mathrm{ҩ}\oplus {\int }_{\mathrm{ⱱ}}^{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}\tilde{\mathcal{P}}\left(\mathrm{ҩ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)d\mathrm{ҩ}\right]$$

$$\oplus {\displaystyle \frac{\begin{array}{c} \\ 1\end{array}}{4{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}}\left[{\int }_{\mathcal{p}}^{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathrm{ʑ}\right)d\mathrm{ʑ}\oplus {\int }_{\mathcal{p}}^{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathrm{ʑ}\right)d\mathrm{ʑ}\right]$$

$${\le }_{\mathbb{F}}{\displaystyle \frac{\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}\right)\oplus \tilde{\mathcal{P}}\left(\mathrm{ⱳ},\mathcal{p}\right)\oplus \tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{q}\right)\oplus \tilde{\mathcal{P}}\left(\mathrm{ⱳ},\mathcal{q}\right)}{4}}.$$

(56)

If we set $\alpha =1$ and $\beta =1$, $\mathrm{ƛ}\left(\mathtt{t}\right)=\mathtt{t}, \mathrm{ƛ}\left(\mathrm{ԟ}\right)=\mathrm{ԟ}$ and $\tilde{\mathcal{P}}$ is $\mathsf{C}$-left-$\mathrm{ƛ}$-pre-invex, then from (30), as a result, there will be inequity:

$$\tilde{\mathcal{P}}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}}, {\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)$$

$$\begin{gathered} {\supseteq }_{\mathbb{F}}{\displaystyle \frac{\begin{array}{c} \\ 1\end{array}}{2}}[{\displaystyle \frac{1}{{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}}{\int }_{\mathrm{ⱱ}}^{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}\tilde{\mathcal{P}}\left(\mathrm{ҩ},{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)d\mathrm{ҩ} \\ \oplus {\displaystyle \frac{1}{{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}}{\int }_{\mathcal{p}}^{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}\tilde{\mathcal{P}}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}},\mathrm{ʑ}\right)d\mathrm{ʑ}] \end{gathered}$$

$${\supseteq }_{\mathbb{F}}{\displaystyle \frac{1}{{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}}{\int }_{\mathrm{ⱱ}}^{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{\int }_{\mathcal{p}}^{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}\tilde{\mathcal{P}}\left(\mathrm{ҩ},\mathrm{ʑ}\right)d\mathrm{ʑ}d\mathrm{ҩ}$$

$${\supseteq }_{\mathbb{F}}{\displaystyle \frac{\begin{array}{c} \\ 1\end{array}}{4{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}}\left[{\int }_{\mathrm{ⱱ}}^{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}\tilde{\mathcal{P}}\left(\mathrm{ҩ},\mathcal{p}\right)d\mathrm{ҩ}\oplus {\int }_{\mathrm{ⱱ}}^{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}\tilde{\mathcal{P}}\left(\mathrm{ҩ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)d\mathrm{ҩ}\right]$$

$$\oplus {\displaystyle \frac{\begin{array}{c} \\ 1\end{array}}{4{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}}\left[{\int }_{\mathcal{p}}^{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathrm{ʑ}\right)d\mathrm{ʑ}\oplus {\int }_{\mathcal{p}}^{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathrm{ʑ}\right)d\mathrm{ʑ}\right]$$

$${\supseteq }_{\mathbb{F}}{\displaystyle \frac{\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}\right)\oplus \tilde{\mathcal{P}}\left(\mathrm{ⱳ},\mathcal{p}\right)\oplus \tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{q}\right)\oplus \tilde{\mathcal{P}}\left(\mathrm{ⱳ},\mathcal{q}\right)}{4}}.$$

(57)

By setting $\mathrm{ƛ}\left(\mathtt{t}\right)=\mathtt{t}, \mathrm{ƛ}\left(\mathrm{ԟ}\right)=\mathrm{ԟ}$ and ${\mathcal{P}}_{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)\ne {\mathcal{P}}^{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)$ with $\mathrm{ɤ}=1$ and $\tilde{\mathcal{P}}$ being $\mathsf{C}$-left-$\mathrm{ƛ}$-pre-invex, following Equation (30), we manage to introduce the forthcoming inequality, as illustrated in [14]:

$$\mathcal{P}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}}, {\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)$$

$$\begin{array}{c}\supseteq {\displaystyle \frac{\Gamma \left(\alpha +1\right)}{{4\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }}}\left[{\mathcal{I}}_{{\mathrm{ⱱ}}^{+}}^{\alpha }\mathcal{P}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)+{\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-}}^{\alpha }\mathcal{P}\left(\mathrm{ⱱ},{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)\right]\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\displaystyle \frac{\Gamma \left(\beta +1\right)}{{4\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}\left[{\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta }\mathcal{P}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)+{\mathcal{I}}_{{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\beta }\mathcal{P}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}},\mathcal{p}\right)\right]\hfill \end{array}$$

$$\begin{gathered} \supseteq {\displaystyle \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{4\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }{\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}[{\mathcal{I}}_{{\mathrm{ⱱ}}^{+},{\mathcal{p}}^{+}}^{\alpha , \beta }\mathcal{P}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)+{\mathcal{I}}_{{\mathrm{ⱱ}}^{+},{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\alpha , \beta }\mathcal{P}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right) \\ +{\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-},{\mathcal{p}}^{+}}^{\alpha , \beta }\mathcal{P}\left(\mathrm{ⱱ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)+{\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-},{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\alpha , \beta }\mathcal{P}\left(\mathrm{ⱱ},\mathcal{p}\right)] \end{gathered}$$

$$\begin{gathered} \supseteq {\displaystyle \frac{\Gamma \left(\alpha +1\right)}{{8\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }}}[{\mathcal{I}}_{{\mathrm{ⱱ}}^{+}}^{\alpha }\mathcal{P}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right)+{\mathcal{I}}_{{\mathrm{ⱱ}}^{+}}^{\alpha }\mathcal{P}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)+{\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-}}^{\alpha }\mathcal{P}\left(\mathrm{ⱱ},\mathcal{p}\right) \\ +{\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-}}^{\alpha }\mathcal{P}\left(\mathrm{ⱱ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)] \end{gathered}$$

$$\begin{gathered} +{\displaystyle \frac{\Gamma \left(\beta +1\right)}{{8\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}[{\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta }\mathcal{P}\left(\mathrm{ⱱ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)+{\mathcal{I}}_{{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\beta }\mathcal{P}\left(\mathrm{ⱱ},\mathcal{p}\right)+{\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta }\mathcal{P}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right) \\ +{\mathcal{I}}_{{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\beta }\mathcal{P}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right)] \end{gathered}$$

$$\supseteq {\displaystyle \frac{\mathcal{P}\left(\mathrm{ⱱ},\mathcal{p}\right)+\mathcal{P}\left(\mathrm{ⱳ},\mathcal{p}\right)+\mathcal{P}\left(\mathrm{ⱱ},\mathcal{q}\right)+\mathcal{P}\left(\mathrm{ⱳ},\mathcal{q}\right)}{4}}.$$

(58)

If we set $\mathrm{ƛ}\left(\mathtt{t}\right)=\mathtt{t}, \mathrm{ƛ}\left(\mathrm{ԟ}\right)=\mathrm{ԟ}$ and ${\mathcal{P}}_{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)\ne {\mathcal{P}}^{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)$ with $\mathrm{ɤ}=1$ and $\tilde{\mathcal{P}}$ is $\mathsf{C}$-left-$\mathrm{ƛ}$-pre-invex, following Equation (30), we manage to introduce the forthcoming inequality, as illustrated in [14]:

$$\mathcal{P}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}}, {\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)$$

$$\supseteq {\displaystyle \frac{\begin{array}{c} \\ 1\end{array}}{2}}\left[{\displaystyle \frac{1}{{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}}{\int }_{\mathrm{ⱱ}}^{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}\mathcal{P}\left(\mathrm{ҩ},{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)d\mathrm{ҩ}+{\displaystyle \frac{1}{{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}}{\int }_{\mathcal{p}}^{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}\mathcal{P}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}},\mathrm{ʑ}\right)d\mathrm{ʑ}\right]$$

$$\supseteq {\displaystyle \frac{1}{{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}}{\int }_{\mathrm{ⱱ}}^{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{\int }_{\mathcal{p}}^{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}\mathcal{P}\left(\mathrm{ҩ},\mathrm{ʑ}\right)d\mathrm{ʑ}d\mathrm{ҩ}$$

$$\supseteq {\displaystyle \frac{\begin{array}{c} \\ 1\end{array}}{4{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}}\left[{\int }_{\mathrm{ⱱ}}^{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}\mathcal{P}\left(\mathrm{ҩ},\mathcal{p}\right)d\mathrm{ҩ}+{\int }_{\mathrm{ⱱ}}^{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}\mathcal{P}\left(\mathrm{ҩ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)d\mathrm{ҩ}\right]$$

$$+{\displaystyle \frac{\begin{array}{c} \\ 1\end{array}}{4{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}}\left[{\int }_{\mathcal{p}}^{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}\mathcal{P}\left(\mathrm{ⱱ},\mathrm{ʑ}\right)d\mathrm{ʑ}+{\int }_{\mathcal{p}}^{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}\mathcal{P}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathrm{ʑ}\right)d\mathrm{ʑ}\right]$$

$$\supseteq {\displaystyle \frac{\mathcal{P}\left(\mathrm{ⱱ},\mathcal{p}\right)+\mathcal{P}\left(\mathrm{ⱳ},\mathcal{p}\right)+\mathcal{P}\left(\mathrm{ⱱ},\mathcal{q}\right)+\mathcal{P}\left(\mathrm{ⱳ},\mathcal{q}\right)}{4}}$$

(59)

By setting $\mathrm{ƛ}\left(\mathtt{t}\right)=\mathtt{t}, \mathrm{ƛ}\left(\mathrm{ԟ}\right)=\mathrm{ԟ}$ and ${\mathcal{P}}_{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)={\mathcal{P}}^{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)$ with $\mathrm{ɤ}=1$, following Equation (30), we manage to introduce the forthcoming inequality, as illustrated in [63]:

$$\mathcal{P}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}}, {\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)$$

$$\begin{array}{c}\le {\displaystyle \frac{\Gamma \left(\alpha +1\right)}{{4\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }}}\left[{\mathcal{I}}_{{\mathrm{ⱱ}}^{+}}^{\alpha } \mathcal{P}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)+{\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-}}^{\alpha }\mathcal{P}\left(\mathrm{ⱱ},{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)\right]\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\displaystyle \frac{\Gamma \left(\beta +1\right)}{{4\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}\left[{\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta }\mathcal{P}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)+{\mathcal{I}}_{{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\beta }\mathcal{P}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}},\mathcal{p}\right)\right]\hfill \end{array}$$

$$\begin{gathered} \le {\displaystyle \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{4\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }{\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}[{\mathcal{I}}_{{\mathrm{ⱱ}}^{+},{\mathcal{p}}^{+}}^{\alpha ,\beta }\mathcal{P}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)+{\mathcal{I}}_{{\mathrm{ⱱ}}^{+},{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\alpha ,\beta }\mathcal{P}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right) \\ +{\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-},{\mathcal{p}}^{+}}^{\alpha ,\beta }\mathcal{P}\left(\mathrm{ⱱ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)+{\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-},{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\alpha ,\beta }\mathcal{P}\left(\mathrm{ⱱ},\mathcal{p}\right)] \end{gathered}$$

$$\begin{gathered} \le {\displaystyle \frac{\Gamma \left(\alpha +1\right)}{{8\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }}}[{\mathcal{I}}_{{\mathrm{ⱱ}}^{+}}^{\alpha }\mathcal{P}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right){\mathcal{I}}_{{\mathrm{ⱱ}}^{+}}^{\alpha }\mathcal{P}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)+{\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-}}^{\alpha }\mathcal{P}\left(\mathrm{ⱱ},\mathcal{p}\right) \\ +{\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-}}^{\alpha }\mathcal{P}\left(\mathrm{ⱱ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)] \end{gathered}$$

$$\begin{gathered} +{\displaystyle \frac{\Gamma \left(\beta +1\right)}{{8\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}[{\mathcal{I}}_{{\mathcal{p}}^{+}}^{ \beta }\mathcal{P}\left(\mathrm{ⱱ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)+{\mathcal{I}}_{{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\beta }\mathcal{P}\left(\mathrm{ⱱ},\mathcal{p}\right)+{\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta }\mathcal{P}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right) \\ +{\mathcal{I}}_{{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\beta }\mathcal{P}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right)] \end{gathered}$$

$$\le {\displaystyle \frac{\mathcal{P}\left(\mathrm{ⱱ},\mathcal{p}\right)+\mathcal{P}\left(\mathrm{ⱳ},\mathcal{p}\right)+\mathcal{P}\left(\mathrm{ⱱ},\mathcal{q}\right)+\mathcal{P}\left(\mathrm{ⱳ},\mathcal{q}\right)}{4}}.$$

(60)

In the subsequent results, we anticipate discovering intriguing findings derived from the product of two $\mathsf{C}$-$\mathrm{ƛ}$-pre-invex $\mathsf{F}$·$\mathsf{N}$·$\mathsf{V}$·$\mathsf{M}$s. These inequalities are recognized as Pachpatte inequalities.

Theorem 8.

Let $\tilde{\mathcal{P}}, \tilde{\mathfrak{G}}:▽\to {\mathbb{F}}_0^{+}$ be two $\mathsf{C}$-$\mathrm{ƛ}$-pre-invex $\mathsf{F}$·$\mathsf{N}$·$\mathsf{V}$·$\mathsf{M}$s on $▽$ and let ${\mathrm{ƛ}}_1,{\mathrm{ƛ}}_2 :\left[0,1\right]\to {\mathbb{R}}^{+}.$ Subsequently, derived from ɤ-levels, we acquire the set of $\mathsf{I}$·$\mathsf{V}$·$\mathsf{M}s$ ${\mathcal{P}}_{\mathrm{ɤ}}:▽\to {\mathbb{R}}_I^{+}\subset {\mathbb{R}}_I$, which are expressed as ${\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ҩ},\mathrm{ʑ}\right)=\left[{\mathcal{P}}_{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right), {\mathcal{P}}^{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)\right]$ and ${\mathfrak{G}}_{\mathrm{ɤ}}\left(\mathrm{ҩ},\mathrm{ʑ}\right)=\left[{\mathfrak{G}}_{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right), {\mathfrak{G}}^{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)\right]$ for all $\left(\mathrm{ҩ},\mathrm{ʑ}\right)\in ▽$ and for all $\mathrm{ɤ}\in \left[0,1\right]$. If ${\tilde{\mathcal{P}}\otimes \tilde{\mathfrak{G}}\in \mathcal{F}\mathfrak{O}}_{▽}$, then the following inequalities hold:

$$\begin{gathered} {\displaystyle \frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }{\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}\left[{\mathcal{I}}_{{\mathrm{ⱱ}}^{+},{\mathcal{p}}^{+}}^{\alpha , \beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\otimes \tilde{\mathfrak{G}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right) \\ \oplus {\mathcal{I}}_{{\mathrm{ⱱ}}^{+},{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\alpha , \beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right)\otimes \tilde{\mathfrak{G}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right)\right] \end{gathered}$$

$$\begin{gathered} \oplus {\displaystyle \frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }{\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}\left[{\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-},{\mathcal{p}}^{+}}^{\alpha , \beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\otimes \tilde{\mathfrak{G}}\left(\mathrm{ⱱ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right) \\ \oplus {\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-},{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\alpha , \beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}\right)\otimes \tilde{\mathfrak{G}}\left(\mathrm{ⱱ},\mathcal{p}\right)\right] \end{gathered}$$

$$\begin{gathered} {\le }_{\mathbb{F}}\tilde{\mathcal{E}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right) \\ +{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)\right]d\mathtt{t}d\mathrm{ԟ} \end{gathered}$$

$$\begin{gathered} \oplus \widetilde{\mathrm{Ѵ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right) \\ +{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)\right]d\mathtt{t}d\mathrm{ԟ} \end{gathered}$$

$$\begin{gathered} \oplus \widetilde{\mathrm{Ѡ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right) \\ +{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)\right]d\mathtt{t}d\mathrm{ԟ} \end{gathered}$$

$$\begin{gathered} \oplus \widetilde{\mathsf{Ϗ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right) \\ +{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)\right]d\mathtt{t}d\mathrm{ԟ} \end{gathered}$$

(61)

If $\tilde{\mathcal{P}}$ and $\tilde{\mathfrak{G}}$ are both $\mathsf{C}$-$\mathrm{ƛ}$-pre-concave $\mathsf{F}$·$\mathsf{N}$·$\mathsf{V}$·$\mathsf{M}$s on $▽$, then the aforementioned inequality can be formulated as follows:

$$\begin{gathered} {\displaystyle \frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }{\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}\left[{\mathcal{I}}_{{\mathrm{ⱱ}}^{+},{\mathcal{p}}^{+}}^{\alpha , \beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\otimes \tilde{\mathfrak{G}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right) \\ \oplus {\mathcal{I}}_{{\mathrm{ⱱ}}^{+},{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\alpha , \beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right)\otimes \tilde{\mathfrak{G}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right)\right] \\ \begin{array}{c}\oplus {\displaystyle \frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }{\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}[{\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-},{\mathcal{p}}^{+}}^{\alpha , \beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\otimes \tilde{\mathfrak{G}}\left(\mathrm{ⱱ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\oplus {\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-},{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\alpha , \beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\otimes \tilde{\mathfrak{G}}\left(\mathrm{ⱱ},\mathcal{p}\right)]\hfill \end{array} \\ {\ge }_{\mathbb{F}}\tilde{\mathcal{E}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right) \\ +{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)\right]d\mathtt{t}d\mathrm{ԟ} \\ \oplus \widetilde{\mathrm{Ѵ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right) \\ +{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)\right]d\mathtt{t}d\mathrm{ԟ} \\ \oplus \widetilde{\mathrm{Ѡ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right) \\ +{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)\right]d\mathtt{t}d\mathrm{ԟ} \\ \oplus \widetilde{\mathsf{Ϗ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right) \\ +{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)\right]d\mathtt{t}d\mathrm{ԟ} \end{gathered}$$

(62)

where

$$\tilde{\mathcal{E}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)=\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}\right)\otimes \tilde{\mathfrak{G}}\left(\mathrm{ⱱ},\mathcal{p}\right)\oplus \tilde{\mathcal{P}}\left(\mathrm{ⱳ},\mathcal{p}\right)\otimes \tilde{\mathfrak{G}}\left(\mathrm{ⱳ},\mathcal{p}\right)\oplus \tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{q}\right)\otimes \tilde{\mathfrak{G}}\left(\mathrm{ⱱ},\mathcal{q}\right)\oplus \tilde{\mathcal{P}}\left(\mathrm{ⱳ},\mathcal{q}\right)\otimes \tilde{\mathfrak{G}}\left(\mathrm{ⱳ},\mathcal{q}\right),$$

$$\widetilde{\mathrm{Ѵ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)=\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}\right)\otimes \tilde{\mathfrak{G}}\left(\mathrm{ⱳ},\mathcal{p}\right)\oplus \tilde{\mathcal{P}}\left(\mathrm{ⱳ},\mathcal{p}\right)\otimes \tilde{\mathfrak{G}}\left(\mathrm{ⱱ},\mathcal{p}\right)\oplus \tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{q}\right)\otimes \tilde{\mathfrak{G}}\left(\mathrm{ⱳ},\mathcal{q}\right)\oplus \tilde{\mathcal{P}}\left(\mathrm{ⱳ},\mathcal{q}\right)\otimes \tilde{\mathfrak{G}}\left(\mathrm{ⱱ},\mathcal{q}\right),$$

$$\widetilde{\mathrm{Ѡ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)=\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}\right)\otimes \tilde{\mathfrak{G}}\left(\mathrm{ⱱ},\mathcal{q}\right)\oplus \tilde{\mathcal{P}}\left(\mathrm{ⱳ},\mathcal{p}\right)\otimes \tilde{\mathfrak{G}}\left(\mathrm{ⱳ},\mathcal{q}\right)\oplus \tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{q}\right)\otimes \tilde{\mathfrak{G}}\left(\mathrm{ⱱ},\mathcal{p}\right)\oplus \tilde{\mathcal{P}}\left(\mathrm{ⱳ},\mathcal{q}\right)\otimes \tilde{\mathfrak{G}}\left(\mathrm{ⱳ},\mathcal{p}\right),$$

$$\widetilde{\mathsf{Ϗ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)=\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}\right)\otimes \tilde{\mathfrak{G}}\left(\mathrm{ⱳ},\mathcal{q}\right)\oplus \tilde{\mathcal{P}}\left(\mathrm{ⱳ},\mathcal{p}\right)\otimes \tilde{\mathfrak{G}}\left(\mathrm{ⱱ},\mathcal{q}\right)\oplus \tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{q}\right)\otimes \tilde{\mathfrak{G}}\left(\mathrm{ⱳ},\mathcal{p}\right)\oplus \tilde{\mathcal{P}}\left(\mathrm{ⱳ},\mathcal{q}\right)\otimes \tilde{\mathfrak{G}}\left(\mathrm{ⱱ},\mathcal{p}\right),$$

and for each $\mathrm{ɤ}\in \left[0,1\right],$ $\tilde{\mathcal{E}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)$, $\widetilde{\mathrm{Ѵ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)$, $\widetilde{\mathrm{Ѡ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)$ and $\widetilde{\mathsf{Ϗ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)$ are defined as follows:

$${\mathcal{E}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)=\left[{\mathcal{E}}_{*}\left(\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right),\mathrm{ɤ}\right), {\mathcal{E}}^{*}\left(\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right),\mathrm{ɤ}\right)\right],$$

$${\mathrm{Ѵ}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)=\left[{\mathrm{Ѵ}}_{*}\left(\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right),\mathrm{ɤ}\right), {\mathrm{Ѵ}}^{*}\left(\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right),\mathrm{ɤ}\right)\right],$$

$${\mathrm{Ѡ}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)=\left[{\mathrm{Ѡ}}_{*}\left(\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right),\mathrm{ɤ}\right), {\mathrm{Ѡ}}^{*}\left(\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right),\mathrm{ɤ}\right)\right],$$

$${\mathsf{Ϗ}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)=\left[{\mathsf{Ϗ}}_{*}\left(\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right),\mathrm{ɤ}\right), {\mathsf{Ϗ}}^{*}\left(\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right),\mathrm{ɤ}\right)\right].$$

Proof. 

Let $\tilde{\mathcal{P}}$ and $\tilde{\mathfrak{G}}$ be two $\mathsf{C}$-${\mathrm{ƛ}}_1$ and ${\mathrm{ƛ}}_2$-pre-invex $\mathsf{F}$·$\mathsf{N}$·$\mathsf{V}$·$\mathsf{M}$s on $\left[\mathrm{ⱱ},\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]\times \left[\mathcal{p},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]$, respectively. Then,

$$\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)$$

$${\le }_{\mathbb{F}}{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right)\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}\right)\oplus {\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right)\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{q}\right)\oplus {\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right)\tilde{\mathcal{P}}\left(\mathrm{ⱳ},\mathcal{p}\right)\oplus {\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right)\tilde{\mathcal{P}}\left(\mathrm{ⱳ},\mathcal{q}\right),$$

$$\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\mathrm{ԟ}{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)$$

$${\le }_{\mathbb{F}}{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right)\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}\right)\oplus {\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right)\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{q}\right)\oplus {\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right)\tilde{\mathcal{P}}\left(\mathrm{ⱳ},\mathcal{p}\right)\oplus {\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right)\tilde{\mathcal{P}}\left(\mathrm{ⱳ},\mathcal{q}\right),$$

$$\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+\mathtt{t}{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)$$

$${\le }_{\mathbb{F}}{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right)\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}\right)\oplus {\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right)\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{q}\right)\oplus {\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right)\tilde{\mathcal{P}}\left(\mathrm{ⱳ},\mathcal{p}\right)\oplus {\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right)\tilde{\mathcal{P}}\left(\mathrm{ⱳ},\mathcal{q}\right),$$

$$\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+\mathtt{t}{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\mathrm{ԟ}{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)$$

$${\le }_{\mathbb{F}}{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right)\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}\right)\oplus {\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right)\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{q}\right)\oplus {\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right)\tilde{\mathcal{P}}\left(\mathrm{ⱳ},\mathcal{p}\right)\oplus {\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right)\tilde{\mathcal{P}}\left(\mathrm{ⱳ},\mathcal{q}\right),$$

and

$$\tilde{\mathfrak{G}}\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\mathrm{ⱳ},\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)$$

$${\le }_{\mathbb{F}}{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)\tilde{\mathfrak{G}}\left(\mathrm{ⱱ},\mathcal{p}\right)\oplus {\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)\tilde{\mathfrak{G}}\left(\mathrm{ⱱ},\mathcal{q}\right)\oplus {\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)\tilde{\mathfrak{G}}\left(\mathrm{ⱳ},\mathcal{p}\right)\oplus {\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)\tilde{\mathfrak{G}}\left(\mathrm{ⱳ},\mathcal{q}\right),$$

$$\tilde{\mathfrak{G}}\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\mathrm{ԟ}{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)$$

$${\le }_{\mathbb{F}}{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)\tilde{\mathfrak{G}}\left(\mathrm{ⱱ},\mathcal{p}\right)\oplus {\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)\tilde{\mathfrak{G}}\left(\mathrm{ⱱ},\mathcal{q}\right)\oplus {\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)\tilde{\mathfrak{G}}\left(\mathrm{ⱳ},\mathcal{p}\right)\oplus {\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)\tilde{\mathfrak{G}}\left(\mathrm{ⱳ},\mathcal{q}\right),$$

$$\tilde{\mathfrak{G}}\left(\mathrm{ⱱ}+\mathtt{t}{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)$$

$${\le }_{\mathbb{F}}{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)\tilde{\mathfrak{G}}\left(\mathrm{ⱱ},\mathcal{p}\right)\oplus {\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)\tilde{\mathfrak{G}}\left(\mathrm{ⱱ},\mathcal{q}\right)\oplus {\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)\tilde{\mathfrak{G}}\left(\mathrm{ⱳ},\mathcal{p}\right)\oplus {\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)\tilde{\mathfrak{G}}\left(\mathrm{ⱳ},\mathcal{q}\right),$$

$$\tilde{\mathfrak{G}}\left(\mathrm{ⱱ}+\mathtt{t}{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\mathrm{ԟ}{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)$$

$${\le }_{\mathbb{F}}{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)\tilde{\mathfrak{G}}\left(\mathrm{ⱱ},\mathcal{p}\right)\oplus {\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)\tilde{\mathfrak{G}}\left(\mathrm{ⱱ},\mathcal{q}\right)\oplus {\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)\tilde{\mathfrak{G}}\left(\mathrm{ⱳ},\mathcal{p}\right)\oplus {\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)\tilde{\mathfrak{G}}\left(\mathrm{ⱳ},\mathcal{q}\right),$$

Since $\tilde{\mathcal{P}}$ and $\tilde{\mathfrak{G}}$ are both $\mathsf{C}$-${\mathrm{ƛ}}_1$ and ${\mathrm{ƛ}}_2$-pre-invex $\mathsf{F}$·$\mathsf{N}$·$\mathsf{V}$·$\mathsf{M}$s on $\left[\mathrm{ⱱ},\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]\times \left[\mathcal{p},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]$, respectively, for any $\mathrm{ɤ}\in \left[0,1\right]$, we have

$${\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\times {\mathfrak{G}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)$$

$$+{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\mathrm{ԟ}{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\times {\mathfrak{G}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\mathrm{ԟ}{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)$$

$$+{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\mathtt{t}{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\times {\mathfrak{G}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\mathtt{t}{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)$$

$$+{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\mathtt{t}{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\mathrm{ԟ}{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\times {\mathfrak{G}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\mathtt{t}{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\mathrm{ԟ}{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)$$

$$\begin{gathered} {\le }_I{\mathcal{E}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)\left[{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right) \\ +{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)\right] \end{gathered}$$

$$\begin{gathered} +{\mathrm{Ѵ}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)\left[{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right) \\ +{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)\right] \end{gathered}$$

$$\begin{gathered} +{\mathrm{Ѡ}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)\left[{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right) \\ +{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)\right] \end{gathered}$$

$$\begin{gathered} +{\mathsf{Ϗ}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)\left[{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right) \\ +{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)\right]. \end{gathered}$$

Taking the multiplication of the above fuzzy inclusion with ${\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}$ and then taking the double integration of the resultant over $\left[0,1\right]\times \left[0,1\right]$ with respect to ($\mathtt{t},\mathtt{ԟ}$), it is the case that

$$\begin{array}{l}{\int }_0^1{\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\times {\mathfrak{G}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)d\mathtt{t}d\mathrm{ԟ}.\\ +{\int }_0^1{\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\mathrm{ԟ}{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\times {\mathfrak{G}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\mathrm{ԟ}{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)d\mathtt{t}d\mathrm{ԟ}\\ +{\int }_0^1{\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\mathtt{t}{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\times {\mathfrak{G}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\mathtt{t}{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)d\mathtt{t}d\mathrm{ԟ}\\ +{\int }_0^1{\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\mathtt{t}{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\mathrm{ԟ}{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\times {\mathfrak{G}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\mathtt{t}{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\mathrm{ԟ}{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)d\mathtt{t}d\mathrm{ԟ}\\ {\le }_I{\mathcal{E}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right){\int }_0^1{\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}[{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+\\ {\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)]d\mathtt{t}d\mathrm{ԟ}\\ +{\mathrm{Ѵ}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right){\int }_0^1{\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}[{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+\\ {\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)]d\mathtt{t}d\mathrm{ԟ}\\ +{\mathrm{Ѡ}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right){\int }_0^1{\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}[{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+\\ {\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)]d\mathtt{t}d\mathrm{ԟ}\\ +{\mathsf{Ϗ}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right){\int }_0^1{\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}[{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1(1-\\ \mathtt{t}){\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)]d\mathtt{t}d\mathrm{ԟ}\end{array}$$

(63)

From the expression on the right side of (63), we obtain

$$\begin{array}{c} {\int }_0^1{\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\times {\mathfrak{G}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)d\mathtt{t}d\mathrm{ԟ}\\ +{\int }_0^1{\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\mathrm{ԟ}{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\times {\mathfrak{G}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\mathrm{ԟ}{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)d\mathtt{t}d\mathrm{ԟ}\\ +{\int }_0^1{\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\mathtt{t}{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\times {\mathfrak{G}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\mathtt{t}{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)d\mathtt{t}d\mathrm{ԟ}\\ +{\int }_0^1{\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\mathtt{t}{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\mathrm{ԟ}{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\times {\mathfrak{G}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\mathtt{t}{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\mathrm{ԟ}{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)d\mathtt{t}d\mathrm{ԟ}\\ ={\displaystyle \frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }{\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}[{\mathcal{I}}_{{\mathrm{ⱱ}}^{+},{\mathcal{p}}^{+}}^{\alpha , \beta }{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{q}\right)\times {\mathfrak{G}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{q}\right)+{\mathcal{I}}_{{\mathrm{ⱱ}}^{+},{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\alpha , \beta }{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right)\times {\mathfrak{G}}_{\mathrm{ɤ}}(\mathrm{ⱱ}+\\ {\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p})].\end{array}$$

(64)

Combining (63) and (64), for each $\mathrm{ɤ}\in \left[0,1\right]$, we have

$$\begin{array}{c}\begin{array}{c} {\displaystyle \frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }{\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}[{\mathcal{I}}_{{\mathrm{ⱱ}}^{+},{\mathcal{p}}^{+}}^{\alpha , \beta }{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\times {\mathfrak{G}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)+{\mathcal{I}}_{{\mathrm{ⱱ}}^{+},{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\alpha , \beta }{\mathcal{P}}_{\mathrm{ɤ}}(\mathrm{ⱱ}+\hfill \\ {\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p})\times {\mathfrak{G}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right)]\hfill \end{array}\hfill \\ \begin{array}{c} {\le }_I{\mathcal{E}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right){\int }_0^1{\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}[{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+\hfill \\ {\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)]d\mathtt{t}d\mathrm{ԟ}\hfill \end{array}\hfill \\ \begin{array}{c} +{\mathrm{Ѵ}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right){\int }_0^1{\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}[{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+\hfill \\ {\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)]d\mathtt{t}d\mathrm{ԟ}\hfill \end{array}\hfill \\ \begin{array}{c} +{\mathrm{Ѡ}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right){\int }_0^1{\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}[{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+\hfill \\ {\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)]d\mathtt{t}d\mathrm{ԟ}\hfill \end{array}\hfill \\ \begin{array}{c} +{\mathsf{Ϗ}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right){\int }_0^1{\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}[{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1(1-\hfill \\ \mathtt{t}){\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)]d\mathtt{t}d\mathrm{ԟ}.\hfill \end{array}\hfill \end{array}$$

Moreover, we have

$$\begin{array}{c}\begin{array}{c}{\displaystyle \frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }{\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}[{\mathcal{I}}_{{\mathrm{ⱱ}}^{+},{\mathcal{p}}^{+}}^{\alpha , \beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\otimes \tilde{\mathfrak{G}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\oplus {\mathcal{I}}_{{\mathrm{ⱱ}}^{+},{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\alpha , \beta }\tilde{\mathcal{P}}(\mathrm{ⱱ}+\hfill \\ {\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p})\otimes \tilde{\mathfrak{G}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right)]\hfill \end{array}\hfill \\ \oplus {\displaystyle \frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }{\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}\left[{\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-},{\mathcal{p}}^{+}}^{\alpha , \beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\otimes \tilde{\mathfrak{G}}\left(\mathrm{ⱱ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\oplus {\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-},{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\alpha , \beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}\right)\otimes \tilde{\mathfrak{G}}\left(\mathrm{ⱱ},\mathcal{p}\right)\right]\hfill \\ \begin{array}{c} {\le }_{\mathbb{F}}\tilde{\mathcal{E}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}[{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+\hfill \\ {\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)]d\mathtt{t}d\mathrm{ԟ}\hfill \end{array}\hfill \\ \begin{array}{c} \oplus \widetilde{\mathrm{Ѵ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}[{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2(1-\hfill \\ \mathtt{t}){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)]d\mathtt{t}d\mathrm{ԟ}\hfill \end{array}\hfill \\ \begin{array}{c} \oplus \widetilde{\mathrm{Ѡ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}[{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+\hfill \\ {\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)]d\mathtt{t}d\mathrm{ԟ}\hfill \end{array}\hfill \\ \begin{array}{c}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\oplus \widetilde{\mathsf{Ϗ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}[{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1(1-\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathtt{t}){\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)]d\mathtt{t}d\mathrm{ԟ}.\hfill \end{array}\hfill \end{array}$$

Hence, the required result. □

Remark 3.

Suppose we assume $\mathrm{ƛ}\left(\mathtt{t}\right)=\mathtt{t}, \mathrm{ƛ}\left(\mathrm{ԟ}\right)=\mathrm{ԟ}$, along with $\alpha =1$ and $\beta =1$. Consequently, based on (61), an inequality emerges, as described in [62]:

$${\displaystyle \frac{1}{{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}}{\int }_{\mathrm{ⱱ}}^{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{\int }_{\mathcal{p}}^{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}\tilde{\mathcal{P}}\left(\mathrm{ҩ},\mathrm{ʑ}\right)\otimes \tilde{\mathfrak{G}}\left(\mathrm{ҩ},\mathrm{ʑ}\right)d\mathrm{ʑ}d\mathrm{ҩ}$$

$${\le }_{\mathbb{F}}{\displaystyle \frac{1}{9}}\tilde{\mathcal{E}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)\oplus {\displaystyle \frac{1}{18}}\left[\widetilde{\mathrm{Ѵ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)\oplus \widetilde{\mathrm{Ѡ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)\right]\oplus {\displaystyle \frac{1}{36}}\widetilde{\mathsf{Ϗ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)$$

(65)

If $\tilde{\mathcal{P}}$ is $\mathsf{C}$-left-$\mathrm{ƛ}$-pre-invex with $\mathrm{ƛ}\left(\mathtt{t}\right)=\mathtt{t}, \mathrm{ƛ}\left(\mathrm{ԟ}\right)=\mathrm{ԟ}$ and one assumes that $\alpha =1$ and $\beta =1$, then from (61), as a result, there will be inequity:

$${\displaystyle \frac{1}{{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}}{\int }_{\mathrm{ⱱ}}^{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{\int }_{\mathcal{p}}^{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}\tilde{\mathcal{P}}\left(\mathrm{ҩ},\mathrm{ʑ}\right)\otimes \tilde{\mathfrak{G}}\left(\mathrm{ҩ},\mathrm{ʑ}\right)d\mathrm{ʑ}d\mathrm{ҩ}$$

$${\supseteq }_{\mathbb{F}}{\displaystyle \frac{1}{9}}\tilde{\mathcal{E}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)\oplus {\displaystyle \frac{1}{18}}\left[\widetilde{\mathrm{Ѵ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)\oplus \widetilde{\mathrm{Ѡ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)\right]\oplus {\displaystyle \frac{1}{36}}\widetilde{\mathsf{Ϗ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)$$

(66)

If $\tilde{\mathcal{P}}$ is $\mathsf{C}$-left-$\mathrm{ƛ}$-pre-invex with ${\mathcal{P}}_{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)\ne {\mathcal{P}}^{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)$ with $\mathrm{ɤ}=1$ and $\mathrm{ƛ}\left(\mathtt{t}\right)=\mathtt{t}, \mathrm{ƛ}\left(\mathrm{ԟ}\right)=\mathrm{ԟ}$ then, following Equation (61), we manage to introduce the forthcoming inequality, as illustrated in [14]:

$$\begin{array}{c}{\displaystyle \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{4\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }{\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}\left[\begin{array}{c}{\mathcal{I}}_{{\mathrm{ⱱ}}^{+},{\mathcal{p}}^{+}}^{\alpha ,\beta }\mathcal{P}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\times \mathfrak{G}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\\ +{\mathcal{I}}_{{\mathrm{ⱱ}}^{+},{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\alpha ,\beta }\mathcal{P}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right)\times \mathfrak{G}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right)\end{array}\right]\hfill \\ +{\displaystyle \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{4\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }{\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}\left[\begin{array}{c}{\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-},{\mathcal{p}}^{+}}^{\alpha ,\beta }\mathcal{P}\left(\mathrm{ⱱ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\times \mathfrak{G}\left(\mathrm{ⱱ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\\ +{\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-},{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\alpha ,\beta }\mathcal{P}\left(\mathrm{ⱱ},\mathcal{p}\right)\times \mathfrak{G}\left(\mathrm{ⱱ},\mathcal{p}\right)\end{array}\right]\hfill \\ \hspace{1em}\supseteq \left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}}\right)\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}}\right)\mathcal{E}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)+{\displaystyle \frac{\alpha }{(\alpha +1)(\alpha +2)}}\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{\beta }{(\beta +1)(\beta +2)}}\right)\mathrm{Ѵ}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{\alpha }{(\alpha +1)(\alpha +2)}}\right){\displaystyle \frac{\beta }{(\beta +1)(\beta +2)}}\mathrm{Ѡ}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)+{\displaystyle \frac{\beta }{(\beta +1)(\beta +2)}}{\displaystyle \frac{\alpha }{(\alpha +1)(\alpha +2)}}\mathsf{Ϗ}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right).\hfill \end{array}$$

(67)

If $\mathrm{ƛ}\left(\mathtt{t}\right)=\mathtt{t}, \mathrm{ƛ}\left(\mathrm{ԟ}\right)=\mathrm{ԟ}$ and ${\mathcal{P}}_{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)\ne {\mathcal{P}}^{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)$ with $\mathrm{ɤ}=1$, then following Equation (61), we manage to introduce the forthcoming inequality, as illustrated in [63]:

$${\displaystyle \frac{1}{{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}}{\int }_{\mathrm{ⱱ}}^{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{\int }_{\mathcal{p}}^{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}\mathcal{P}\left(\mathrm{ҩ},\mathrm{ʑ}\right)\times \mathfrak{G}\left(\mathrm{ҩ},\mathrm{ʑ}\right)d\mathrm{ʑ}d\mathrm{ҩ}$$

$${\le }_{\mathrm{I}}{\displaystyle \frac{1}{9}}\mathcal{E}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)+{\displaystyle \frac{1}{18}}\left[\mathrm{Ѵ}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)+\mathrm{Ѡ}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)\right]+{\displaystyle \frac{1}{36}}\mathsf{Ϗ}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)$$

(68)

If ${\mathcal{P}}_{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)={\mathcal{P}}^{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)$ and ${\mathfrak{G}}_{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)={\mathfrak{G}}^{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)$ with $\mathrm{ɤ}=1$ and $\mathrm{ƛ}\left(\mathtt{t}\right)=\mathtt{t}, \mathrm{ƛ}\left(\mathrm{ԟ}\right)=\mathrm{ԟ}$, then following Equation (61), we manage to introduce the forthcoming inequality, as illustrated in [63]:

$$\begin{array}{c}{\displaystyle \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{4\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }{\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}\left[\begin{array}{c}{\mathcal{I}}_{{\mathrm{ⱱ}}^{+},{\mathcal{p}}^{+}}^{\alpha ,\beta }\mathcal{P}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\times \mathfrak{G}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\\ +{\mathcal{I}}_{{\mathrm{ⱱ}}^{+},{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\alpha ,\beta }\mathcal{P}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right)\times \mathfrak{G}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right)\end{array}\right]\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\displaystyle \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{4\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }{\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}\left[\begin{array}{c}{\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-},{\mathcal{p}}^{+}}^{\alpha ,\beta }\mathcal{P}\left(\mathrm{ⱱ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\times \mathfrak{G}\left(\mathrm{ⱱ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\\ +{\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-},{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\alpha ,\beta }\mathcal{P}\left(\mathrm{ⱱ},\mathcal{p}\right)\times \mathfrak{G}\left(\mathrm{ⱱ},\mathcal{p}\right)\end{array}\right]\hfill \\ \hspace{1em}\le \left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}}\right)\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}}\right)\mathcal{E}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)+{\displaystyle \frac{\alpha }{(\alpha +1)(\alpha +2)}}\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{\beta }{(\beta +1)(\beta +2)}}\right)\mathrm{Ѵ}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{\alpha }{(\alpha +1)(\alpha +2)}}\right){\displaystyle \frac{\beta }{(\beta +1)(\beta +2)}}\mathrm{Ѡ}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)+{\displaystyle \frac{\beta }{(\beta +1)(\beta +2)}}{\displaystyle \frac{\alpha }{(\alpha +1)(\alpha +2)}}\mathsf{Ϗ}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right).\hfill \end{array}$$

(69)

Theorem 9.

Let $\tilde{\mathcal{P}}, \tilde{\mathfrak{G}}:▽\to {\mathbb{F}}_0^{+}$ be a $\mathsf{C}$-$\mathrm{ƛ}$-pre-invex $\mathsf{F}$·$\mathsf{N}$·$\mathsf{V}$·$\mathsf{M}$ on $▽$ and let $\mathrm{ƛ}:\left[0,1\right]\to {\mathbb{R}}^{+}$. Subsequently, derived from ɤ-levels, we acquire the set of $\mathsf{I}$·$\mathsf{V}$·$\mathsf{M}s$ ${\mathcal{P}}_{\mathrm{ɤ}}:▽\to {\mathbb{R}}_I^{+}\subset {\mathbb{R}}_I$, which are expressed as ${\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ҩ},\mathrm{ʑ}\right)=\left[{\mathcal{P}}_{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right), {\mathcal{P}}^{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)\right]$ and ${\mathfrak{G}}_{\mathrm{ɤ}}\left(\mathrm{ҩ},\mathrm{ʑ}\right)=\left[{\mathfrak{G}}_{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right), {\mathfrak{G}}^{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)\right]$ for all $\left(\mathrm{ҩ},\mathrm{ʑ}\right)\in ▽$ and for all $\mathrm{ɤ}\in \left[0,1\right]$. If ${\tilde{\mathcal{P}}\otimes \tilde{\mathfrak{G}}\in \mathcal{F}\mathfrak{O}}_{▽}$, then the following inequalities hold:

$$\begin{array}{c} {\displaystyle \frac{1}{2\alpha \beta {{\mathrm{ƛ}}_1}^2\left(\frac{1}{2}\right){{\mathrm{ƛ}}_2}^2\left(\frac{1}{2}\right)}}\tilde{\mathcal{P}}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}}, {\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)\otimes \tilde{\mathfrak{G}}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}}, {\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)\hfill \\ {\le }_{\mathbb{F}}{\displaystyle \frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{2\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }{\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}\left[\begin{array}{c}{\mathcal{I}}_{{\mathrm{ⱱ}}^{+},{\mathcal{p}}^{+}}^{\alpha , \beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\otimes \tilde{\mathfrak{G}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\\ \oplus {\mathcal{I}}_{{\mathrm{ⱱ}}^{+},{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\alpha , \beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right)\otimes \tilde{\mathfrak{G}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right)\end{array}\right]\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\oplus {\displaystyle \frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{2{\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }{\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}\left[\begin{array}{c}{\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-},{\mathcal{p}}^{+}}^{\alpha , \beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\otimes \tilde{\mathfrak{G}}\left(\mathrm{ⱱ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\\ \oplus {\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-},{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\alpha , \beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}\right)\otimes \tilde{\mathfrak{G}}\left(\mathrm{ⱱ},\mathcal{p}\right)\end{array}\right]\hfill \\ \begin{array}{c} \oplus \tilde{\mathcal{E}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}[{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)\right]+{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1(1-\hfill \\ \mathrm{ԟ})\left[{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)\right]]d\mathtt{t}d\mathrm{ԟ}\hfill \end{array}\hfill \\ \begin{array}{c} \oplus \widetilde{\mathrm{Ѵ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}[{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)\right]+{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1(1-\hfill \\ \mathrm{ԟ})\left[{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)\right]]d\mathtt{t}d\mathrm{ԟ}\hfill \end{array}\hfill \\ \begin{array}{c} \oplus \widetilde{\mathrm{Ѡ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}[{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)\right]+{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1(1-\hfill \\ \mathrm{ԟ})\left[{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)\right]]d\mathtt{t}d\mathrm{ԟ}\hfill \end{array}\hfill \\ \begin{array}{c}\hspace{1em}\hspace{1em}\hspace{1em}\oplus \widetilde{\mathsf{Ϗ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}[{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)\right]+{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1(1-\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{ԟ})\left[{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)\right]]d\mathtt{t}d\mathrm{ԟ}.\hfill \end{array}\hfill \end{array}$$

(70)

If $\tilde{\mathcal{P}}$ and $\tilde{\mathfrak{G}}$ are both coordinate $\mathrm{ƛ}$-pre-concave $\mathsf{F}$·$\mathsf{N}$·$\mathsf{V}$·$\mathsf{M}$s on $▽$, then the inequality above can be expressed as follows:

$$\begin{array}{c} {\displaystyle \frac{1}{2\alpha \beta {{\mathrm{ƛ}}_1}^2\left(\frac{1}{2}\right){{\mathrm{ƛ}}_2}^2\left(\frac{1}{2}\right)}}\tilde{\mathcal{P}}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}}, {\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)\otimes \tilde{\mathfrak{G}}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}}, {\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)\hfill \\ \begin{array}{c}{\ge }_{\mathbb{F}}{\displaystyle \frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{2\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }{\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}[{\mathcal{I}}_{{\mathrm{ⱱ}}^{+},{\mathcal{p}}^{+}}^{\alpha , \beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\otimes \tilde{\mathfrak{G}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\oplus {\mathcal{I}}_{{\mathrm{ⱱ}}^{+},{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\alpha , \beta }\tilde{\mathcal{P}}(\mathrm{ⱱ}+\hfill \\ {\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p})\otimes \tilde{\mathfrak{G}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right)]\hfill \end{array}\hfill \\ \begin{array}{c}\hspace{1em}\hspace{1em}\hspace{1em}\oplus {\displaystyle \frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{2{\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }{\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}[{\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-},{\mathcal{p}}^{+}}^{\alpha , \beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\otimes \tilde{\mathfrak{G}}\left(\mathrm{ⱱ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\oplus {\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-},{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\alpha , \beta }\tilde{\mathcal{P}}\left(\mathrm{ⱱ},\mathcal{p}\right)\otimes \hfill \\ \tilde{\mathfrak{G}}\left(\mathrm{ⱱ},\mathcal{p}\right)]\hfill \end{array}\hfill \\ \begin{array}{c} \oplus \tilde{\mathcal{E}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}[{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)\right]+{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1(1-\hfill \\ \mathrm{ԟ})\left[{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)\right]]d\mathtt{t}d\mathrm{ԟ}\hfill \end{array}\hfill \\ \begin{array}{c} \oplus \widetilde{\mathrm{Ѵ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}[{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)\right]+{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1(1-\hfill \\ \mathrm{ԟ})\left[{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)\right]]d\mathtt{t}d\mathrm{ԟ}\hfill \end{array}\hfill \\ \begin{array}{c} \oplus \widetilde{\mathrm{Ѡ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}[{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)\right]+{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1(1-\hfill \\ \mathrm{ԟ})\left[{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)\right]]d\mathtt{t}d\mathrm{ԟ}\hfill \end{array}\hfill \\ \begin{array}{c}\hspace{1em}\hspace{1em}\hspace{1em}\oplus \widetilde{\mathsf{Ϗ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}[{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)\right]+{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1(1-\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{ԟ})\left[{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)\right]]d\mathtt{t}d\mathrm{ԟ}.\hfill \end{array}\hfill \end{array}$$

(71)

where $\tilde{\mathcal{E}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)$, $\widetilde{\mathrm{Ѵ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)$, $\widetilde{\mathrm{Ѡ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)$, and $\widetilde{\mathrm{Ϗ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)$ are given in Theorem 8.

Proof. 

Since $\tilde{\mathcal{P}},\tilde{\mathfrak{G}}:▽\to {\mathbb{F}}_0$ are two $\mathrm{ƛ}$-pre-invex $\mathsf{F}$·$\mathsf{N}$·$\mathsf{V}$·$\mathsf{M}$s, from inequality (18) and for each $\mathrm{ɤ}\in \left[0,1\right],$ we have

$$\begin{array}{c} {\mathcal{P}}_{\mathrm{ɤ}}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}},{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)\times {\mathfrak{G}}_{\mathrm{ɤ}}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}},{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)\hfill \\ ={\mathcal{P}}_{\mathrm{ɤ}}\left({\displaystyle \frac{\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}}+{\displaystyle \frac{\mathrm{ⱱ}+\mathtt{t}{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}},{\displaystyle \frac{\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}+{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)\times {\mathfrak{G}}_{\mathrm{ɤ}}\left({\displaystyle \frac{\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}}+{\displaystyle \frac{\mathrm{ⱱ}+\mathtt{t}{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}},{\displaystyle \frac{\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}+{\displaystyle \frac{\mathcal{p}+\mathrm{ԟ}{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)\hfill \\ {\le }_I{{\mathrm{ƛ}}_1}^2\left({\displaystyle \frac{1}{2}}\right){{\mathrm{ƛ}}_2}^2\left({\displaystyle \frac{1}{2}}\right)\times \left[\begin{array}{c}{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)+{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\mathtt{t}{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\\ +{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\mathrm{ԟ}{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)+{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\mathtt{t}{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\mathrm{ԟ}{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\end{array}\right]\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times \left[\begin{array}{c}{\mathfrak{G}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)+{\mathfrak{G}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\mathtt{t}{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\\ +{\mathfrak{G}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\mathrm{ԟ}{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)+{\mathfrak{G}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\mathtt{t}{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\mathrm{ԟ}{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\end{array}\right]\hfill \\ {\le }_I{{\mathrm{ƛ}}_1}^2\left({\displaystyle \frac{1}{2}}\right){{\mathrm{ƛ}}_2}^2\left({\displaystyle \frac{1}{2}}\right)\times \left[\begin{array}{c}{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\times {\mathfrak{G}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\\ +{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\mathtt{t}{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\times {\mathfrak{G}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\mathtt{t}{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\\ +{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\mathrm{ԟ}{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\times {\mathfrak{G}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\mathrm{ԟ}{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\\ +{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\mathtt{t}{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\mathrm{ԟ}{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\times {\mathfrak{G}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\mathtt{t}{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\mathrm{ԟ}{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\end{array}\right]\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{{\mathrm{ƛ}}_1}^2\left({\displaystyle \frac{1}{2}}\right){{\mathrm{ƛ}}_2}^2\left({\displaystyle \frac{1}{2}}\right)\times \left[\begin{array}{c}{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)\right]\\ +{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)\right]\\ \begin{array}{c}+{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)\right]\\ +{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)\right]\end{array}\end{array}\right]{\mathcal{E}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)\hfill \\ +{{\mathrm{ƛ}}_1}^2\left({\displaystyle \frac{1}{2}}\right){{\mathrm{ƛ}}_2}^2\left({\displaystyle \frac{1}{2}}\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times \left[\begin{array}{c}{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)\right]\\ +{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)\right]\\ +{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)\right]\\ +{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)\right]\end{array}\right]{\mathrm{Ѵ}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)\hfill \\ +{{\mathrm{ƛ}}_1}^2\left({\displaystyle \frac{1}{2}}\right){{\mathrm{ƛ}}_2}^2\left({\displaystyle \frac{1}{2}}\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times \left[\begin{array}{c}{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)\right]\\ +{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)\right]\\ +{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)\right]\\ +{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)\right]\end{array}\right]{\mathrm{Ѡ}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)\hfill \\ +{{\mathrm{ƛ}}_1}^2\left({\displaystyle \frac{1}{2}}\right){{\mathrm{ƛ}}_2}^2\left({\displaystyle \frac{1}{2}}\right)\hfill \\ \times \left[\begin{array}{c}{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)\right]\\ +{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)\right]\\ \begin{array}{c}+{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)\right]\\ +{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)\right]\end{array}\end{array}\right]{\mathsf{Ϗ}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right).\hfill \end{array}$$

Multiplying the above fuzzy inclusion with ${\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}$ and then taking the double integration of the resultant over $\left[0,1\right]\times \left[0,1\right]$ with respect to ($\mathtt{t},\mathtt{ԟ}$), we have

$$\begin{array}{c} {\int }_0^1{\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}{\mathcal{P}}_{\mathrm{ɤ}}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}},{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)\times {\mathfrak{G}}_{\mathrm{ɤ}}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}},{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)d\mathtt{t}d\mathrm{ԟ}\hfill \\ \begin{array}{c}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\le }_I{{\mathrm{ƛ}}_1}^2\left({\displaystyle \frac{1}{2}}\right){{\mathrm{ƛ}}_2}^2\left({\displaystyle \frac{1}{2}}\right)\times \hfill \\ {\int }_0^1{\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[\begin{array}{c}{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\times {\mathfrak{G}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\\ +{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\mathtt{t}{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\times {\mathfrak{G}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\mathtt{t}{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\left(1-\mathrm{ԟ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\\ +{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\mathrm{ԟ}{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\times {\mathfrak{G}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\left(1-\mathtt{t}\right){\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\mathrm{ԟ}{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\\ +{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\mathtt{t}{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\mathrm{ԟ}{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\times {\mathfrak{G}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+\mathtt{t}{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+\mathrm{ԟ}{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\end{array}\right]d\mathtt{t}d\mathrm{ԟ}\hfill \end{array}\hfill \\ \begin{array}{c} +{{\mathrm{ƛ}}_1}^2\left({\displaystyle \frac{1}{2}}\right){{\mathrm{ƛ}}_2}^2\left({\displaystyle \frac{1}{2}}\right){\mathcal{E}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)\times \hfill \\ {\int }_0^1{\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[\begin{array}{c}{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)\right]\\ +{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)\right]\\ \begin{array}{c}+{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)\right]\\ +{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)\right]\end{array}\end{array}\right]d\mathtt{t}d\mathrm{ԟ}\hfill \end{array}\hfill \\ \begin{array}{c} +{{\mathrm{ƛ}}_1}^2\left({\displaystyle \frac{1}{2}}\right){{\mathrm{ƛ}}_2}^2\left({\displaystyle \frac{1}{2}}\right){\mathrm{Ѵ}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)\times \hfill \\ {\int }_0^1{\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[\begin{array}{c}{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)\right]\\ +{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)\right]\\ +{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)\right]\\ +{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)\right]\end{array}\right]d\mathtt{t}d\mathrm{ԟ}\hfill \end{array}\hfill \\ +{{\mathrm{ƛ}}_1}^2\left({\displaystyle \frac{1}{2}}\right){{\mathrm{ƛ}}_2}^2\left({\displaystyle \frac{1}{2}}\right){\mathrm{Ѡ}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times {\int }_0^1{\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[\begin{array}{c}{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)\right]\\ +{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)\right]\\ +{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)\right]\\ +{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)\right]\end{array}\right]d\mathtt{t}d\mathrm{ԟ}\hfill \\ +{{\mathrm{ƛ}}_1}^2\left({\displaystyle \frac{1}{2}}\right){{\mathrm{ƛ}}_2}^2\left({\displaystyle \frac{1}{2}}\right){\mathsf{Ϗ}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)\hfill \\ \times {\int }_0^1{\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[\begin{array}{c}{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)\right]\\ +{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)\right]\\ \begin{array}{c}+{\mathrm{ƛ}}_1\left(1-\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)\right]\\ +{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)\right]\end{array}\end{array}\right]d\mathtt{t}d\mathrm{ԟ},\hfill \end{array}$$

which implies that

$$\begin{array}{c} {\displaystyle \frac{1}{\alpha \beta }}{\mathcal{P}}_{\mathrm{ɤ}}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}}, {\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)\times {\mathfrak{G}}_{\mathrm{ɤ}}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}}, {\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)\hfill \\ \begin{array}{c}{\le }_{\mathbb{F}}{\displaystyle \frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right){{\mathrm{ƛ}}_1}^2\left(\frac{1}{2}\right){{\mathrm{ƛ}}_2}^2\left(\frac{1}{2}\right)}{{\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }{\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}[{\mathcal{I}}_{{\mathrm{ⱱ}}^{+},{\mathcal{p}}^{+}}^{\alpha , \beta }{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\times {\mathfrak{G}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)+{\mathcal{I}}_{{\mathrm{ⱱ}}^{+},{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\alpha , \beta }{\mathcal{P}}_{\mathrm{ɤ}}(\mathrm{ⱱ}+\hfill \\ {\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p})\times {\mathfrak{G}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right)]\hfill \end{array}\hfill \\ +{\displaystyle \frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right){{\mathrm{ƛ}}_1}^2\left(\frac{1}{2}\right){{\mathrm{ƛ}}_2}^2\left(\frac{1}{2}\right)}{{\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }{\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}\left[{\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-},{\mathcal{p}}^{+}}^{\alpha , \beta }{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathcal{q}\right)\times {\mathfrak{G}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathcal{q}\right)+{\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-},{\mathcal{q}}^{-}}^{\alpha , \beta }{\mathcal{P}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathcal{p}\right)\times {\mathfrak{G}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathcal{p}\right)\right]\hfill \\ \begin{array}{c}+2{{\mathrm{ƛ}}_1}^2\left({\displaystyle \frac{1}{2}}\right){{\mathrm{ƛ}}_2}^2\left({\displaystyle \frac{1}{2}}\right){\mathcal{E}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}[{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)\right]+\hfill \\ {\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)\right]]d\mathtt{t}d\mathrm{ԟ}\hfill \end{array}\hfill \\ \begin{array}{c}\hspace{1em}\hspace{1em}\hspace{1em}+2{{\mathrm{ƛ}}_1}^2\left({\displaystyle \frac{1}{2}}\right){{\mathrm{ƛ}}_2}^2\left({\displaystyle \frac{1}{2}}\right){\mathrm{Ѵ}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}[{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)\right]+\hfill \\ {\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)\right]]d\mathtt{t}d\mathrm{ԟ}\hfill \end{array}\hfill \\ \begin{array}{c} +2{{\mathrm{ƛ}}_1}^2\left({\displaystyle \frac{1}{2}}\right){{\mathrm{ƛ}}_2}^2\left({\displaystyle \frac{1}{2}}\right){\mathrm{Ѡ}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}[{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)\right]+\hfill \\ {\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)\right]]d\mathtt{t}d\mathrm{ԟ}\hfill \end{array}\hfill \\ \begin{array}{c}+2{{\mathrm{ƛ}}_1}^2\left({\displaystyle \frac{1}{2}}\right){{\mathrm{ƛ}}_2}^2\left({\displaystyle \frac{1}{2}}\right){\mathsf{Ϗ}}_{\mathrm{ɤ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}[{\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)\right]+\hfill \\ {\mathrm{ƛ}}_1\left(\mathtt{t}\right){\mathrm{ƛ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ƛ}}_2\left(1-\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ƛ}}_2\left(\mathtt{t}\right){\mathrm{ƛ}}_2\left(1-\mathrm{ԟ}\right)\right]]d\mathtt{t}d\mathrm{ԟ},\hfill \end{array}\hfill \end{array}$$

as $\mathrm{ɤ}\in \left[0,1\right]$, following simplification, we arrive at the desired conclusion. □

Remark 4.

If one assumes that $\mathrm{ƛ}\left(\mathtt{t}\right)=\mathtt{t}, \mathrm{ƛ}\left(\mathrm{ԟ}\right)=\mathrm{ԟ}$ and $\alpha =1$ and $\beta =1$, then from (70), as a result, there will be inequity; see [62]:

$$4\tilde{\mathcal{P}}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}},{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)\otimes \tilde{\mathfrak{G}}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}},{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)$$

$${\le }_{\mathbb{F}}{\displaystyle \frac{1}{{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}}{\int }_{\mathrm{ⱱ}}^{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{\int }_{\mathcal{p}}^{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}\tilde{\mathcal{P}}\left(\mathrm{ҩ},\mathrm{ʑ}\right)\otimes \tilde{\mathfrak{G}}\left(\mathrm{ҩ},\mathrm{ʑ}\right)d\mathrm{ʑ}d\mathrm{ҩ}\oplus {\displaystyle \frac{5}{36}}\tilde{\mathcal{E}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)$$

$$\oplus {\displaystyle \frac{7}{36}}\left[\widetilde{\mathrm{Ѵ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)\oplus \widetilde{\mathrm{Ѡ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)\right]\oplus {\displaystyle \frac{2}{9}}\widetilde{\mathsf{Ϗ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right).$$

(72)

If $\tilde{\mathcal{P}}$ is $\mathsf{C}$-left-$\mathrm{ƛ}$-pre-invex with $\mathrm{ƛ}\left(\mathtt{t}\right)=\mathtt{t}, \mathrm{ƛ}\left(\mathrm{ԟ}\right)=\mathrm{ԟ}$ and one assumes that $\alpha =1$ and $\beta =1$, then from (70), as a result, there will be inequity:

$$4\tilde{\mathcal{P}}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}},{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)\otimes \tilde{\mathfrak{G}}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}},{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)$$

$${\supseteq }_{\mathbb{F}}{\displaystyle \frac{1}{{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}}{\int }_{\mathrm{ⱱ}}^{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{\int }_{\mathcal{p}}^{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}\tilde{\mathcal{P}}\left(\mathrm{ҩ},\mathrm{ʑ}\right)\otimes \tilde{\mathfrak{G}}\left(\mathrm{ҩ},\mathrm{ʑ}\right)d\mathrm{ʑ}d\mathrm{ҩ}\oplus {\displaystyle \frac{5}{36}}\tilde{\mathcal{E}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)$$

$$\oplus {\displaystyle \frac{7}{36}}\left[\widetilde{\mathrm{Ѵ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)\oplus \widetilde{\mathrm{Ѡ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)\right]\oplus {\displaystyle \frac{2}{9}}\widetilde{\mathsf{Ϗ}}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right).$$

(73)

If ${\mathcal{P}}_{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)\ne {\mathcal{P}}^{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)$ with $\mathrm{ƛ}\left(\mathtt{t}\right)=\mathtt{t}, \mathrm{ƛ}\left(\mathrm{ԟ}\right)=\mathrm{ԟ}$ and $\mathrm{ɤ}=1$, then following Equation (70), we manage to introduce the forthcoming inequality, as illustrated in [14]:

$$4\mathcal{P}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}},{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)\times \mathfrak{G}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}},{\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)$$

$$\supseteq {\displaystyle \frac{1}{{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right){\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}}{\int }_{\mathrm{ⱱ}}^{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{\int }_{\mathcal{p}}^{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}\mathcal{P}\left(\mathrm{ҩ},\mathrm{ʑ}\right)\times \mathfrak{G}\left(\mathrm{ҩ},\mathrm{ʑ}\right)d\mathrm{ʑ}d\mathrm{ҩ}+{\displaystyle \frac{5}{36}}\mathcal{E}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)$$

$$+{\displaystyle \frac{7}{36}}\left[\mathrm{Ѵ}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)+\mathrm{Ѡ}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)\right]+{\displaystyle \frac{2}{9}}\mathsf{Ϗ}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right).$$

(74)

If ${\mathcal{P}}_{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)\ne {\mathcal{P}}^{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)$ with $\mathrm{ɤ}=1$ and $\mathrm{ƛ}\left(\mathtt{t}\right)=\mathtt{t}, \mathrm{ƛ}\left(\mathrm{ԟ}\right)=\mathrm{ԟ}$, then following Equation (70), we manage to introduce the forthcoming inequality, as illustrated in [14]:

$$\begin{array}{c}4\mathcal{P}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}}, {\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)\times \mathfrak{G}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}}, {\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)\supseteq \hfill \\ {\displaystyle \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{4\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }{\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}\left[\begin{array}{c}{\mathcal{I}}_{{\mathrm{ⱱ}}^{+},{\mathcal{p}}^{+}}^{\alpha ,\beta }\mathcal{P}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\times \mathfrak{G}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\\ +{\mathcal{I}}_{{\mathrm{ⱱ}}^{+},{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\alpha ,\beta }\mathcal{P}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right)\times \mathfrak{G}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right)\\ +{\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-},{\mathcal{p}}^{+}}^{\alpha ,\beta }\mathcal{P}\left(\mathrm{ⱱ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\times \mathfrak{G}\left(\mathrm{ⱱ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\\ +{\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-},{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\alpha ,\beta }\mathcal{P}\left(\mathrm{ⱱ},\mathcal{p}\right)\times \mathfrak{G}\left(\mathrm{ⱱ},\mathcal{p}\right)\end{array}\right]\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\left[{\displaystyle \frac{\alpha }{2\left(\alpha +1\right)\left(\alpha +2\right)}}+{\displaystyle \frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}}\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}}\right)\right]\mathcal{E}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\left[{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}}\right)+{\displaystyle \frac{\alpha }{(\alpha +1)(\alpha +2)}}{\displaystyle \frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}}\right]\mathrm{Ѵ}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\left[{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}}\right)+{\displaystyle \frac{\alpha }{(\alpha +1)(\alpha +2)}}{\displaystyle \frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}}\right]\mathrm{Ѡ}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\left[{\displaystyle \frac{1}{4}}-{\displaystyle \frac{\alpha }{(\alpha +1)(\alpha +2)}}{\displaystyle \frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}}\right]\mathsf{Ϗ}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right).\hfill \end{array}$$

(75)

If ${\mathcal{P}}_{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)={\mathcal{P}}^{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)$ and ${\mathfrak{G}}_{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)={\mathfrak{G}}^{*}\left(\left(\mathrm{ҩ},\mathrm{ʑ}\right),\mathrm{ɤ}\right)$ with $\mathrm{ɤ}=1$ and $\mathrm{ƛ}\left(\mathtt{t}\right)=\mathtt{t}, \mathrm{ƛ}\left(\mathrm{ԟ}\right)=\mathrm{ԟ}$, then following Equation (70), we manage to introduce the forthcoming inequality, as illustrated in [63]:

$$\begin{array}{l}4\mathcal{P}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}}, {\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)\times \mathfrak{G}\left({\displaystyle \frac{2\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{2}}, {\displaystyle \frac{2\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}{2}}\right)\\ \le {\displaystyle \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{4\left[{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)\right]}^{\alpha }{\left[{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right]}^{\beta }}}\left[\begin{array}{c}{\mathcal{I}}_{{\mathrm{ⱱ}}^{+},{\mathcal{p}}^{+}}^{\alpha , \beta }\mathcal{P}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\times \mathfrak{G}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\\ +{\mathcal{I}}_{{\mathrm{ⱱ}}^{+},{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\alpha , \beta }\mathcal{P}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right)\times \mathfrak{G}\left(\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\mathcal{p}\right)\\ +{\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-},{\mathcal{p}}^{+}}^{\alpha , \beta }\mathcal{P}\left(\mathrm{ⱱ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\times \mathfrak{G}\left(\mathrm{ⱱ},\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)\right)\\ +{\mathcal{I}}_{{\mathrm{ⱱ}+{\mathrm{\hslash }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}^{-},{\mathcal{p}+{\mathrm{\hslash }}_2\left(\mathcal{q},\mathcal{p}\right)}^{-}}^{\alpha , \beta }\mathcal{P}\left(\mathrm{ⱱ},\mathcal{p}\right)\times \mathfrak{G}\left(\mathrm{ⱱ},\mathcal{p}\right)\end{array}\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\left[{\displaystyle \frac{\alpha }{2\left(\alpha +1\right)\left(\alpha +2\right)}}+{\displaystyle \frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}}\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}}\right)\right]\mathcal{E}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\left[{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}}\right)+{\displaystyle \frac{\alpha }{(\alpha +1)(\alpha +2)}}{\displaystyle \frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}}\right]\mathrm{Ѵ}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)\\ +\left[{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}}\right)+{\displaystyle \frac{\alpha }{(\alpha +1)(\alpha +2)}}{\displaystyle \frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}}\right]\mathrm{Ѡ}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right)\\ +\left[{\displaystyle \frac{1}{4}}-{\displaystyle \frac{\alpha }{(\alpha +1)(\alpha +2)}}{\displaystyle \frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}}\right]\mathsf{Ϗ}\left(\mathrm{ⱱ},\mathrm{ⱳ},\mathcal{p},\mathcal{q}\right).\end{array}$$

(76)

4. Conclusions

This work studies various inequalities related to a novel class of pre-invexity via $\mathsf{C}$-$\mathrm{ƛ}$-pre-invex type $\mathsf{F}$·$\mathsf{N}$·$\mathsf{V}$·$\mathsf{M}$s via fuzzy-set-valued functions. First, under the full-order relation, we define the $\mathsf{C}$-$\mathrm{ƛ}$-pre-invex fuzzy mappings and investigate some of their induced features. Through the use of arbitrary non-negative functions and related bifunctions of Hermite, Hadamard, and Fejér-type inequalities, we construct unique forms and expand greatly on previously published results. Several unique instances of these inequities are also covered. Several numerical examples are provided to further show the correctness of the results. The notions and concepts presented in this paper can be used to investigate additional types of convex inequalities, with possible applications in problems such as optimization and differential equations with convex shapes attached.