Some New Estimations of Left and Right Interval Fractional Pachpatte’s Type Integral Inequalities via Rectangle Plane

Abstract

Inequalities involving fractional operators have been an active area of research, which is crucial in establishing bounds, estimates, and stability conditions for solutions to fractional integrals. In this paper, we initially presented a new class that is known as coordinated left and right $\hslash$-pre-invex interval-valued mappings ($\mathcal{C}$·$\mathsf{L}$·$\mathsf{R}$-$\hslash$-pre-invex ${\rm I}$·$\mathsf{V}$-$\mathsf{M}$), as well classical convex and nonconvex are also obtained. This newly defined class enabled us to derive novel inequalities, such as Hermite–Hadamard and Pachpatte’s type inequalities. Furthermore, the obtained results allowed us to recapture several special cases of known results for different parameter choices, which can be applications of the main results. Finally, we discussed the validity of the main outcomes.

1. Introduction

The theory of inequalities is fundamental in numerous scientific and engineering fields and is extensively taught due to its practical applications. Mathematical inequalities are used in system design and analysis, engineering, signal processing, and optimization problems [1,2]. They offer a robust framework for analyzing and comprehending the behavior of solutions to numerical and partial differential equations. Fractional analysis, an innovative extension of classical analysis to non-integer orders, plays a significant role in this context. In studying physical systems described by fractional differential equations, fractional integral inequalities are utilized to obtain energy estimates, see [3,4,5,6].

Convex functions have been a central topic in mathematics for a significant period. Their growing popularity is likely due to their essential role in optimization, machine learning, and various applications across scientific and engineering disciplines [7,8,9,10,11]. There is a profound and intricate relationship between the theory of inequalities and convex functions. Convex functions have been crucial in the discovery and development of numerous important and useful inequalities. The primary definition of a convex function on convex sets is as follows.

If $\mathcal{G}:K\to \mathfrak{N}$ is a convex function defined on the interval $K$ of real numbers, and$\theta , \lambda \in K$ with $\theta <\lambda$, then

$$\mathcal{G}\left(\kappa \theta +\left(1-\kappa \right)\lambda \right)\le \kappa \mathcal{G}\left(\theta \right)+\left(1-\kappa \right)\mathcal{G}\left(\lambda \right).$$

(1)

Jensen’s inequality is a foundational and widely recognized result for convex functions, with various related forms and extensions. Its extensive applications highlight its importance in numerous scientific, engineering, and computational fields, see [12,13]. Convex functions serve as the cornerstone of mathematical inequalities and are essential tools in contemporary research.

Fractional calculus offers a framework for describing and modeling systems with intricate dynamics, memory effects, and anomalous behaviors that classical calculus may not adequately capture. It is a specialized branch of applied mathematics that aids in the comprehension and analysis of complex real-world phenomena, see [14,15,16]. Fractional integral and derivative operators are vital in connecting mathematical models to real-world problems. These operators have substantially contributed to the development of new concepts in fractional analysis, broadening their application areas [17,18,19]. Fractional integrals and derivatives differ in terms of singularity, locality, and kernel. Fractional integrals, in particular, are powerful mathematical tools which are valuable for addressing a wide range of issues in science and engineering, see [20,21]. The development of fractional calculus has led to the introduction and analysis of various fractional integral operators, each with distinct properties and applications, see [22,23]. The Riemann–Liouville fractional integral is one of the most fundamental and widely utilized operators in this field. Additionally, operators introduced by Caputo, Fabrizio, and tempered operators add further depth and versatility to the theory. Due to the significance of fractional calculus, many mathematicians are interested in fractional operators and integral inequalities. Kashuri et al. [24,25] proved trapezoidal-type inequalities for differentiable convex functions with applications to matrices and the q-digamma function. Junjua et al. [26,27] obtained trapezoidal-type inequalities for exponential convex functions, applying them to certain real means. Furthermore, several articles [28,29,30] have demonstrated fractional trapezoidal-type inequalities for functions with convex second derivatives in the absolute value. Klasoom et al. [31,32] examined various modifications of trapezoidal-type inequalities for twice differentiable convex functions using (p,q)-integrals. For different fractional versions of various inequalities, see [33,34].

Due to the broad spectrum of fields where set-valued analysis is applicable, researchers have formulated integral inequalities using various operators and order relations within the realm of interval-valued mappings (I.V.F.). Taking inspiration from this approach, Zhao et al. [35,36] recently employed the conventional integral operator to establish inequality (1.2) within the framework of interval inclusion relations. The authors of [37] introduced pre-invex functions on the plane with interval values and derived multiple double inequalities. Wannalookkhee et al. [38] utilized quantum integrals to derive a double inequality on the plane and for other applications. Kalsoom et al. [39] developed a Hermite–Hadamard-type (H.H) inequality associated with coordinated higher-order generalized pre and quasi-invex mappings by utilizing the concept of quantum integrals. Utilizing two intriguing identities for functions of two variables, Akkurt et al. [40] innovatively formulated Hermite–Hadamard (H.H.)-type fractional integral inequalities for double fractional integrals. Shi et al. [41] established H.H. and its symmetric variation using interval-valued mappings (${\rm I}$·$\mathsf{V}$-$\mathsf{M}$) and two forms of generalized convex functions. Saeed et al. [42] devised H.H. and Pachpatte-type integral inequalities employing coordinated up and down convex mappings with fuzzy-number values. Wu et al. [43] derived three fundamental integral identities based on the first- and second-order derivatives of a given function using fractional integrals with an exponential kernel. Ahmad et al. [44] introduced fractional operators with non-singular kernels to support H.H., Hermite–Hadamard–Fejer, Dragomir–Agarwal, and Pachpatte-type inequalities for convex functions. Khan et al. [45,46,47,48,49] formulated specific Hermite–Hadamard (H.H.) and its various variations of inequalities for exponential trigonometric convex mappings with fuzzy-number values, employing fuzzy fractional integral operators with exponential kernels.

In this paper, our focus lies on examining pseudo left–right interval order connections. Recent studies in this direction have yielded notable results: Saeed et al. [50] explored coordinated h-convex mappings, leading to the derivation of three well-known inequalities within the framework of pseudo order relations. Stojiljkovic et al. [51,52] utilized Katugampola integrals to introduce a new class of p-convex mappings, establishing numerous generalized bounds for double inequalities through the utilization of left–right order relations. Khan et al. [53] employed convex interval-valued functions via log convex functions based on the pseudo order relation, revealing several novel properties and inequalities. For more information related to fractional integral operator and coordinated inequalities, see [54,55,56,57,58,59].

The rest of the paper is structured as follows: Section 2 reviews some basic notations that will be used throughout. In Section 3, we first introduce a new convex class on coordinates known as $\mathcal{C}$·$\mathsf{L}$·$\mathsf{R}$-$\mathrm{\hslash }$- pre-invex ${\rm I}$·$\mathsf{V}$-$\mathsf{M}$. By using fractional integral operators, various types of inequalities have been proved as well as Hermite–Hadamard and Pachpatte’s type integral inequalities. Moreover, useful examples are also provided to check the validations of our main outcomes. For the applications of these results, our results will allow us to recover several special cases of known results for different parameter choices. In Section 4, we will draw some conclusions and provide future recommendations for interested readers.

2. Preliminaries

Let $\mathbb{R}$ be the set of real numbers and ${\mathbb{R}}_I$ containing all bounded and closed intervals within $\mathbb{R}$. $\mathrm{Ꙍ}\in {\mathbb{R}}_I$ should be defined as follows:

$$\mathrm{Ꙍ}=\left[{\mathrm{Ꙍ}}_{\ast }, {\mathrm{Ꙍ}}^{\ast }\right]=\left\{\mathsf{y}\in \mathbb{R}|{\mathrm{Ꙍ}}_{\ast }\le \mathsf{y}\le {\mathrm{Ꙍ}}^{\ast }\right\}, \left({\mathrm{Ꙍ}}_{\ast }, {\mathrm{Ꙍ}}^{\ast }\in \mathbb{R}\right).$$

(2)

It is argued that $\mathrm{Ꙍ}$ is degenerate if ${\mathrm{Ꙍ}}_{\ast }={\mathrm{Ꙍ}}^{\ast }$. The interval $\left[{\mathrm{Ꙍ}}_{\ast }, {\mathrm{Ꙍ}}^{\ast }\right]$ is referred to as positive if ${\mathrm{Ꙍ}}_{\ast }\ge 0$,${\mathbb{R}}_I^{+}$ represents the set of all positive intervals and is defined as ${\mathbb{R}}_I^{+}=\left\{\left[{\mathrm{Ꙍ}}_{\ast }, {\mathrm{Ꙍ}}^{\ast }\right]:\left[{\mathrm{Ꙍ}}_{\ast }, {\mathrm{Ꙍ}}^{\ast }\right]\in {\mathbb{R}}_I \mathrm{a}\mathrm{n}\mathrm{d} {\mathrm{Ꙍ}}_{\ast }\ge 0\right\}$.

Let $\varrho \in \mathbb{R}$, $\mathrm{Ꙍ},\mathcal{S}\in {\mathbb{R}}_I$ be defined by, with $\mathcal{S}=\left[{\mathcal{S}}_{\ast }, {\mathcal{S}}^{\ast }\right]$ and $\mathrm{Ꙍ}=\left[{\mathrm{Ꙍ}}_{\ast }, {\mathrm{Ꙍ}}^{\ast }\right]$, we may define the interval arithmetic as follows:

• Scaler multiplication:

$$\varrho .\mathrm{Ꙍ}=\{\begin{array}{c}\left[\varrho {\mathrm{Ꙍ}}_{\ast }, {\varrho \mathrm{Ꙍ}}^{\ast }\right] \mathrm{if} \varrho >0,\\ \left\{0\right\} \mathrm{if} \varrho =0,\\ \left[\varrho {\mathrm{Ꙍ}}^{\ast }{,\varrho \mathrm{Ꙍ}}_{\ast }\right] \mathrm{if} \varrho <0. \end{array}$$

(3)

• Minkowski difference:

$$\begin{array}{c} \\ \left[{\mathcal{S}}_{\ast }, {\mathcal{S}}^{\ast }\right]-\left[{\mathrm{Ꙍ}}_{\ast }, {\mathrm{Ꙍ}}^{\ast }\right]=\left[{\mathcal{S}}_{\ast }-{\mathrm{Ꙍ}}_{\ast }, {\mathcal{S}}^{\ast }-{\mathrm{Ꙍ}}^{\ast }\right], \end{array}$$

(4)

• Addition:

$$\begin{array}{c} \\ \left[{\mathcal{S}}_{\ast }, {\mathcal{S}}^{\ast }\right]+\left[{\mathrm{Ꙍ}}_{\ast }, {\mathrm{Ꙍ}}^{\ast }\right]=\left[{\mathcal{S}}_{\ast }+{\mathrm{Ꙍ}}_{\ast }, {\mathcal{S}}^{\ast }+{\mathrm{Ꙍ}}^{\ast }\right],\end{array}$$

(5)

• Multiplication:

$$\left[{\mathcal{S}}_{\ast }, {\mathcal{S}}^{\ast }\right]\times \left[{\mathrm{Ꙍ}}_{\ast }, {\mathrm{Ꙍ}}^{\ast }\right]=\left[min\left\{{\mathcal{S}}_{\ast }{\mathrm{Ꙍ}}_{\ast }, {\mathcal{S}}^{\ast }{\mathrm{Ꙍ}}_{\ast }, {\mathcal{S}}_{\ast }{\mathrm{Ꙍ}}^{\ast }, {\mathcal{S}}^{\ast }{\mathrm{Ꙍ}}^{\ast }\right\}, max\left\{{\mathcal{S}}_{\ast }{\mathrm{Ꙍ}}_{\ast }, {\mathcal{S}}^{\ast }{\mathrm{Ꙍ}}_{\ast }, {\mathcal{S}}_{\ast }{\mathrm{Ꙍ}}^{\ast }, {\mathcal{S}}^{\ast }{\mathrm{Ꙍ}}^{\ast }\right\}\right].$$

(6)

The inclusion $“\supseteq ”$ means that

$\mathrm{Ꙍ}\supseteq \mathcal{S}$ if and only if, $\left[{\mathrm{Ꙍ}}_{\ast }, {\mathrm{Ꙍ}}^{\ast }\right]\supseteq \left[{\mathcal{S}}_{\ast }, {\mathcal{S}}^{\ast }\right]$, and if and only if

$${\mathrm{Ꙍ}}_{\ast }\le {\mathcal{S}}_{\ast }, {\mathcal{S}}^{\ast }\le {\mathrm{Ꙍ}}^{\ast }.$$

(7)

Remark 1

([58]). (i) The relation ${“\le }_p”$ is defined on ${\mathbb{R}}_I$ by

$$\left[{\mathcal{S}}_{\ast }, {\mathcal{S}}^{\ast }\right]{\le }_p\left[{\mathrm{Ꙍ}}_{\ast }, {\mathrm{Ꙍ}}^{\ast }\right] \mathrm{if} \mathrm{and} \mathrm{only} \mathrm{if} {\mathcal{S}}_{\ast }{\le \mathrm{Ꙍ}}_{\ast }, {\mathcal{S}}^{\ast }{\le \mathrm{Ꙍ}}^{\ast },$$

(8)

for all $\left[{\mathcal{S}}_{\ast }, {\mathcal{S}}^{\ast }\right], \left[{\mathrm{Ꙍ}}_{\ast }, {\mathrm{Ꙍ}}^{\ast }\right]\in {\mathbb{R}}_I,$ and it is a pseudo order relation. The relation $\left[{\mathcal{S}}_{\ast }, {\mathcal{S}}^{\ast }\right]{\le }_p\left[{\mathrm{Ꙍ}}_{\ast }, {\mathrm{Ꙍ}}^{\ast }\right]$ coincident to $\left[{\mathcal{S}}_{\ast }, {\mathcal{S}}^{\ast }\right]\le \left[{\mathrm{Ꙍ}}_{\ast }, {\mathrm{Ꙍ}}^{\ast }\right]$ on ${\mathbb{R}}_I$ when it is ${“\le }_p”$.

(ii) It can be easily seen that $“{\le }_p”$ looks like “left and right” on the real line $\mathbb{R},$ so we call $“{\le }_p”$ is “left and right” (or “LR” order, in short).

Definition 1

([55,56]). Let $\mathcal{G}:\left[\mathrm{ȵ}, \mathrm{ȶ}\right]\to {\mathbb{R}}_I^{+}$ be an interval-valued mapping ($I\mathsf{V}\mathsf{M}$) and $\mathcal{G}\in {\mathcal{I}\mathcal{R}}_{\left[\mathrm{ȵ}, \mathrm{ȶ}\right]}$. Then, the interval Riemann–Liouville-type integrals of $\mathcal{G}$ are defined as

$${\mathcal{I}}_{{\mathrm{ȵ}}^{+}}^{\mathfrak{i}}\mathcal{G}\left(y\right)=\frac{1}{\Gamma \left(\mathfrak{i}\right)}{\int }_{\mathrm{ȵ}}^y{\left(y-\mathtt{t}\right)}^{\mathfrak{i}-1}\mathcal{G}\left(\mathtt{t}\right)d\mathtt{t} \left(y>\mathrm{ȵ}\right),$$

(9)

$${\mathcal{I}}_{{\mathrm{ȶ}}^{-}}^{\mathfrak{i}}\mathcal{G}\left(y\right)=\frac{1}{\Gamma \left(\mathfrak{i}\right)}{\int }_y^{\mathrm{ȶ}}{\left(\mathtt{t}-y\right)}^{\mathfrak{i}-1}\mathcal{G}\left(\mathtt{t}\right)d\mathtt{t} (y<\mathrm{ȶ}),$$

(10)

where $\mathfrak{i}>0$ and $\Gamma$ is the gamma function.

Theorem 1

([59]). Let $\hslash :\left[0, 1\right]\to {\mathbb{R}}^{+}$ and $\mathcal{G}:\left[ⱬ, ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right]\to {\mathbb{R}}_I^{+}$ be a $\mathsf{L}$·$\mathsf{R}$-$\hslash$-pre-invex ${\rm I}$·$\mathsf{V}$-$\mathsf{M}$ on $\left[ⱬ, ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right],$ given by $\mathcal{G}\left(y\right)=\left[{\mathcal{G}}_{\ast }\left(y\right), {\mathcal{G}}^{\ast }\left(y\right)\right]$ for all $y\in \left[ⱬ, ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right]$. If $\mathcal{G}\in L\left(\left[ⱬ, ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right],{\mathbb{R}}_I^{+}\right)$, then

$$\begin{gathered} \frac{1}{\mathfrak{i}\hslash \left(\frac{1}{2}\right)}\mathcal{G}\left(\frac{2ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right){\le }_p\frac{\Gamma \left(\mathfrak{i}\right)}{{\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{i}}}\left[{\mathcal{I}}_{{ⱬ}^{+}}^{\mathfrak{i}}\mathcal{G}\left(ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+{\mathcal{I}}_{{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{i}}\mathcal{G}\left(ⱬ\right)\right] \\ {\le }_p\left[\mathcal{G}\left(ⱬ\right)+\mathcal{G}\left(\theta \right)\right]{\int }_0^1{\upsilon }^{\mathfrak{i}-1}\left[\hslash \left(\upsilon \right)+\hslash \left(1-\upsilon \right)\right]d\upsilon . \end{gathered}$$

(11)

Interval double Aumann’s type integrals are defined as follows for coordinated $I\mathsf{V}\mathsf{M}$ ($\mathcal{C}$-$I\mathsf{V}\mathsf{M}$) $\mathcal{G}\left(x,y\right)$.

Theorem 2

([57]). Let $\mathcal{G}:\Omega =\left[\mathrm{ȵ}, \mathrm{ȶ}\right]\times \left[ⱬ, \theta \right]\subset {\mathbb{R}}^2\to {\mathbb{R}}_I^{ }$ be an ${\rm I}$·$\mathsf{V}$-$\mathsf{M}$ on coordinates, given by $\mathcal{G}\left(x,y\right)=\left[{\mathcal{G}}_{\ast }\left(x,y\right), {\mathcal{G}}^{\ast }\left(x,y\right)\right]$ for all $\left(x,y\right)\in \Omega =\left[\mathrm{ȵ}, \mathrm{ȶ}\right]\times \left[ⱬ, \theta \right]$. Then, $\mathcal{G}$ is double integrable ($FD$-integrable) over $\Omega$ if and only if ${\mathcal{G}}_{\ast }\left(x,y\right)$ and ${\mathcal{G}}^{\ast }\left(x,y\right)$ both are $D$-integrable over $\Omega .$ Moreover, if $\mathcal{G}$ is $FD$-integrable over $\Omega ,$ then

$$\left(ID\right){\int }_{\mathrm{ȵ}}^{\mathrm{ȶ}}{\int }_{ⱬ}^{\theta }\mathcal{G}\left(x,y\right)dydx=\left[\left(D\right){\int }_{\mathrm{ȵ}}^{\mathrm{ȶ}}{\int }_{ⱬ}^{\theta }{\mathcal{G}}_{\ast }\left(x,y\right)dydx, \left(D\right){\int }_{\mathrm{ȵ}}^{\mathrm{ȶ}}{\int }_{ⱬ}^{\theta }{\mathcal{G}}^{\ast }\left(x,y\right)dydx\right]$$

(12)

The family of all $FD$-integrables of ${\rm I}$·$\mathsf{V}$-$\mathsf{M}$s over coordinates and $D$-integrable functions over coordinates are denoted by ${\mathcal{F}\mathfrak{O}}_{\Omega }$ and ${\mathfrak{O}}_{\Omega }.$

Here is the main definition of the fuzzy Riemann–Liouville fractional integral on the coordinates of the function $\mathcal{G}\left(x,y\right)$ by:

Definition 2

([56]). Let $\mathcal{G}:\Omega \to {\mathbb{R}}_I^{ }$ and $\mathcal{G}\in {\mathcal{F}\mathfrak{O}}_{\Omega }$. The double fuzzy interval Riemann–Liouville-type integrals ${\mathcal{I}}_{{\mathrm{ȵ}}^{+},{ⱬ}^{+}}^{\mathfrak{i},\mathfrak{j}}, {\mathcal{I}}_{{\mathrm{ȵ}}^{+},{\theta }^{-}}^{\mathfrak{i},\mathfrak{j}}, {\mathcal{I}}_{{\mathrm{ȶ}}^{-},{ⱬ}^{+}}^{\mathfrak{i},\mathfrak{j}}, {\mathcal{I}}_{{\mathrm{ȶ}}^{-},{\theta }^{-}}^{\mathfrak{i},\mathfrak{j}}$ of $\mathcal{G}$ order $\mathfrak{i},\mathfrak{j}>0$ are defined by

$${\mathcal{I}}_{{\mathrm{ȵ}}^{+},{ⱬ}^{+}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(x,y\right)=\frac{1}{\Gamma \left(\mathfrak{i}\right)\Gamma \left(\mathfrak{j}\right)}{\int }_{\mathrm{ȵ}}^x{\int }_{ⱬ}^y{{\left(x-\mathtt{t}\right)}^{\mathfrak{i}-1}\left(y-\mathtt{s}\right)}^{\mathfrak{j}-1}\mathcal{G}\left(\mathtt{t},\mathtt{s}\right)d\mathtt{s}d\mathtt{t}, \left(x>\mathrm{ȵ}, y>ⱬ\right),$$

(13)

$${\mathcal{I}}_{{\mathrm{ȵ}}^{+},{\theta }^{-}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(x,y\right)=\frac{1}{\Gamma \left(\mathfrak{i}\right)\Gamma \left(\mathfrak{j}\right)}{\int }_{\mathrm{ȵ}}^x{\int }_y^{\theta }{{\left(x-\mathtt{t}\right)}^{\mathfrak{i}-1}\left(\mathtt{s}-y\right)}^{\mathfrak{j}-1}\mathcal{G}\left(\mathtt{t},\mathtt{s}\right)d\mathtt{s}d\mathtt{t}, (x>\mathrm{ȵ}, y<\theta ),$$

(14)

$${\mathcal{I}}_{{\mathrm{ȶ}}^{-},{ⱬ}^{+}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(x,y\right)=\frac{1}{\Gamma \left(\mathfrak{i}\right)\Gamma \left(\mathfrak{j}\right)}{\int }_x^{\mathrm{ȶ}}{\int }_{ⱬ}^y{{\left(\mathtt{t}-x\right)}^{\mathfrak{i}-1}\left(y-\mathtt{s}\right)}^{\mathfrak{j}-1}\mathcal{G}\left(\mathtt{t},\mathtt{s}\right)d\mathtt{s}d\mathtt{t}, \left(x<\mathrm{ȶ}, y>ⱬ\right),$$

(15)

$${\mathcal{I}}_{{\mathrm{ȶ}}^{-},{\theta }^{-}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(x,y\right)=\frac{1}{\Gamma \left(\mathfrak{i}\right)\Gamma \left(\mathfrak{j}\right)}{\int }_x^{\mathrm{ȶ}}{\int }_y^{\theta }{{\left(\mathtt{t}-x\right)}^{\mathfrak{i}-1}\left(\mathtt{s}-y\right)}^{\mathfrak{j}-1}\mathcal{G}\left(\mathtt{t},\mathtt{s}\right)d\mathtt{s}d\mathtt{t}, (x<\mathrm{ȶ}, y<\theta ).$$

(16)

Here is the newly defined concept of $\mathcal{C}$·$\mathsf{L}$·$\mathsf{R}$-$\hslash$-pre-invexity over fuzzy number space in the codomain via fuzzy relation given by

Definition 3.

The ${\rm I}$·$\mathsf{V}$-$\mathsf{M}$ $\mathcal{G}:\Omega =\left[\mathrm{ȵ}, \mathrm{ȶ}\right]\times \left[ⱬ,\theta \right]\to {\mathbb{R}}_I^{+}$ is referred to be $\mathcal{C}$·$\mathsf{L}$·$\mathsf{R}$-$\hslash$-pre-invex ${\rm I}$·$\mathsf{V}$-$\mathsf{M}$ on $\Omega$ with respect to bifurcations $\left({\mathrm{ʎ}}_1, {\mathrm{ʎ}}_2\right)$ if

$$\begin{gathered} \mathcal{G}\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right) \\ {\le }_p\hslash \left(\upsilon \right)\hslash \left(\kappa \right)\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)+\hslash \left(\upsilon \right)\hslash \left(1-\kappa \right)\mathcal{G}\left(\mathrm{ȵ},\theta \right)+\hslash \left(1-\upsilon \right)\hslash \left(\kappa \right)\mathcal{G}\left(\mathrm{ȶ},ⱬ\right) \\ +\hslash \left(1-\upsilon \right)\hslash \left(1-\kappa \right)\mathcal{G}\left(\mathrm{ȶ},\theta \right), \end{gathered}$$

(17)

for all$\left(\mathrm{ȵ}, \mathrm{ȶ}\right), \left(ⱬ,\theta \right)\in \Omega ,$and $\upsilon ,\kappa \in \left[0, 1\right],$ where ${\mathrm{ʎ}}_1: \left[\mathrm{ȵ}, \mathrm{ȶ}\right]\times \left[\mathrm{ȵ}, \mathrm{ȶ}\right]\to \left[\mathrm{ȵ}, \mathrm{ȶ}\right]$ and ${\mathrm{ʎ}}_2: \left[ⱬ,\theta \right]\times \left[ⱬ,\theta \right]\to \left[ⱬ,\theta \right]$. If the inequality (17) is reversed, then $\mathcal{G}$ is referred to be $\mathcal{C}$·$\mathsf{L}$·$\mathsf{R}$-$\mathbf{\hslash }$-pre-concave ${\rm I}$·$\mathsf{V}$-$\mathsf{M}$ on $\Omega$.

Lemma 1.

Let $\mathcal{G}:\Omega \to {\mathbb{R}}_I^{+}$ be a coordinated ${\rm I}$·$\mathsf{V}$-$\mathsf{M}$ on $\Omega$. Then, $\mathcal{G}$ is $\mathcal{C}$·$\mathsf{L}$·$\mathsf{R}$-$\hslash$- pre-invex ${\rm I}$·$\mathsf{V}$-$\mathsf{M}$ on $\Omega$ if and only if there exist two $\mathcal{C}$·$\mathsf{L}$·$\mathsf{R}$-$\hslash$-pre-invex ${\rm I}$·$\mathsf{V}$-$\mathsf{M}$s ${\mathcal{G}}_x:\left[ⱬ,\theta \right]\to {\mathbb{R}}_I^{+}$, ${\mathcal{G}}_x\left(w\right)=\mathcal{G}\left(x,w\right)$ and ${\mathcal{G}}_y:\left[\mathrm{ȵ},\mathrm{ȶ}\right]\to {\mathbb{R}}_I^{+}$, ${\mathcal{G}}_y\left(z\right)=\mathcal{G}\left(z,y\right)$.

Theorem 3.

Let $\mathcal{G}:\Omega \to {\mathbb{R}}_I^{+}$ be an ${\rm I}$·$\mathsf{V}$-$\mathsf{M}$ on $\Omega$, given by

$$\mathcal{G}\left(x,y\right)=\left[{\mathcal{G}}_{\ast }\left(x,y\right), {\mathcal{G}}^{\ast }\left(x,y\right)\right],$$

(18)

for all $\left(x,y\right)\in \Omega$. Then, $\mathcal{G}$ is $\mathcal{C}$·$\mathsf{L}$·$\mathsf{R}$-$\hslash$-pre-invex ${\rm I}$·$\mathsf{V}$-$\mathsf{M}$ on $\Omega ,$ if and only if, ${\mathcal{G}}_{\ast }\left(x,y\right)$ and ${\mathcal{G}}^{\ast }\left(x,y\right)$ both are $\mathcal{C}$·$\mathsf{L}$·$\mathsf{R}$-$\hslash$-pre-invex.

Proof. 

Assume that, ${\mathcal{G}}_{\ast }\left(x,y\right)$ and ${\mathcal{G}}^{\ast }\left(x,y\right)$ are $\mathcal{C}$·$\mathsf{L}$·$\mathsf{R}$-$\hslash$-pre-invex and $\mathbf{\hslash }$-pre-invex on $\Omega$, respectively. Then, from Equation (17), for all$\left(\mathrm{ȵ}, \mathrm{ȶ}\right), \left(ⱬ,\theta \right)\in \Omega , \upsilon$, and $\kappa \in \left[0, 1\right]$, we have

$$\begin{gathered} {\mathcal{G}}_{\ast }\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right) \\ \le \hslash \left(\upsilon \right){\hslash \left(\kappa \right)\mathcal{G}}_{\ast }\left(\mathrm{ȵ},ⱬ\right)+\hslash \left(\upsilon \right)\hslash \left(1-\kappa \right){\mathcal{G}}_{\ast }\left(\mathrm{ȵ},\theta \right)+{\hslash \left(\kappa \right)\hslash \left(1-\upsilon \right)\mathcal{G}}_{\ast }\left(\mathrm{ȶ},ⱬ\right)+ \\ \hslash \left(1-\upsilon \right)\hslash \left(1-\kappa \right){\mathcal{G}}_{\ast }\left(\mathrm{ȶ},\theta \right), \end{gathered}$$

(19)

and

$$\begin{gathered} {\mathcal{G}}^{\ast }\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right) \\ \le \hslash \left(\upsilon \right)\hslash \left(\kappa \right){\mathcal{G}}^{\ast }\left(\mathrm{ȵ},ⱬ\right)+\hslash \left(\upsilon \right)\hslash \left(1-\kappa \right){\mathcal{G}}^{\ast }\left(\mathrm{ȵ},\theta \right)+\hslash \left(\kappa \right)\hslash \left(1-\upsilon \right){\mathcal{G}}^{\ast }\left(\mathrm{ȶ},ⱬ\right)+ \\ \hslash \left(1-\upsilon \right)\hslash \left(1-\kappa \right){\mathcal{G}}^{\ast }\left(\mathrm{ȶ},\theta \right), \end{gathered}$$

(20)

Then, by Equations (18), (3), and (5), we obtain

$$\begin{gathered} \mathcal{G}\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right) \\ =\left[{\mathcal{G}}_{\ast }\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right), {\mathcal{G}}^{\ast }\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+ \\ \left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\right] \\ {\le }_p\hslash \left(\upsilon \right)\hslash \left(\kappa \right)\left[{\mathcal{G}}_{\ast }\left(\mathrm{ȵ},ⱬ\right), {\mathcal{G}}^{\ast }\left(\mathrm{ȵ},ⱬ\right)\right]+\hslash \left(\upsilon \right)\hslash \left(1-\kappa \right)[{\mathcal{G}}_{\ast }\left(\mathrm{ȵ},\theta \right), {\mathcal{G}}^{\ast }\left(\mathrm{ȵ},\theta \right)] \\ +\hslash \left(\kappa \right)\hslash \left(1-\upsilon \right)\left[{\mathcal{G}}_{\ast }\left(\mathrm{ȶ},ⱬ\right), {\mathcal{G}}^{\ast }\left(\mathrm{ȶ},ⱬ\right)\right]+\hslash \left(1-\upsilon \right)\hslash \left(1-\kappa \right)\left[{\mathcal{G}}_{\ast }\left(\mathrm{ȶ},\theta \right), {\mathcal{G}}^{\ast }\left(\mathrm{ȶ},\theta \right)\right] \end{gathered}$$

From (19) and (20), we have

$$\begin{gathered} \mathcal{G}\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right) \\ {\le }_p\hslash \left(\upsilon \right)\hslash \left(\kappa \right)\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)+\hslash \left(\upsilon \right)\hslash \left(1-\kappa \right)\mathcal{G}\left(\mathrm{ȵ},\theta \right)+\hslash \left(1-\upsilon \right)\hslash \left(1-\kappa \right)\mathcal{G}\left(\mathrm{ȶ},ⱬ\right)+ \\ \hslash \left(1-\upsilon \right)\hslash \left(1-\kappa \right)\mathcal{G}\left(\mathrm{ȶ},\theta \right), \end{gathered}$$

hence, $\mathcal{G}$ is $\mathcal{C}$·$\mathsf{L}$·$\mathsf{R}$-$\hslash$-pre-invex ${\rm I}$·$\mathsf{V}$-$\mathsf{M}$ on $\Omega$.

Conversely, let $\mathcal{G}$ be $\mathcal{C}$·$\mathsf{L}$·$\mathsf{R}$-$\hslash$- pre-invex ${\rm I}$·$\mathsf{V}$-$\mathsf{M}$ on $\Omega .$ Then, for all$\left(\mathrm{ȵ}, \mathrm{ȶ}\right), \left(ⱬ,\theta \right)\in \Omega , \upsilon$, and $\kappa \in \left[0, 1\right],$ we have

$$\begin{gathered} \mathcal{G}\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right) \\ {\le }_p\hslash \left(\upsilon \right)\hslash \left(\kappa \right)\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)+\hslash \left(\upsilon \right)\hslash \left(1-\kappa \right)\mathcal{G}\left(\mathrm{ȵ},\theta \right)+\hslash \left(1-\upsilon \right)\hslash \left(\kappa \right)\mathcal{G}\left(\mathrm{ȶ},ⱬ\right)+\hslash \left(1-\upsilon \right)\hslash (1-\kappa )\mathcal{G}\left(\mathrm{ȶ},\theta \right). \end{gathered}$$

Therefore, again from Equation (18), we have

$$\begin{gathered} \mathcal{G}\left(\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\right) \\ =\left[{\mathcal{G}}_{\ast }\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right), {\mathcal{G}}^{\ast }\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ+ \\ \left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\right]. \end{gathered}$$

(21)

Again, by Equations (3) and (5), we obtain

$$\begin{gathered} \hslash \left(\upsilon \right)\hslash \left(\kappa \right)\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)+\hslash \left(\upsilon \right)\hslash \left(1-\kappa \right)\mathcal{G}\left(\mathrm{ȵ},\theta \right)+\hslash \left(1-\upsilon \right)\hslash \left(\kappa \right)\mathcal{G}\left(\mathrm{ȶ},ⱬ\right)+\hslash \left(1-\upsilon \right)\hslash \left(1-\kappa \right)\mathcal{G}\left(\mathrm{ȶ},\theta \right) \\ =\hslash \left(\upsilon \right)\hslash \left(\kappa \right)\left[{\mathcal{G}}_{\ast }\left(\mathrm{ȵ},ⱬ\right), {\mathcal{G}}^{\ast }\left(\mathrm{ȵ},ⱬ\right)\right]+\hslash \left(\upsilon \right)\hslash \left(1-\kappa \right)\left[{\mathcal{G}}_{\ast }\left(\mathrm{ȵ},\theta \right), {\mathcal{G}}^{\ast }\left(\mathrm{ȵ},\theta \right)\right] \\ +\hslash \left(\kappa \right)\hslash \left(1-\upsilon \right)\left[{\mathcal{G}}_{\ast }\left(\mathrm{ȵ},ⱬ\right), {\mathcal{G}}^{\ast }\left(\mathrm{ȵ},ⱬ\right)\right]+\hslash \left(1-\upsilon \right)\hslash \left(1-\kappa \right)\left[{\mathcal{G}}_{\ast }\left(\mathrm{ȵ},\theta \right), {\mathcal{G}}^{\ast }\left(\mathrm{ȵ},\theta \right)\right], \end{gathered}$$

(22)

for all $\mathcal{x},\omega \in \Omega$ and $\upsilon \in \left[0, 1\right].$ Then, from (21) and (22), we have for all $\mathcal{x},\omega \in \Omega$ and $\upsilon \in \left[0, 1\right]$such that

$$\begin{gathered} {\mathcal{G}}_{\ast }\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right) \\ \le \hslash \left(\upsilon \right)\hslash \left(\kappa \right){\mathcal{G}}_{\ast }\left(\mathrm{ȵ},ⱬ\right)+\hslash \left(\upsilon \right)\hslash \left(1-\kappa \right){\mathcal{G}}_{\ast }\left(\mathrm{ȵ},\theta \right)+\hslash \left(1-\upsilon \right)\hslash \left(\kappa \right){\mathcal{G}}_{\ast }\left(\mathrm{ȶ},ⱬ\right)+\hslash \left(1-\upsilon \right)\hslash \left(1- \\ \kappa \right){\mathcal{G}}_{\ast }\left(\mathrm{ȶ},\theta \right), \end{gathered}$$

and

$$\begin{gathered} {\mathcal{G}}^{\ast }\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right) \\ \le \hslash \left(\upsilon \right)\hslash \left(\kappa \right){\mathcal{G}}^{\ast }\left(\mathrm{ȵ},ⱬ\right)+\hslash \left(\upsilon \right)\hslash \left(1-\kappa \right){\mathcal{G}}^{\ast }\left(\mathrm{ȵ},\theta \right)+\hslash \left(1-\upsilon \right)\hslash \left(\kappa \right){\mathcal{G}}^{\ast }\left(\mathrm{ȶ},ⱬ\right)+ \\ \hslash \left(1-\upsilon \right)\hslash \left(1-\kappa \right){\mathcal{G}}^{\ast }\left(\mathrm{ȶ},\theta \right), \end{gathered}$$

Hence, the result follows. □

Example 1.

We consider the ${\rm I}$·$\mathsf{V}$-$\mathsf{M}$ $\mathcal{G}:\left[0, 1\right]\times \left[0, 1\right]\to {\mathbb{R}}_I^{+}$ defined by,

$$\mathcal{G}\left(x\right)=\left[xy,\left(6+e^x\right)\left(6+e^y\right)\right].$$

(23)

Since endpoint functions ${\mathcal{G}}_{\ast }\left(x,y\right),$ ${\mathcal{G}}^{\ast }\left(x,y\right)$ are $\mathcal{C}$-$\mathbf{\hslash }$-pre-invex functions. Hence, $\mathcal{G}\left(x,y\right)$ is $\mathcal{C}$·$\mathsf{L}$·$\mathsf{R}$-$\hslash$-pre-invex ${\rm I}$·$\mathsf{V}$-$\mathsf{M}$.

From Lemma 1 and Example 1, we can easily note that each $\mathsf{L}\mathsf{R}$-$\hslash$-pre-invex ${\rm I}$·$\mathsf{V}$-$\mathsf{M}$ is $\mathcal{C}$·$\mathsf{L}$·$\mathsf{R}$-$\hslash$- pre-invex ${\rm I}$·$\mathsf{V}$-$\mathsf{M}$. However, the converse is not true.

Remark 2.

Assuming that${\mathcal{G}}_{\ast }\left(x,y\right)={\mathcal{G}}^{\ast }\left(x,y\right)$, then $\mathcal{G}$ is referred to be as a classical $\mathcal{C}$·$\mathsf{L}$·$\mathsf{R}$-$\hslash$-pre-invex function if $\mathcal{G}$ meets the stated inequality here:

$$\begin{gathered} \mathcal{G}\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right) \\ \le \hslash \left(\upsilon \right){\hslash \left(\kappa \right)\mathcal{G}}_{\ast }\left(\mathrm{ȵ},ⱬ\right)+\hslash \left(\upsilon \right)\hslash \left(1-\kappa \right){\mathcal{G}}_{\ast }\left(\mathrm{ȵ},\theta \right)+{\hslash \left(\kappa \right)\hslash \left(1-\upsilon \right)\mathcal{G}}_{\ast }\left(\mathrm{ȵ},ⱬ\right) \\ +\hslash \left(1-\upsilon \right)\hslash \left(1-\kappa \right){\mathcal{G}}_{\ast }\left(\mathrm{ȵ},\theta \right). \end{gathered}$$

(24)

Assuming that $\hslash \left(\upsilon \right)=\upsilon , \hslash \left(\kappa \right)=\kappa$ and ${\mathcal{G}}_{\ast }\left(x,y\right)={\mathcal{G}}^{\ast }\left(x,y\right)$, then $\mathcal{G}$ is referred to be as a classical $\mathcal{C}$-pre-invex function if $\mathcal{G}$ meets the stated inequality here:

$$\begin{gathered} \mathcal{G}\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right) \\ \le \upsilon \kappa \mathcal{G}\left(\mathrm{ȵ},ⱬ\right)+\upsilon \left(1-\kappa \right)\mathcal{G}\left(\mathrm{ȵ},\theta \right)+\left(1-\upsilon \right)\kappa \mathcal{G}\left(\mathrm{ȶ},ⱬ\right)+\left(1-\upsilon \right)\left(1-\kappa \right)\mathcal{G}\left(\mathrm{ȶ},\theta \right). \end{gathered}$$

(25)

Let $\hslash \left(\upsilon \right)=\upsilon , \hslash \left(\kappa \right)=\kappa$. If ${\mathcal{G}}_{\ast }\left(x,y\right)$ and ${\mathcal{G}}^{\ast }\left(x,y\right)$ are affine and pre-concave functions with ${\mathcal{G}}_{\ast }\left(x,y\right)\ne {\mathcal{G}}^{\ast }\left(x,y\right)$, respectively, then the stated inequality we have, see [35],

$$\begin{gathered} \mathcal{G}\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\supseteq \upsilon \kappa \mathcal{G}\left(\mathrm{ȵ},ⱬ\right)+\upsilon \left(1-\kappa \right)\mathcal{G}\left(\mathrm{ȵ},\theta \right)+ \\ \left(1-\upsilon \right)\kappa \mathcal{G}\left(\mathrm{ȶ},ⱬ\right)+\left(1-\upsilon \right)\left(1-\kappa \right)\mathcal{G}\left(\mathrm{ȶ},\theta \right), \end{gathered}$$

(26)

is true.

Definition 4.

Let $\mathcal{G}:\Omega \to {\mathbb{R}}_I^{+}$ be a ${\rm I}$·$\mathsf{V}$-$\mathsf{M}$ on $\Omega$. Then, we obtain

$$\mathcal{G}\left(x,y\right)=\left[{\mathcal{G}}_{\ast }\left(x,y\right), {\mathcal{G}}^{\ast }\left(x,y\right)\right],$$

for all $\left(x,y\right)\in \Omega$. Then, $\mathcal{G}$ is left-$\mathcal{C}$·$\mathsf{L}$·$\mathsf{R}$-$\hslash$- pre-invex (pre-concave) ${\rm I}$·$\mathsf{V}$-$\mathsf{M}$ on $\Omega ,$ if and only if, ${\mathcal{G}}_{\ast }\left(x,y\right)$ and ${\mathcal{G}}^{\ast }\left(x,y\right)$ are $\mathcal{C}$-$\hslash$- pre-invex (pre-concave) and affine functions on $\Omega$, respectively.

Definition 5.

Let $\mathcal{G}:\Omega \to {\mathbb{R}}_I^{+}$ be a ${\rm I}$·$\mathsf{V}$-$\mathsf{M}$ on $\Omega$. Then, we obtain

$$\mathcal{G}\left(x,y\right)=\left[{\mathcal{G}}_{\ast }\left(x,y\right),{\mathcal{G}}^{\ast }\left(x,y\right)\right],$$

for all $\left(x,y\right)\in \Omega$. Then, $\mathcal{G}$ is right-$\mathcal{C}$·$\mathsf{L}$·$\mathsf{R}$-$\hslash$- pre-invex (pre-concave) ${\rm I}$·$\mathsf{V}$-$\mathsf{M}$ on $\Omega ,$ if and only if, ${\mathcal{G}}_{\ast }\left(x,y\right)$ and ${\mathcal{G}}^{\ast }\left(x,y\right)$ are $\mathcal{C}$-$\mathit{\hslash }$-affine and $\mathit{\hslash }$-(pre-concave) functions on $\Omega$, respectively.

Theorem 4.

Let $\Omega$ be a coordinated invex set, and let $\mathcal{G}:\Omega \to {\mathbb{R}}_I^{+}$ be a ${\rm I}$·$\mathsf{V}$-$\mathsf{M}$, defined by

$$\mathcal{G}\left(x,y\right)=\left[{\mathcal{G}}_{\ast }\left(x,y\right), {\mathcal{G}}^{\ast }\left(x,y\right)\right],$$

for all $\left(x,y\right)\in \Omega$. Then, $\mathcal{G}$ is $\mathcal{C}$·$\mathsf{L}$·$\mathsf{R}$-$\mathbf{\hslash }$-pre-concave ${\rm I}$·$\mathsf{V}$-$\mathsf{M}$ on $\Omega ,$ if and only if, ${\mathcal{G}}_{\ast }\left(x,y\right)$ and ${\mathcal{G}}^{\ast }\left(x,y\right)$ are $\mathcal{C}$-$\mathbf{\hslash }$-pre-concave and $\mathcal{C}$-$\hslash$-pre-concave functions, respectively.

Proof. 

The demonstration of proof of Theorem 4 is similar to the demonstration proof of Theorem 3. □

Example 2.

We consider the ${\rm I}$·$\mathsf{V}$-$\mathsf{M}$s $\mathcal{G}:\left[0, 1\right]\times \left[0, 1\right]\to {\mathbb{R}}_I^{+}$defined by,

$$\mathcal{G}\left(x,y\right)=\left[\left(6-e^x\right)\left(6-e^y\right),40xy\right].$$

(27)

Then, we have since endpoint functions ${\mathcal{G}}_{\ast }\left(x,y\right),$ ${\mathcal{G}}^{\ast }\left(x,y\right)$ which are both $\mathcal{C}$-$\mathbf{\hslash }$- pre-concave functions. Hence $\mathcal{G}\left(x,y\right)$ is $\mathcal{C}$·$\mathsf{L}$·$\mathsf{R}$-$\mathbf{\hslash }$- pre-concave ${\rm I}$·$\mathsf{V}$-$\mathsf{M}$.

In the next results, to avoid confusion, we will not include the symbols $\left(R\right)$, $\left(IR\right)$, and $\left(ID\right)$ before the integral sign.

3. Main Results

In this section, Hermite–Hadamard and Pachpatte-type inequalities for interval-value functions are given. We first present an inequality of Hermite–Hadamard via $\mathcal{C}$·$\mathsf{L}$·$\mathsf{R}$-$\mathbf{\hslash }$-pre-convex ${\rm I}$·$\mathsf{V}$-$\mathsf{M}$s.

Theorem 5.

Let $\mathcal{G}:\Omega \to {\mathbb{R}}_I^{+}$ be a $\mathcal{C}$·$\mathsf{L}$·$\mathsf{R}$-$\hslash$-pre-invex ${\rm I}$·$\mathsf{V}$-$\mathsf{M}$ on $\Omega$, where $\mathcal{G}\left(x,y\right)=\left[{\mathcal{G}}_{\ast }\left(x,y\right), {\mathcal{G}}^{\ast }\left(x,y\right)\right]$ for all $\left(x,y\right)\in \Omega$ and let $\hslash :\left[0, 1\right]\to {\mathbb{R}}^{+}$. If ${\mathcal{G}\in \mathcal{F}\mathfrak{O}}_{\Omega }$, then the following inequalities holds:

$$\begin{gathered} \frac{1}{{\hslash }^2\left(\frac{1}{2}\right)}\mathcal{G}\left(\frac{2\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2}, \frac{2ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right) \\ {\le }_p\frac{\Gamma \left(\mathfrak{i}+1\right)}{{2\hslash \left(\frac{1}{2}\right)\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}}\left[{\mathcal{I}}_{{\mathrm{ȵ}}^{+}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),\frac{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right) \\ +{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ},\frac{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right)\right] \\ +\frac{\Gamma \left(\mathfrak{j}+1\right)}{{2\hslash \left(\frac{1}{2}\right)\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[{\mathcal{I}}_{{ⱬ}^{+}}^{\mathfrak{j}}\mathcal{G}\left(\frac{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right) \\ +{\mathcal{I}}_{{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{j}}\mathcal{G}\left(\frac{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2},ⱬ\right)\right] \\ {\le }_p\frac{\Gamma \left(\mathfrak{i}+1\right)\Gamma \left(\mathfrak{j}+1\right)}{{\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}{\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[{\mathcal{I}}_{{\mathrm{ȵ}}^{+},{ⱬ}^{+}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right) \\ +{\mathcal{I}}_{{\mathrm{ȵ}}^{+},{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right) \\ +{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-},{ⱬ}^{+}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-},{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)\right] \\ {\le }_p\frac{\mathfrak{j}\Gamma \left(\mathfrak{i}+1\right)}{{\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}}\left[\begin{array}{c}{\mathcal{I}}_{{\mathrm{ȵ}}^{+}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right)+{\mathcal{I}}_{{\mathrm{ȵ}}^{+}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\\ +{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)+{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\end{array}\right]\times \\ {\int }_0^1{\kappa }^{\mathfrak{j}-1}\left[\hslash \left(\kappa \right)+\hslash \left(1-\kappa \right)\right]d\kappa \\ +\frac{\mathfrak{i}\Gamma \left(\mathfrak{j}+1\right)}{{\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[\begin{array}{c}{\mathcal{I}}_{{ⱬ}^{+}}^{\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+{\mathcal{I}}_{{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right)\\ +{\mathcal{I}}_{{ⱬ}^{+}}^{\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+{\mathcal{I}}_{{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right)\end{array}\right]\times \\ {\int }_0^1{\upsilon }^{\mathfrak{i}-1}\hslash \left(\upsilon \right)+\hslash \left(1-\upsilon \right)d\upsilon {\le }_p\mathfrak{i}\mathfrak{j}\left[\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)+\mathcal{G}\left(\mathrm{ȶ},ⱬ\right)+\mathcal{G}\left(\mathrm{ȵ},\theta \right)+\mathcal{G}\left(\mathrm{ȶ},\theta \right)\right]\times \\ {\int }_0^1{\kappa }^{\mathfrak{j}-1}\left[\hslash \left(\kappa \right)+\hslash \left(1-\kappa \right)\right]d\kappa {\int }_0^1{\upsilon }^{\mathfrak{i}-1}\left[\hslash \left(\upsilon \right)+\hslash \left(1-\upsilon \right)\right]d\upsilon . \end{gathered}$$

(28)

If $\mathcal{G}$ is $\mathcal{C}$·$\mathsf{L}$·$\mathsf{R}$-$\mathbf{\hslash }$-pre-concave, then the inequality (28) is reversed such that

$$\begin{gathered} \frac{1}{{\hslash }^2\left(\frac{1}{2}\right)}\mathcal{G}\left(\frac{2\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2}, \frac{2ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right) \\ {\ge }_p\frac{\Gamma \left(\mathfrak{i}+1\right)}{{2\hslash \left(\frac{1}{2}\right)\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}}\left[{\mathcal{I}}_{{\mathrm{ȵ}}^{+}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),\frac{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right) \\ +{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ},\frac{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right)\right] \\ +\frac{\Gamma \left(\mathfrak{j}+1\right)}{{2\hslash \left(\frac{1}{2}\right)\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[{\mathcal{I}}_{{ⱬ}^{+}}^{\mathfrak{j}}\mathcal{G}\left(\frac{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right) \\ +{\mathcal{I}}_{{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{j}}\mathcal{G}\left(\frac{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2},ⱬ\right)\right] \\ {\ge }_p\frac{\Gamma \left(\mathfrak{i}+1\right)\Gamma \left(\mathfrak{j}+1\right)}{{\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}{\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[\begin{array}{c}{\mathcal{I}}_{{\mathrm{ȵ}}^{+},{ⱬ}^{+}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+{\mathcal{I}}_{{\mathrm{ȵ}}^{+},{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right)\\ +{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-},{ⱬ}^{+}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-},{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)\end{array}\right] \\ {\ge }_p\frac{\mathfrak{j}\Gamma \left(\mathfrak{i}+1\right)}{{\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}}\left[\begin{array}{c}{\mathcal{I}}_{{\mathrm{ȵ}}^{+}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right)+{\mathcal{I}}_{{\mathrm{ȵ}}^{+}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\\ +{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)+{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\end{array}\right]\times \\ {\int }_0^1{\kappa }^{\mathfrak{j}-1}\left[\hslash \left(\kappa \right)+\hslash \left(1-\kappa \right)\right]d\kappa \\ +\frac{\mathfrak{i}\Gamma \left(\mathfrak{j}+1\right)}{{\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[\begin{array}{c}{\mathcal{I}}_{{ⱬ}^{+}}^{\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+{\mathcal{I}}_{{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right)\\ +{\mathcal{I}}_{{ⱬ}^{+}}^{\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+{\mathcal{I}}_{{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right)\end{array}\right]\times \\ {\int }_0^1{\upsilon }^{\mathfrak{i}-1}\left[\hslash \left(\upsilon \right)+\hslash \left(1-\upsilon \right)\right]d\upsilon \\ {\ge }_p\mathfrak{i}\mathfrak{j}\left[\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)+\mathcal{G}\left(\mathrm{ȶ},ⱬ\right)+\mathcal{G}\left(\mathrm{ȵ},\theta \right)+\mathcal{G}\left(\mathrm{ȶ},\theta \right)\right] \\ \times {\int }_0^1{\kappa }^{\mathfrak{j}-1}\left[\hslash \left(\kappa \right)+\hslash \left(1-\kappa \right)\right]d\kappa {\int }_0^1{\upsilon }^{\mathfrak{i}-1}\left[\hslash \left(\upsilon \right)+\hslash \left(1-\upsilon \right)\right]d\upsilon . \end{gathered}$$

(29)

Proof. 

Let $\mathcal{G}:\left[\mathrm{ȵ}, \mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right]\to {\mathbb{R}}_I^{+}$ be a $\mathcal{C}$·$\mathsf{L}$·$\mathsf{R}$-$\hslash$- pre-invex ${\rm I}$·$\mathsf{V}$-$\mathsf{M}$. Then, by hypothesis, we have

$$\begin{gathered} \frac{1}{{\hslash }^2\left(\frac{1}{2}\right)}\mathcal{G}\left(\frac{2\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2},\frac{2ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right){\le }_p\mathcal{G}\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ \\ +\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+\mathcal{G}\left(\mathrm{ȵ}+\upsilon {\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), \left(1-\upsilon \right)ⱬ+\upsilon \theta \right). \end{gathered}$$

By using Theorem 4, we have

$$\begin{array}{c}\frac{1}{{\hslash }^2\left(\frac{1}{2}\right)}{\mathcal{G}}_{\ast }\left(\frac{2\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2},\frac{2ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right) \\ \le {\mathcal{G}}_{\ast }\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+{\mathcal{G}}_{\ast }\left(\mathrm{ȵ}+\upsilon {\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\kappa {\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right), \\ \frac{1}{{\hslash }^2\left(\frac{1}{2}\right)}{\mathcal{G}}^{\ast }\left(\frac{2\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2},\frac{2ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right) \\ \le {\mathcal{G}}^{\ast }\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+{\mathcal{G}}^{\ast }\left(\mathrm{ȵ}+\upsilon {\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\kappa {\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right).\end{array}$$

By using Lemma 1, we have

$$\begin{array}{c}\frac{1}{\hslash \left(\frac{1}{2}\right)}{\mathcal{G}}_{\ast }\left(x,\frac{2ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right)\le {\mathcal{G}}_{\ast }\left(x, ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+{\mathcal{G}}_{\ast }\left(x, ⱬ+\kappa {\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right), \\ \frac{1}{\hslash \left(\frac{1}{2}\right)}{\mathcal{G}}^{\ast }\left(x,\frac{2ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right)\le {\mathcal{G}}^{\ast }\left(x, ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+{\mathcal{G}}^{\ast }\left(x, ⱬ+\kappa {\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right),\end{array}$$

(30)

and

$$\begin{array}{c}\frac{1}{\hslash \left(\frac{1}{2}\right)}{\mathcal{G}}_{\ast }\left(\frac{2\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2},y\right)\le {\mathcal{G}}_{\ast }\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), y\right)+{\mathcal{G}}_{\ast }\left(\mathrm{ȵ}+\upsilon {\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), y\right), \\ \frac{1}{\hslash \left(\frac{1}{2}\right)}{\mathcal{G}}^{\ast }\left(\frac{2\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2},y\right)\le {\mathcal{G}}^{\ast }\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), y\right)+{\mathcal{G}}^{\ast }\left(\mathrm{ȵ}+\upsilon {\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), y\right).\end{array}$$

(31)

From (30) and (31), we have

$$\begin{gathered} \frac{1}{\hslash \left(\frac{1}{2}\right)}\left[{\mathcal{G}}_{\ast }\left(x,\frac{2ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right),{\mathcal{G}}^{\ast }\left(x,\frac{2ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right)\right] \\ {\le }_p\left[{\mathcal{G}}_{\ast }\left(x, ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right),{\mathcal{G}}^{\ast }\left(x, ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\right] \\ +\left[{\mathcal{G}}_{\ast }\left(x, ⱬ+\kappa {\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right),{\mathcal{G}}^{\ast }\left(x, ⱬ+\kappa {\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\right], \end{gathered}$$

and

$$\begin{gathered} \frac{1}{\hslash \left(\frac{1}{2}\right)}\left[{\mathcal{G}}_{\ast }\left(\frac{2\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2},y\right),{\mathcal{G}}^{\ast }\left(\frac{2\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2},y\right)\right] \\ {\le }_p\left[{\mathcal{G}}_{\ast }\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), y\right),{\mathcal{G}}^{\ast }\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), y\right)\right] \\ +\left[{\mathcal{G}}_{\ast }\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), y\right),{\mathcal{G}}^{\ast }\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), y\right)\right], \end{gathered}$$

It follows that

$$\frac{1}{\hslash \left(\frac{1}{2}\right)}\mathcal{G}\left(x,\frac{2ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right){\le }_p\mathcal{G}\left(x, ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+\mathcal{G}\left(x, ⱬ+\kappa {\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right),$$

(32)

and

$$\frac{1}{\hslash \left(\frac{1}{2}\right)}\mathcal{G}\left(\frac{2\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2},y\right){\le }_p\mathcal{G}\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), y\right)+\mathcal{G}\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), y\right).$$

(33)

Since $\mathcal{G}\left(x,.\right)$ and $\mathcal{G}\left(.,y\right)$, both are $\mathcal{C}$·$\mathsf{L}$·$\mathsf{R}$-$\mathsf{L}\mathsf{R}$-$\hslash$- pre-invex -${\rm I}$·$\mathsf{V}$-$\mathsf{M}$s, then, we have

$$\begin{gathered} \frac{1}{\mathfrak{j}\hslash \left(\frac{1}{2}\right)}{\mathcal{G}}_x\left(\frac{2ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right){\le }_p\frac{\Gamma \left(\mathfrak{j}\right)}{{\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[{\mathcal{I}}_{{ⱬ}^{+}}^{\mathfrak{j}}{\mathcal{G}}_x\left(ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+{\mathcal{I}}_{{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{j}}{\mathcal{G}}_x\left(ⱬ\right)\right] \\ {\le }_p\left[{\mathcal{G}}_x\left(ⱬ\right)+{\mathcal{G}}_x\left(\theta \right)\right]{\int }_0^1{\kappa }^{\mathfrak{j}-1}\left[\hslash \left(\kappa \right)+\hslash \left(1-\kappa \right)\right]d\kappa \end{gathered}$$

(34)

and

$$\begin{gathered} \frac{1}{\mathfrak{i}\hslash \left(\frac{1}{2}\right)}{\mathcal{G}}_y\left(\frac{2\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2}\right){\le }_p\frac{\Gamma \left(\mathfrak{i}\right)}{{\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}}\left[{\mathcal{I}}_{{\mathrm{ȵ}}^{+}}^{\mathfrak{i}}{\mathcal{G}}_y\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right) \\ +{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-}}^{\mathfrak{i}}{\mathcal{G}}_y\left(\mathrm{ȵ}\right)\right] \\ {\le }_p\left[{\mathcal{G}}_y\left(\mathrm{ȵ}\right)+{\mathcal{G}}_y\left(\mathrm{ȶ}\right)\right]{\int }_0^1{\upsilon }^{\mathfrak{i}-1}\hslash \left(\upsilon \right)+\hslash \left(1-\upsilon \right)d\upsilon \end{gathered}$$

(35)

Since ${\mathcal{G}}_x\left(w\right)=\mathcal{G}\left(x,w\right)$, then (34) can be written as

$$\begin{gathered} \frac{1}{\mathfrak{j}\hslash \left(\frac{1}{2}\right)}\mathcal{G}\left(x,\frac{2ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right){\le }_p\frac{\Gamma \left(\mathfrak{j}\right)}{{\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[{\mathcal{I}}_{{ⱬ}^{+}}^{\mathfrak{i}}\mathcal{G}\left(x,ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+{\mathcal{I}}_{{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{i}}\mathcal{G}\left(x,ⱬ\right)\right] \\ {\le }_p\left[\mathcal{G}\left(x,ⱬ\right)+\mathcal{G}\left(x,ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\right]{\int }_0^1{\kappa }^{\mathfrak{j}-1}\left[\hslash \left(\kappa \right)+\hslash \left(1-\kappa \right)\right]d\kappa . \end{gathered}$$

(36)

That is

$$\begin{gathered} \frac{1}{\mathfrak{j}\hslash \left(\frac{1}{2}\right)}\mathcal{G}\left(x,\frac{2ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right){\le }_p\frac{1}{{\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[ {\int }_{ⱬ}^{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{\left(ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right) \\ -\kappa \right)}^{\mathfrak{j}-1}\mathcal{G}\left(x,\kappa \right)d\kappa +{\int }_{ⱬ}^{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{\left(\kappa -ⱬ\right)}^{\mathfrak{j}-1}\mathcal{G}\left(x,\kappa \right)d\kappa \right] \\ {\le }_p\left[\mathcal{G}\left(x,ⱬ\right)+\mathcal{G}\left(x,\theta \right)\right]{\int }_0^1{\kappa }^{\mathfrak{j}-1}\left[\hslash \left(\kappa \right)+\hslash \left(1-\kappa \right)\right]d\kappa . \end{gathered}$$

Multiplying the double inequality (36) by $\frac{{\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)-x\right)}^{\mathfrak{i}-1}}{{\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}}$ and integrating with respect to $x$ over $\left[\mathrm{ȵ}, \mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right],$ we have

$$\begin{gathered} \frac{1}{{\mathfrak{j}\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}\hslash \left(\frac{1}{2}\right)}{\int }_{\mathrm{ȵ}}^{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}\mathcal{G}\left(x,\frac{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right){\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)-x\right)}^{\mathfrak{i}-1}dx \\ {\le }_p\frac{1}{{{\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}{\int }_{\mathrm{ȵ}}^{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{\int }_{ⱬ}^{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)-x\right)}^{\mathfrak{i}-1}{\left(ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)- \\ \kappa \right)}^{\mathfrak{j}-1}\mathcal{G}\left(x,\kappa \right)d\kappa dx+{\int }_{\mathrm{ȵ}}^{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{\int }_{ⱬ}^{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)-x\right)}^{\mathfrak{i}-1}{\left(\kappa - \\ ⱬ\right)}^{\mathfrak{j}-1}\mathcal{G}\left(x,\kappa \right)d\kappa dx \\ {\le }_p\frac{1}{{\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}}\left[{\int }_{\mathrm{ȵ}}^{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)-x\right)}^{\mathfrak{i}-1}\mathcal{G}\left(x,ⱬ\right)dx+{\int }_{\mathrm{ȵ}}^{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)- \\ x\right)}^{\mathfrak{i}-1}\mathcal{G}\left(x,ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)dx\right]{\int }_0^1{\kappa }^{\mathfrak{j}-1}\left[\hslash \left(\kappa \right)+\hslash \left(1-\kappa \right)\right]d\kappa . \end{gathered}$$

(37)

Again, multiplying the double inequality (36) by $\frac{{\left(x-\mathrm{ȵ}\right)}^{\mathfrak{i}-1}}{{\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}}$ and integrating with respect to $x$ over $\left[\mathrm{ȵ}, \mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right],$ we have

$$\begin{gathered} \frac{1}{{\mathfrak{j}\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}\hslash \left(\frac{1}{2}\right)}{\int }_{\mathrm{ȵ}}^{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}\mathcal{G}\left(x,\frac{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right){\left(x-\mathrm{ȵ}\right)}^{\mathfrak{i}-1}dx \\ {\le }_p\frac{1}{{\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}{\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}{\int }_{\mathrm{ȵ}}^{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{\int }_{ⱬ}^{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{\left(x-\mathrm{ȵ}\right)}^{\mathfrak{i}-1}{\left(ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right) \\ -\kappa \right)}^{\mathfrak{j}-1}\mathcal{G}\left(x,\kappa \right)d\kappa dx \\ +\frac{1}{{\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}{\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}{\int }_{\mathrm{ȵ}}^{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{\int }_{ⱬ}^{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{\left(x-\mathrm{ȵ}\right)}^{\mathfrak{i}-1}{\left(\kappa -ⱬ\right)}^{\mathfrak{j}-1}\mathcal{G}\left(x,\kappa \right)d\kappa dx \\ {\le }_p\frac{1}{{\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}}\left[{\int }_{\mathrm{ȵ}}^{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{\left(x-\mathrm{ȵ}\right)}^{\mathfrak{i}-1}\mathcal{G}\left(x,ⱬ\right)dx+{\int }_{\mathrm{ȵ}}^{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{\left(x-\mathrm{ȵ}\right)}^{\mathfrak{i}-1}\mathcal{G}\left(x,ⱬ+ \\ {\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)dx\right]{\int }_0^1{\kappa }^{\mathfrak{j}-1}\left[\hslash \left(\kappa \right)+\hslash \left(1-\kappa \right)\right]d\kappa . \end{gathered}$$

(38)

From (37), we have

$$\begin{gathered} \frac{\Gamma \left(\mathfrak{i}+1\right)}{{2\hslash \left(\frac{1}{2}\right)\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}}\left[{\mathcal{I}}_{{\mathrm{ȵ}}^{+}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),\frac{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right)\right] \\ {\le }_p\frac{\Gamma \left(\mathfrak{i}+1\right)\Gamma \left(\mathfrak{j}+1\right)}{{\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}{\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[{\mathcal{I}}_{{\mathrm{ȵ}}^{+},{ⱬ}^{+}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right) \\ +{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-},{ⱬ}^{+}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right)\right] \\ {\le }_p\frac{\mathfrak{j}\Gamma \left(\mathfrak{i}+1\right)}{{\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}}\left[{\mathcal{I}}_{{\mathrm{ȵ}}^{+}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right)+{\mathcal{I}}_{{\mathrm{ȵ}}^{+}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ+ \\ {\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\right]{\int }_0^1{\kappa }^{\mathfrak{j}-1}\left[\hslash \left(\kappa \right)+\hslash \left(1-\kappa \right)\right]d\kappa . \end{gathered}$$

(39)

From (38), we have

$$\begin{gathered} \frac{\Gamma \left(\mathfrak{i}+1\right)}{{2\hslash \left(\frac{1}{2}\right)\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}}\left[{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ},\frac{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right)\right] \\ {\le }_p\frac{\Gamma \left(\mathfrak{i}+1\right)\Gamma \left(\mathfrak{j}+1\right)}{{\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}{\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-},{ⱬ}^{+}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-},{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)\right] \\ {\le }_p\frac{\mathfrak{j}\Gamma \left(\mathfrak{i}+1\right)}{{\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}}\left[{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)+{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\right]{\int }_0^1{\kappa }^{\mathfrak{j}-1}\left[\hslash \left(\kappa \right)+ \\ \hslash \left(1-\kappa \right)\right]d\kappa \end{gathered}$$

(40)

Similarly, since ${\mathcal{G}}_y\left(z\right)=\mathcal{G}\left(z,y\right)$ then, from (35), (41), and (42), we have

$$\begin{gathered} \frac{\Gamma \left(\mathfrak{j}+1\right)}{{2\hslash \left(\frac{1}{2}\right)\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[{\mathcal{I}}_{{ⱬ}^{+}}^{\mathfrak{j}}\mathcal{G}\left(\frac{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2}, ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\right] \\ {\le }_p\frac{\Gamma \left(\mathfrak{i}+1\right)\Gamma \left(\mathfrak{j}+1\right)}{{\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}{\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[{\mathcal{I}}_{{\mathrm{ȵ}}^{+},{ⱬ}^{+}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-},{ⱬ}^{+}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ+ \\ {\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\right] \\ {\le }_p\frac{\mathfrak{i}\Gamma \left(\mathfrak{j}+1\right)}{{\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[{\mathcal{I}}_{{ⱬ}^{+} }^{\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+{\mathcal{I}}_{{ⱬ}^{+}}^{\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\right]. \end{gathered}$$

(41)

and

$$\begin{gathered} \frac{\Gamma \left(\mathfrak{j}+1\right)}{{2\hslash \left(\frac{1}{2}\right)\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{i}}}\left[{\mathcal{I}}_{{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{j}}\mathcal{G}\left(\frac{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2},ⱬ\right)\right] \\ {\le }_p\frac{\Gamma \left(\mathfrak{i}+1\right)\Gamma \left(\mathfrak{j}+1\right)}{{\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}{\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[{\mathcal{I}}_{{\mathrm{ȵ}}^{+},{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right)+{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-},{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)\right] \\ {\le }_p\frac{\mathfrak{i}\Gamma \left(\mathfrak{j}+1\right)}{{\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[{\mathcal{I}}_{{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)+{\mathcal{I}}_{{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right)\right]. \end{gathered}$$

(42)

The second, third, and fourth inequalities of (28) result from combining inequalities (41) and (42).

Now, we have the left portion of the inequality (11):

$$\begin{gathered} \frac{1}{{\hslash }^2\left(\frac{1}{2}\right)}\mathcal{G}\left(\frac{2\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2},\frac{2ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right){\le }_p\frac{\Gamma \left(\mathfrak{j}+1\right)}{{\hslash \left(\frac{1}{2}\right)\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[{\mathcal{I}}_{{ⱬ}^{+}}^{\mathfrak{j}}\mathcal{G}\left(\frac{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+ \\ {\mathcal{I}}_{{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{j}}\mathcal{G}\left(\frac{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2},ⱬ\right)\right], \end{gathered}$$

(43)

and

$$\begin{gathered} \frac{1}{{\hslash }^2\left(\frac{1}{2}\right)}\mathcal{G}\left(\frac{2\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2},\frac{2ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right){\le }_p\frac{\Gamma \left(\mathfrak{i}+1\right)}{{\hslash \left(\frac{1}{2}\right)\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}}\left[{\mathcal{I}}_{{\mathrm{ȵ}}^{+}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),\frac{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right)+ \\ {\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ},\frac{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right)\right] \end{gathered}$$

(44)

The following inequality results from combining inequalities (43) and (44):

$$\begin{gathered} \frac{1}{{\hslash }^2\left(\frac{1}{2}\right)}\mathcal{G}\left(\frac{2\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2},\frac{2ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right){\le }_p\frac{\Gamma \left(\mathfrak{i}+1\right)}{{\hslash \left(\frac{1}{2}\right)\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}}\left[{\mathcal{I}}_{{\mathrm{ȵ}}^{+}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),\frac{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right)+ \\ {\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ},\frac{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right)\right] \\ +\frac{\Gamma \left(\mathfrak{j}+1\right)}{{\hslash \left(\frac{1}{2}\right)\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[{\mathcal{I}}_{{ⱬ}^{+}}^{\mathfrak{j}}\mathcal{G}\left(\frac{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+{\mathcal{I}}_{{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{j}}\mathcal{G}\left(\frac{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2},ⱬ\right)\right]. \end{gathered}$$

Similarly, given the set of Ι of ${\rm I}$·$\mathsf{V}$-$\mathsf{M}$s $\mathcal{G}:\Omega \to {\mathbb{R}}_I^{+}$, the inequality can be expressed as follows:

$$\begin{gathered} \frac{1}{{\hslash }^2\left(\frac{1}{2}\right)}\mathcal{G}\left(\frac{2\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2},\frac{2ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right) \\ {\le }_p\frac{\Gamma \left(\mathfrak{i}+1\right)}{{\hslash \left(\frac{1}{2}\right)\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}}\left[{\mathcal{I}}_{{\mathrm{ȵ}}^{+}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),\frac{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right) \\ +{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ},\frac{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right)\right] \\ +\frac{\Gamma \left(\mathfrak{j}+1\right)}{{\hslash \left(\frac{1}{2}\right)\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[{\mathcal{I}}_{{ⱬ}^{+}}^{\mathfrak{j}}\mathcal{G}\left(\frac{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right) \\ +{\mathcal{I}}_{{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{j}}\mathcal{G}\left(\frac{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2},ⱬ\right)\right]. \end{gathered}$$

(45)

This gives the first inequality in (28).

Again, we have the right-hand side of the inequality (11):

$$\begin{gathered} \frac{\Gamma \left(\mathfrak{j}\right)}{{\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[{\mathcal{I}}_{{ⱬ}^{+}}^{\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+{\mathcal{I}}_{{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)\right]{\le }_p\left[\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)+\mathcal{G}\left(\mathrm{ȵ},\theta \right)\right]\times \\ {\int }_0^1{\kappa }^{\mathfrak{j}-1}\left[\hslash \left(\kappa \right)+\hslash \left(1-\kappa \right)\right]d\kappa \end{gathered}$$

(46)

$$\begin{gathered} \frac{\Gamma \left(\mathfrak{j}\right)}{{\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[{\mathcal{I}}_{{ⱬ}^{+}}^{\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right) \\ +{\mathcal{I}}_{{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right)\right]{\le }_p\left[\mathcal{G}\left(\mathrm{ȶ},ⱬ\right)+\mathcal{G}\left(\mathrm{ȶ},\theta \right)\right] \\ \times {\int }_0^1{\kappa }^{\mathfrak{j}-1}\left[\hslash \left(\kappa \right)+\hslash \left(1-\kappa \right)\right]d\kappa \end{gathered}$$

(47)

$$\begin{gathered} \frac{\Gamma \left(\mathfrak{i}\right)}{{\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}}\left[{\mathcal{I}}_{{\mathrm{ȵ}}^{+}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right)+{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)\right]{\le }_p\left[\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)+\mathcal{G}\left(\mathrm{ȶ},ⱬ\right)\right]\times \\ {\int }_0^1{\upsilon }^{\mathfrak{i}-1}\hslash \left(\upsilon \right)+\hslash \left(1-\upsilon \right)d\upsilon \end{gathered}$$

(48)

$$\begin{gathered} \frac{\Gamma \left(\mathfrak{i}\right)}{{\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}}\left[{\mathcal{I}}_{{\mathrm{ȵ}}^{+}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right) \\ +{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\right]{\le }_p\left[\mathcal{G}\left(\mathrm{ȵ},\theta \right)+\mathcal{G}\left(\mathrm{ȶ},\theta \right)\right] \\ \times {\int }_0^1{\upsilon }^{\mathfrak{i}-1}\hslash \left(\upsilon \right)+\hslash \left(1-\upsilon \right)d\upsilon \end{gathered}$$

(49)

By summing inequalities (46) to (49), and then multiplying the result by $\mathfrak{i}\mathfrak{j}$, we obtain

$$\begin{gathered} \frac{\mathfrak{j}\Gamma \left(\mathfrak{i}+1\right)}{{\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}}\left[{\mathcal{I}}_{{\mathrm{ȵ}}^{+}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right)+{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)+{\mathcal{I}}_{{\mathrm{ȵ}}^{+}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ+ \\ {\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\right] \\ +\frac{\mathfrak{i}\Gamma \left(\mathfrak{j}+1\right)}{{\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[\begin{array}{c}{\mathcal{I}}_{{ⱬ}^{+}}^{\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+{\mathcal{I}}_{{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)\\ +{\mathcal{I}}_{{ⱬ}^{+}}^{\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+{\mathcal{I}}_{{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right)\end{array}\right] \\ {\le }_p\left[\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)+\mathcal{G}\left(\mathrm{ȵ},\theta \right)+\mathcal{G}\left(\mathrm{ȶ},ⱬ\right)+\mathcal{G}\left(\mathrm{ȶ},\theta \right)\right] \\ \times {\int }_0^1{\kappa }^{\mathfrak{j}-1}\left[\hslash \left(\kappa \right)+\hslash \left(1-\kappa \right)\right]d\kappa {\int }_0^1{\upsilon }^{\mathfrak{i}-1}\hslash \left(\upsilon \right)+\hslash \left(1-\upsilon \right)d\upsilon . \end{gathered}$$

(50)

The final inequality of (28) marks the conclusion of the derivation.□

Example 3.

We assume the ${\rm I}$·$\mathsf{V}$-$\mathsf{M}$s $\mathcal{G}:\left[0, 2\right]\times \left[0, 2\right]\to {\mathbb{R}}_I^{+}$defined by,

$$\mathcal{G}\left(x,y\right)=\left[\left(2-\sqrt{x}\right)\left(2-\sqrt{y}\right),2\left(2-\sqrt{x}\right)\left(2-\sqrt{y}\right)\right],$$

(51)

then, for each $\gamma \in \left[0, 1\right],$ we have. Since the endpoint functions ${\mathcal{G}}_{\ast }\left(x,y\right),$ ${\mathcal{G}}^{\ast }\left(x,y\right)$ are $\hslash$-pre-invex functions with respect to ${\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)=$ and ${\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)$. Hence, $\tilde{\mathcal{G}}\left(x,y\right)$ is $\mathcal{C}$·$\mathsf{L}$·$\mathsf{R}$-$\hslash$-pre-invex ${\rm I}$·$\mathsf{V}$-$\mathsf{M}$.

$$\begin{gathered} \mathcal{G}\left(\frac{2\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2}, \frac{2ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right)=\left[1,2\right] \\ \frac{\Gamma \left(\mathfrak{i}+1\right)}{{4\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}}\left[{\mathcal{I}}_{{\mathrm{ȵ}}^{+}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),\frac{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right)+{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ},\frac{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right)\right] \\ +\frac{\Gamma \left(\mathfrak{j}+1\right)}{{4\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[{\mathcal{I}}_{{ⱬ}^{+}}^{\mathfrak{j}}\mathcal{G}\left(\frac{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+{\mathcal{I}}_{{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{j}}\mathcal{G}\left(\frac{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2},ⱬ\right)\right] \\ =2-\frac{\sqrt{2}}{4}-\frac{\sqrt{2}}{8}\mathrm{ʎ}·\left[1,2\right] \\ \frac{\Gamma \left(\mathfrak{i}+1\right)\Gamma \left(\mathfrak{j}+1\right)}{{4\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}{\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[{\mathcal{I}}_{{\mathrm{ȵ}}^{+},{ⱬ}^{+}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+{\mathcal{I}}_{{\mathrm{ȵ}}^{+},{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right) \\ +{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-},{ⱬ}^{+}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-},{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)\right] \\ =\frac{33}{8}-\sqrt{2}-\frac{\sqrt{2}}{2}\mathrm{ʎ}+\frac{\mathrm{ʎ}}{8}+\frac{{\mathrm{ʎ}}^2}{32}·\left[1,2\right] \\ \frac{\Gamma \left(\mathfrak{i}+1\right)}{{8\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}}\left[{\mathcal{I}}_{{\mathrm{ȵ}}^{+}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right)+{\mathcal{I}}_{{\mathrm{ȵ}}^{+}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ},ⱬ\right) \\ +{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\right] \\ +\frac{\Gamma \left(\mathfrak{j}+1\right)}{{8\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[\begin{array}{c}{\mathcal{I}}_{{ⱬ}^{+}}^{\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+{\mathcal{I}}_{{ⱬ}^{+}}^{\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\\ +{\mathcal{I}}_{{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)+{\mathcal{I}}_{{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right)\end{array}\right] \\ =\frac{34\sqrt{2}+\left(\sqrt{2}-4\right)\mathrm{ʎ}-24}{8\sqrt{2}}·\left[1,2\right] \\ \frac{\mathcal{G}\left(ⱬ,\mathrm{ȶ}\right)+\mathcal{G}\left(\sigma ,\mathrm{ȶ}\right)+\mathcal{G}\left(ⱬ,\theta \right)+\mathcal{G}\left(\sigma ,\theta \right)}{4}=\left(\frac{9}{2}-2\sqrt{2}\right)·\left[1,2\right]. \end{gathered}$$

That is

$$\begin{gathered} \left[1,2\right]{\le }_{p}2-\frac{\sqrt{2}}{4}-\frac{\sqrt{2}}{8}\mathrm{ʎ}·\left[1,2\right]{\le }_p\frac{33}{8}-\sqrt{2}-\frac{\sqrt{2}}{2}\mathrm{ʎ}+\frac{\mathrm{ʎ}}{8}+\frac{{\mathrm{ʎ}}^2}{32}· \\ \left[1,2\right]{\le }_p\frac{34\sqrt{2}+\left(\sqrt{2}-4\right)\mathrm{ʎ}-24}{8\sqrt{2}}·\left[1,2\right]{\le }_p\left(\frac{9}{2}-2\sqrt{2}\right)·\left[1,2\right]. \end{gathered}$$

Hence, Theorem 5 has been verified.

Remark 3.

Assuming that $\mathfrak{i}=1$ and $\mathfrak{j}=1$, and $\hslash \left(\upsilon \right)=\upsilon , \hslash \left(\kappa \right)=\kappa$, then from (28), the following inequality emerges, see [53]:

$$\begin{gathered} \mathcal{G}\left(\frac{2\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2}, \frac{2ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right) \\ {\le }_p\frac{\begin{array}{c} \\ 1\end{array}}{2}\left[\frac{1}{{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{\int }_{\mathrm{ȵ}}^{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}\mathcal{G}\left(x,\frac{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right)dx \\ +\frac{1}{{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{\int }_{ⱬ}^{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}\mathcal{G}\left(\frac{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2},y\right)dy\right] \\ {\le }_p\frac{1}{\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}{\int }_{\mathrm{ȵ}}^{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{\int }_{ⱬ}^{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}\mathcal{G}\left(x,y\right)dydx \\ {\le }_p\frac{\begin{array}{c} \\ 1\end{array}}{4\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}\left[{\int }_{\mathrm{ȵ}}^{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}\mathcal{G}\left(x,ⱬ\right)dx+{\int }_{\mathrm{ȵ}}^{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}\mathcal{G}\left(x,\theta \right)dx\right] \\ +\frac{\begin{array}{c} \\ 1\end{array}}{4\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}\left[{\int }_{ⱬ}^{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}\mathcal{G}\left(\mathrm{ȵ},y\right)dy+{\int }_{ⱬ}^{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}\mathcal{G}\left(\mathrm{ȶ},y\right)dy\right] \\ {\le }_p\frac{\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)+\mathcal{G}\left(\mathrm{ȶ},ⱬ\right)+\mathcal{G}\left(\mathrm{ȵ},\theta \right)+\mathcal{G}\left(\mathrm{ȶ},\theta \right)}{4}. \end{gathered}$$

(52)

Assuming that $\mathfrak{i}=1$ and $\mathfrak{j}=1$, $\hslash \left(\upsilon \right)=\upsilon , \hslash \left(\kappa \right)=\kappa$, and $\mathcal{G}$ is left-$\mathcal{C}$·$\mathsf{L}$·$\mathsf{R}$-$\mathbf{\hslash }$- pre-invex, then from (28), the following inequality emerges, see [35]:

$$\begin{gathered} \mathcal{G}\left(\frac{2\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2}, \frac{2ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right) \\ \supseteq \frac{\begin{array}{c} \\ 1\end{array}}{2}\left[\frac{1}{{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{\int }_{\mathrm{ȵ}}^{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}\mathcal{G}\left(x,\frac{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right)dx \\ +\frac{1}{{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{\int }_{ⱬ}^{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}\mathcal{G}\left(\frac{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2},y\right)dy\right] \\ \supseteq \frac{1}{\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}{\int }_{\mathrm{ȵ}}^{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{\int }_{ⱬ}^{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}\mathcal{G}\left(x,y\right)dydx \\ \supseteq \frac{\begin{array}{c} \\ 1\end{array}}{4\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}\left[{\int }_{\mathrm{ȵ}}^{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}\mathcal{G}\left(x,ⱬ\right)dx+{\int }_{\mathrm{ȵ}}^{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}\mathcal{G}\left(x,\theta \right)dx\right] \\ +\frac{\begin{array}{c} \\ 1\end{array}}{4\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}\left[{\int }_{ⱬ}^{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}\mathcal{G}\left(\mathrm{ȵ},y\right)dy+{\int }_{ⱬ}^{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}\mathcal{G}\left(\mathrm{ȶ},y\right)dy\right] \\ \supseteq \frac{\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)+\mathcal{G}\left(\mathrm{ȶ},ⱬ\right)+\mathcal{G}\left(\mathrm{ȵ},\theta \right)+\mathcal{G}\left(\mathrm{ȶ},\theta \right)}{4}. \end{gathered}$$

(53)

If $\hslash \left(\upsilon \right)=\upsilon , \hslash \left(\kappa \right)=\kappa$, and ${\mathcal{G}}_{\ast }\left(x,y\right)\ne {\mathcal{G}}^{\ast }\left(x,y\right)$, then we can derive the following inequality from (28); for further information, see [53]:

$$\begin{gathered} \mathcal{G}\left(\frac{2\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2}, \frac{2ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right) \\ {\le }_p\frac{\Gamma \left(\mathfrak{i}+1\right)}{{4\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}}\left[{\mathcal{I}}_{{\mathrm{ȵ}}^{+}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),\frac{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right)+{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ},\frac{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right)\right]+ \\ \frac{\Gamma \left(\mathfrak{j}+1\right)}{{4\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[{\mathcal{I}}_{{ⱬ}^{+}}^{\mathfrak{j}}\mathcal{G}\left(\frac{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+{\mathcal{I}}_{{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{j}}\mathcal{G}\left(\frac{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2},ⱬ\right)\right] \\ {\le }_p\frac{\Gamma \left(\mathfrak{i}+1\right)\Gamma \left(\mathfrak{j}+1\right)}{{4\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}{\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}} \left[{\mathcal{I}}_{{\mathrm{ȵ}}^{+},{ⱬ}^{+}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+{\mathcal{I}}_{{\mathrm{ȵ}}^{+},{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+ \\ {\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right)+{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-},{ⱬ}^{+}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-},{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)\right] \\ {\le }_p\frac{\Gamma \left(\mathfrak{i}+1\right)}{{8\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}}\left[{\mathcal{I}}_{{\mathrm{ȵ}}^{+}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right)+{\mathcal{I}}_{{\mathrm{ȵ}}^{+}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+ \\ {\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)+{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\right] \\ +\frac{\Gamma \left(\mathfrak{j}+1\right)}{{8\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[{\mathcal{I}}_{{ⱬ}^{+} }^{\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+{\mathcal{I}}_{{ⱬ+ \\ {\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)+{\mathcal{I}}_{{ⱬ}^{+}}^{\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ+ \\ {\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+{\mathcal{I}}_{{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right)\right] \\ {\le }_p\frac{\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)+\mathcal{G}\left(\mathrm{ȶ},ⱬ\right)+\mathcal{G}\left(\mathrm{ȵ},\theta \right)+\mathcal{G}\left(\mathrm{ȶ},\theta \right)}{4}. \end{gathered}$$

(54)

If $\hslash \left(\upsilon \right)=\upsilon , \hslash \left(\kappa \right)=\kappa$, and ${\mathcal{G}}_{\ast }\left(x,y\right)\ne {\mathcal{G}}^{\ast }\left(x,y\right)$, then we can derive the following inequality from (28); for further information, see [58]:

$$\begin{gathered} \mathcal{G}\left(\frac{2\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2}, \frac{2ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right) \\ {\le }_p\frac{\begin{array}{c} \\ 1\end{array}}{2}\left[\frac{1}{{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{\int }_{\mathrm{ȵ}}^{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}\mathcal{G}\left(x,\frac{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right)dx+ \\ \frac{1}{{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{\int }_{ⱬ}^{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}\mathcal{G}\left(\frac{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2},y\right)dy\right]{\le }_p\frac{1}{\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)} {\int }_{\mathrm{ȵ}}^{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{\int }_{ⱬ}^{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}\mathcal{G}\left(x,y\right)dydx \\ {\le }_p\frac{\begin{array}{c} \\ 1\end{array}}{4\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}\left[{\int }_{\mathrm{ȵ}}^{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}\mathcal{G}\left(x,ⱬ\right)dx+{\int }_{\mathrm{ȵ}}^{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}\mathcal{G}\left(x,\theta \right)dx\right]+ \\ \frac{\begin{array}{c} \\ 1\end{array}}{4\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}\left[{\int }_{ⱬ}^{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}\mathcal{G}\left(\mathrm{ȵ},y\right)dy+{\int }_{ⱬ}^{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}\mathcal{G}\left(\mathrm{ȶ},y\right)dy\right]{\le }_p\frac{\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)+\mathcal{G}\left(\mathrm{ȶ},ⱬ\right)+\mathcal{G}\left(\mathrm{ȵ},\theta \right)+\mathcal{G}\left(\mathrm{ȶ},\theta \right)}{4}. \end{gathered}$$

(55)

If $\mathcal{G}$ is $\mathcal{C}$·$\mathsf{L}$·$\mathsf{R}$-$\mathbf{\hslash }$- pre-invex with $\hslash \left(\upsilon \right)=\upsilon , \hslash \left(\kappa \right)=\kappa$, and ${\mathcal{G}}_{\ast }\left(x,y\right)={\mathcal{G}}^{\ast }\left(x,y\right)$, then we can derive the following classical inequality from (28):

$$\begin{gathered} \mathcal{G}\left(\frac{2\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2}, \frac{2ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right) \\ \le \frac{\Gamma \left(\mathfrak{i}+1\right)}{{4\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}}\left[{\mathcal{I}}_{{\mathrm{ȵ}}^{+}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),\frac{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right) \\ +{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ},\frac{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right)\right] \\ +\frac{\Gamma \left(\mathfrak{j}+1\right)}{{4\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[{\mathcal{I}}_{{ⱬ}^{+}}^{\mathfrak{j}}\mathcal{G}\left(\frac{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right) \\ +{\mathcal{I}}_{{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{j}}\mathcal{G}\left(\frac{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2},ⱬ\right)\right] \\ \le \frac{\Gamma \left(\mathfrak{i}+1\right)\Gamma \left(\mathfrak{j}+1\right)}{{4\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}{\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[{\mathcal{I}}_{{\mathrm{ȵ}}^{+},{ⱬ}^{+}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right) \\ +{\mathcal{I}}_{{\mathrm{ȵ}}^{+},{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right) \\ +{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-},{ⱬ}^{+}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-},{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)\right] \\ \le \frac{\Gamma \left(\mathfrak{i}+1\right)}{{8\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}}\left[{\mathcal{I}}_{{\mathrm{ȵ}}^{+}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right)+\mathcal{G}{\mathcal{I}}_{{\mathrm{ȵ}}^{+}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right) \\ +{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)+{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-}}^{\mathfrak{i}}\mathcal{G}\left(\mathrm{ȵ},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\right] \\ +\frac{\Gamma \left(\mathfrak{j}+1\right)}{{8\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[{\mathcal{I}}_{{ⱬ}^{+} }^{\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+{\mathcal{I}}_{{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ\right) \\ +{\mathcal{I}}_{{ⱬ}^{+}}^{\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+{\mathcal{I}}_{{\theta }^{-}}^{\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right)\right] \\ \le \frac{\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)+\mathcal{G}\left(\mathrm{ȶ},ⱬ\right)+\mathcal{G}\left(\mathrm{ȵ},\theta \right)+\mathcal{G}\left(\mathrm{ȶ},\theta \right)}{4}. \end{gathered}$$

(56)

In the upcoming results, we will discover intriguing findings derived from the product of two $\mathcal{C}$·$\mathsf{L}$·$\mathsf{R}$-$\hslash$-pre-invex ${\rm I}$·$\mathsf{V}$-$\mathsf{M}$s. These findings are commonly referred to as Pachpatte’s inequalities.

Theorem 6.

Let $\mathcal{G}, \mathcal{J}:\Omega \to {\mathbb{R}}_I^{+}$ be two $\mathcal{C}$·$\mathsf{L}$·$\mathsf{R}$-$\hslash$- pre-invex ${\rm I}$·$\mathsf{V}$-$\mathsf{M}$s on $\Omega$, given by $\mathcal{G}\left(x,y\right)=\left[{\mathcal{G}}_{\ast }\left(x,y\right), {\mathcal{G}}^{\ast }\left(x,y\right)\right]$ and $\mathcal{J}\left(x,y\right)=\left[{\mathcal{J}}_{\ast }\left(x,y\right), {\mathcal{J}}^{\ast }\left(x,y\right)\right]$ for all $\left(x,y\right)\in \Omega$ and let ${\hslash }_1,{\hslash }_2:\left[0, 1\right]\to {\mathbb{R}}^{+}$. If ${\mathcal{G}\times \mathcal{J}\in \mathcal{F}\mathfrak{O}}_{\Omega }$, then the following inequalities holds:

$$\begin{gathered} \frac{\Gamma \left(\mathfrak{i}\right)\Gamma \left(\mathfrak{j}\right)}{{\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}{\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[{\mathcal{I}}_{{\mathrm{ȵ}}^{+},{ⱬ}^{+}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\times \mathcal{J}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ+ \\ {\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+{\mathcal{I}}_{{\mathrm{ȵ}}^{+},{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right)\times \mathcal{J}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right)\right] \\ +\frac{\Gamma \left(\mathfrak{i}\right)\Gamma \left(\mathfrak{j}\right)}{{\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}{\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-},{ⱬ}^{+}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\times \mathcal{J}\left(\mathrm{ȵ},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+ \\ {\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-},{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)\times \mathcal{J}\left(\mathrm{ȵ},ⱬ\right)\right] \\ {\le }_p\mathcal{M}\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right){\int }_0^1{\upsilon }^{\mathfrak{i}-1}{\kappa }^{\mathfrak{j}-1}\left[{\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(1-\kappa \right){\hslash }_2\left(1-\kappa \right) \\ +{\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(\kappa \right) \\ +{\hslash }_1\left(\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(1-\kappa \right){\hslash }_2\left(1-\kappa \right) \\ +{\hslash }_1\left(\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(\kappa \right)\right]d\upsilon d\kappa \\ +P\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right){\int }_0^1{\upsilon }^{\mathfrak{i}-1}{\kappa }^{\mathfrak{j}-1}\left[{\hslash }_1\left(\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(1-\kappa \right){\hslash }_2\left(1-\kappa \right)+ \\ {\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(1-\kappa \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_1\left(\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(\kappa \right)+ \\ {\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(\kappa \right)\right]d\upsilon d\kappa \\ +\mathcal{N}\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right){\int }_0^1{\upsilon }^{\mathfrak{i}-1}{\kappa }^{\mathfrak{j}-1}\left[{\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(1-\kappa \right) \\ +{\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(1-\kappa \right){\hslash }_2\left(\kappa \right) \\ +{\hslash }_1\left(\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(1-\kappa \right){\hslash }_2\left(\kappa \right) \\ +{\hslash }_1\left(\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(1-\kappa \right)\right]d\upsilon d\kappa \\ +Q\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right){\int }_0^1{\upsilon }^{\mathfrak{i}-1}{\kappa }^{\mathfrak{j}-1}\left[{\hslash }_1\left(\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_1\left(\upsilon \right){\hslash }_2\left(1- \\ \upsilon \right){\hslash }_1\left(1-\kappa \right){\hslash }_2\left(\kappa \right)+{\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_1\left(\upsilon \right){\hslash }_2\left(1- \\ \upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(1-\kappa \right)\right]d\upsilon d\kappa . \end{gathered}$$

(57)

If $\mathcal{G}$ is $\mathcal{C}$·$\mathsf{L}$·$\mathsf{R}$-$\mathbf{\hslash }$-pre-concave, then the inequality (57) is reversed such that

$$\begin{gathered} \frac{\Gamma \left(\mathfrak{i}\right)\Gamma \left(\mathfrak{j}\right)}{{\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}{\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[{\mathcal{I}}_{{\mathrm{ȵ}}^{+},{ⱬ}^{+}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\times \mathcal{J}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+{\mathcal{I}}_{{\mathrm{ȵ}}^{+},{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+ \\ {\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right)\times \mathcal{J}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right)\right] \\ +\frac{\Gamma \left(\mathfrak{i}\right)\Gamma \left(\mathfrak{j}\right)}{{\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}{\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-},{ⱬ}^{+}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\times \mathcal{J}\left(\mathrm{ȵ},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-},{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)\times \mathcal{J}\left(\mathrm{ȵ},ⱬ\right)\right] \\ {\ge }_p\mathcal{M}\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right){\int }_0^1{\upsilon }^{\mathfrak{i}-1}{\kappa }^{\mathfrak{j}-1}\left[{\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(1-\kappa \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(\kappa \right) \\ +{\hslash }_1\left(\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(1-\kappa \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_1\left(\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(\kappa \right)\right]d\upsilon d\kappa \\ +P\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right){\int }_0^1{\upsilon }^{\mathfrak{i}-1}{\kappa }^{\mathfrak{j}-1}\left[{\hslash }_1\left(\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(1-\kappa \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(1-\kappa \right){\hslash }_2\left(1-\kappa \right)+ \\ {\hslash }_1\left(\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(\kappa \right)+{\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(\kappa \right)\right]d\upsilon d\kappa \\ +\mathcal{N}\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right){\int }_0^1{\upsilon }^{\mathfrak{i}-1}{\kappa }^{\mathfrak{j}-1}\left[{\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(1-\kappa \right){\hslash }_2\left(\kappa \right) \\ +{\hslash }_1\left(\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(1-\kappa \right){\hslash }_2\left(\kappa \right)+{\hslash }_1\left(\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(1-\kappa \right)\right]d\upsilon d\kappa \\ +Q\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right){\int }_0^1{\upsilon }^{\mathfrak{i}-1}{\kappa }^{\mathfrak{j}-1}\left[{\hslash }_1\left(\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_1\left(\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(1-\kappa \right){\hslash }_2\left(\kappa \right)+ \\ {\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_1\left(\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(1-\kappa \right)\right]d\upsilon d\kappa . \end{gathered}$$

(58)

where

$$\begin{gathered} \mathcal{M}\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right)=\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)\times \mathcal{J}\left(\mathrm{ȵ},ⱬ\right)+\mathcal{G}\left(\mathrm{ȶ},ⱬ\right)\times \mathcal{J}\left(\mathrm{ȶ},ⱬ\right)+\mathcal{G}\left(\mathrm{ȵ},\theta \right)\times \mathcal{J}\left(\mathrm{ȵ},\theta \right)+ \\ \mathcal{G}\left(\mathrm{ȶ},\theta \right)\times \mathcal{J}\left(\mathrm{ȶ},\theta \right),P\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right)=\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)\times \mathcal{J}\left(\mathrm{ȶ},ⱬ\right)+\mathcal{G}\left(\mathrm{ȶ},ⱬ\right)\times \mathcal{J}\left(\mathrm{ȵ},ⱬ\right)+ \\ \mathcal{G}\left(\mathrm{ȵ},\theta \right)\times \mathcal{J}\left(\mathrm{ȶ},\theta \right)+\mathcal{G}\left(\mathrm{ȶ},\theta \right)\times \mathcal{J}\left(\mathrm{ȵ},\theta \right), \\ \mathcal{N}\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right)=\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)\times \mathcal{J}\left(\mathrm{ȵ},\theta \right)+\mathcal{G}\left(\mathrm{ȶ},ⱬ\right)\times \mathcal{J}\left(\mathrm{ȶ},\theta \right)+\mathcal{G}\left(\mathrm{ȵ},\theta \right)\times \mathcal{J}\left(\mathrm{ȵ},ⱬ\right) \\ +\mathcal{G}\left(\mathrm{ȶ},\theta \right)\times \mathcal{J}\left(\mathrm{ȶ},ⱬ\right), \\ Q\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right)=\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)\times \mathcal{J}\left(\mathrm{ȶ},\theta \right)+\mathcal{G}\left(\mathrm{ȶ},ⱬ\right)\times \mathcal{J}\left(\mathrm{ȵ},\theta \right)+\mathcal{G}\left(\mathrm{ȵ},\theta \right)\times \mathcal{J}\left(\mathrm{ȶ},ⱬ\right)+ \\ \mathcal{G}\left(\mathrm{ȶ},\theta \right)\times \mathcal{J}\left(\mathrm{ȵ},ⱬ\right), \end{gathered}$$

and $\mathcal{M}\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right)$, $P\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right)$, $\mathcal{N}\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right)$, and $Q\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right)$ are defined as follows:

$$\begin{gathered} \mathcal{M}\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right)=\left[{\mathcal{M}}_{\ast }\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right), {\mathcal{M}}^{\ast }\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right)\right], \\ P\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right)=\left[P_{\ast }\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right), P^{\ast }\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right)\right], \\ \mathcal{N}\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right)=\left[{\mathcal{N}}_{\ast }\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right), {\mathcal{N}}^{\ast }\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right)\right], \\ Q\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right)=\left[Q_{\ast }\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right), Q^{\ast }\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right)\right]. \end{gathered}$$

Proof. 

Let $\mathcal{G}$ and $\mathcal{J}$ be two $\mathcal{C}$·$\mathsf{L}$·$\mathsf{R}$-${\hslash }_1$ and $\mathcal{C}$·$\mathsf{L}$·$\mathsf{R}$-${\hslash }_2$- pre-invex ${\rm I}$·$\mathsf{V}$-$\mathsf{M}$s on $\left[\mathrm{ȵ}, \mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right]\times \left[ⱬ, ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right]$, respectively. Then,

$$\begin{gathered} \mathcal{G}\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right) \\ {\le }_p{\hslash }_1\left(\upsilon \right){\hslash }_1\left(\kappa \right)\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)+{\hslash }_1\left(\upsilon \right){\hslash }_1\left(1-\kappa \right)\mathcal{G}\left(\mathrm{ȵ},\theta \right)+{\hslash }_1\left(1-\upsilon \right){\hslash }_1\left(\kappa \right)\mathcal{G}\left(\mathrm{ȶ},ⱬ\right) \\ +{\hslash }_1\left(1-\upsilon \right){\hslash }_1\left(1-\kappa \right)\mathcal{G}\left(\mathrm{ȶ},\theta \right), \\ \mathcal{G}\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\kappa {\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right) \\ {\le }_p{\hslash }_1\left(\upsilon \right){\hslash }_1\left(1-\kappa \right)\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)+{\hslash }_1\left(\upsilon \right){\hslash }_1\left(\kappa \right)\mathcal{G}\left(\mathrm{ȵ},\theta \right)+{\hslash }_1\left(1-\upsilon \right){\hslash }_1\left(1-\kappa \right)\mathcal{G}\left(\mathrm{ȶ},ⱬ\right) \\ +{\hslash }_1\left(1-\upsilon \right){\hslash }_1\left(\kappa \right)\mathcal{G}\left(\mathrm{ȶ},\theta \right), \\ \mathcal{G}\left(\mathrm{ȵ}+\upsilon {\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right) \\ {\le }_p{\hslash }_1\left(1-\upsilon \right){\hslash }_1\left(\kappa \right)\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)+{\hslash }_1\left(1-\upsilon \right){\hslash }_1\left(1-\kappa \right)\mathcal{G}\left(\mathrm{ȵ},\theta \right)+{\hslash }_1\left(\upsilon \right){\hslash }_1\left(\kappa \right)\mathcal{G}\left(\mathrm{ȶ},ⱬ\right) \\ +{\hslash }_1\left(\upsilon \right){\hslash }_1\left(1-\kappa \right)\mathcal{G}\left(\mathrm{ȶ},\theta \right), \\ \mathcal{G}\left(\mathrm{ȵ}+\upsilon {\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\kappa {\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right) \\ {\le }_p{\hslash }_1\left(1-\upsilon \right){\hslash }_1\left(1-\kappa \right)\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)+{\hslash }_1\left(1-\upsilon \right){\hslash }_1\left(\kappa \right)\mathcal{G}\left(\mathrm{ȵ},\theta \right)+{\hslash }_1\left(\upsilon \right){\hslash }_1\left(1-\kappa \right)\mathcal{G}\left(\mathrm{ȶ},ⱬ\right) \\ +{\hslash }_1\left(\upsilon \right){\hslash }_1\left(\kappa \right)\mathcal{G}\left(\mathrm{ȶ},\theta \right), \end{gathered}$$

and

$$\begin{gathered} \mathcal{J}\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right) \\ {\le }_p{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)\mathcal{J}\left(\mathrm{ȵ},ⱬ\right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)\mathcal{J}\left(\mathrm{ȵ},\theta \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)\mathcal{J}\left(\mathrm{ȶ},ⱬ\right) \\ +{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)\mathcal{J}\left(\mathrm{ȶ},\theta \right), \\ \mathcal{J}\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\kappa {\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right) \\ {\le }_p{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)\mathcal{J}\left(\mathrm{ȵ},ⱬ\right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)\mathcal{J}\left(\mathrm{ȵ},\theta \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)\mathcal{J}\left(\mathrm{ȶ},ⱬ\right) \\ +{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)\mathcal{J}\left(\mathrm{ȶ},\theta \right), \\ \mathcal{J}\left(\mathrm{ȵ}+\upsilon {\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right) \\ {\le }_p{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)\mathcal{J}\left(\mathrm{ȵ},ⱬ\right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)\mathcal{J}\left(\mathrm{ȵ},\theta \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)\mathcal{J}\left(\mathrm{ȶ},ⱬ\right) \\ +{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)\mathcal{J}\left(\mathrm{ȶ},\theta \right), \\ \mathcal{J}\left(\mathrm{ȵ}+\upsilon {\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\kappa {\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right) \\ {\le }_p{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)\mathcal{J}\left(\mathrm{ȵ},ⱬ\right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)\mathcal{J}\left(\mathrm{ȵ},\theta \right) \\ +{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)\mathcal{J}\left(\mathrm{ȶ},ⱬ\right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)\mathcal{J}\left(\mathrm{ȶ},\theta \right), \end{gathered}$$

Since $\mathcal{G}$ and $\mathcal{J}$ both are $\mathcal{C}$·$\mathsf{L}$·$\mathsf{R}$-${\mathrm{\hslash }}_1$ and $\mathcal{C}$·$\mathsf{L}$·$\mathsf{R}$-${\mathrm{\hslash }}_2$- pre-invex ${\rm I}$·$\mathsf{V}$-$\mathsf{M}$s on $\left[\mathrm{ȵ},\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right]\times \left[\mathrm{ⱬ},\mathrm{ⱬ}+{\mathrm{ʎ}}_2\left(\theta ,\mathrm{ⱬ}\right)\right]$, we have

$$\begin{gathered} \mathcal{G}\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right) \\ \times \mathcal{J}\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right) \\ +\mathcal{G}\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\kappa {\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\times \mathcal{J}\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\kappa {\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right) \\ +\mathcal{G}\left(\mathrm{ȵ}+\upsilon {\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right) \\ \times \mathcal{J}\left(\mathrm{ȵ}+\upsilon {\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right) \\ +\mathcal{G}\left(\mathrm{ȵ}+\upsilon {\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\kappa {\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\times \mathcal{J}\left(\mathrm{ȵ}+\upsilon {\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\kappa {\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right) \\ {\le }_p\mathcal{M}\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right)\left[{\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(1-\kappa \right){\hslash }_2\left(1-\kappa \right) \\ +{\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(\kappa \right) \\ +{\hslash }_1\left(\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(1-\kappa \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_1\left(\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(\kappa \right)\right] \\ +P\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right)\left[{\hslash }_1\left(\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(1-\kappa \right){\hslash }_2\left(1-\kappa \right) \\ +{\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(1-\kappa \right){\hslash }_2\left(1-\kappa \right) \\ +{\hslash }_1\left(\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(\kappa \right)+{\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(\kappa \right)\right] \\ +\mathcal{N}\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right)\left[{\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(1-\kappa \right) \\ +{\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(1-\kappa \right){\hslash }_2\left(\kappa \right) \\ +{\hslash }_1\left(\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(1-\kappa \right){\hslash }_2\left(\kappa \right)+{\hslash }_1\left(\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(1-\kappa \right)\right] \\ +Q\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right)\left[{\hslash }_1\left(\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_1\left(\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(1-\kappa \right){\hslash }_2\left(\kappa \right) \\ +{\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(1-\kappa \right) \\ +{\hslash }_1\left(\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(1-\kappa \right)\right]. \end{gathered}$$

Taking the multiplication of the above fuzzy inclusion with ${\upsilon }^{\mathfrak{i}-1}{\kappa }^{\mathfrak{j}-1}$ and then taking the double integration of the resultant over $\left[0, 1\right]\times \left[0, 1\right]$ with respect to ($\upsilon , \kappa$) such that

$$\begin{gathered} {\int }_0^1{\int }_0^1{\upsilon }^{\mathfrak{i}-1}{\kappa }^{\mathfrak{j}-1}\mathcal{G}\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\times \mathcal{J}\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)d\upsilon d\kappa \\ +{\int }_0^1{\int }_0^1{\upsilon }^{\mathfrak{i}-1}{\kappa }^{\mathfrak{j}-1}\mathcal{G}\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\kappa {\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\times \mathcal{J}\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\kappa {\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)d\upsilon d\kappa \\ +{\int }_0^1{\int }_0^1{\upsilon }^{\mathfrak{i}-1}{\kappa }^{\mathfrak{j}-1}\mathcal{G}\left(\mathrm{ȵ}+\upsilon {\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\times \mathcal{J}\left(\mathrm{ȵ}+\upsilon {\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)d\upsilon d\kappa \\ +{\int }_0^1{\int }_0^1{\upsilon }^{\mathfrak{i}-1}{\kappa }^{\mathfrak{j}-1}\mathcal{G}\left(\mathrm{ȵ}+\upsilon {\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\kappa {\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\times \mathcal{J}\left(\mathrm{ȵ}+\upsilon {\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\kappa {\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)d\upsilon d\kappa \\ {\le }_p\mathcal{M}\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right){\int }_0^1{\int }_0^1{\upsilon }^{\mathfrak{i}-1}{\kappa }^{\mathfrak{j}-1}\left[{\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(1-\kappa \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(\kappa \right) \\ +{\hslash }_1\left(\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(1-\kappa \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_1\left(\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(\kappa \right)\right]d\upsilon d\kappa \\ +P\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right){\int }_0^1{\int }_0^1{\upsilon }^{\mathfrak{i}-1}{\kappa }^{\mathfrak{j}-1}\left[{\hslash }_1\left(\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(1-\kappa \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(1-\kappa \right){\hslash }_2\left(1-\kappa \right) \\ +{\hslash }_1\left(\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(\kappa \right)+{\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(\kappa \right)\right]d\upsilon d\kappa \\ +\mathcal{N}\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right){\int }_0^1{\int }_0^1{\upsilon }^{\mathfrak{i}-1}{\kappa }^{\mathfrak{j}-1}\left[{\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(1-\kappa \right){\hslash }_2\left(\kappa \right) \\ +{\hslash }_1\left(\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(1-\kappa \right){\hslash }_2\left(\kappa \right)+{\hslash }_1\left(\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(1-\kappa \right)\right]d\upsilon d\kappa \\ +Q\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right){\int }_0^1{\int }_0^1{\upsilon }^{\mathfrak{i}-1}{\kappa }^{\mathfrak{j}-1}\left[{\hslash }_1\left(\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_1\left(\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(1-\kappa \right){\hslash }_2\left(\kappa \right) \\ +{\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_1\left(\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(1-\kappa \right)\right]d\upsilon d\kappa \end{gathered}$$

(59)

From the right hand side of (59), we have

$$\begin{gathered} {\int }_0^1{\int }_0^1{\upsilon }^{\mathfrak{i}-1}{\kappa }^{\mathfrak{j}-1}\mathcal{G}\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right) \\ \times \mathcal{J}\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)d\upsilon d\kappa \\ +{\int }_0^1{\int }_0^1{\upsilon }^{\mathfrak{i}-1}{\kappa }^{\mathfrak{j}-1}\mathcal{G}\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\kappa {\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right) \\ \times \mathcal{J}\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\kappa {\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)d\upsilon d\kappa \\ +{\int }_0^1{\int }_0^1{\upsilon }^{\mathfrak{i}-1}{\kappa }^{\mathfrak{j}-1}\mathcal{G}\left(\mathrm{ȵ}+\upsilon {\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\times \mathcal{J}\left(\mathrm{ȵ}+\upsilon {\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+ \\ \left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)d\upsilon d\kappa \\ +{\int }_0^1{\int }_0^1{\upsilon }^{\mathfrak{i}-1}{\kappa }^{\mathfrak{j}-1}\mathcal{G}\left(\mathrm{ȵ}+\upsilon {\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\kappa {\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right) \\ \times \mathcal{J}\left(\mathrm{ȵ}+\upsilon {\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\kappa {\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)d\upsilon d\kappa \\ =\frac{\Gamma \left(\mathfrak{i}\right)\Gamma \left(\mathfrak{j}\right)}{{\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}{\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[\begin{array}{c}{\mathcal{I}}_{{\mathrm{ȵ}}^{+},{ⱬ}^{+}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\times J\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\\ +{\mathcal{I}}_{{\mathrm{ȵ}}^{+},{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right)\times J\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right)\end{array}\right]. \end{gathered}$$

(60)

Combining (59) and (60), we have

$$\begin{gathered} \frac{\Gamma \left(\mathfrak{i}\right)\Gamma \left(\mathfrak{j}\right)}{{\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}{\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[{\mathcal{I}}_{{\mathrm{ȵ}}^{+},{ⱬ}^{+}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\times \mathcal{J}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ+ \\ {\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+{\mathcal{I}}_{{\mathrm{ȵ}}^{+},{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right)\times \mathcal{J}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right)\right] \\ {\le }_p\mathcal{M}\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right){\int }_0^1{\int }_0^1{\upsilon }^{\mathfrak{i}-1}{\kappa }^{\mathfrak{j}-1}\left[{\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(1-\kappa \right){\hslash }_2\left(1-\kappa \right) \\ +{\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(\kappa \right) \\ +{\hslash }_1\left(\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(1-\kappa \right){\hslash }_2\left(1-\kappa \right) \\ +{\hslash }_1\left(\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(\kappa \right)\right]d\upsilon d\kappa \\ +P\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right){\int }_0^1{\int }_0^1{\upsilon }^{\mathfrak{i}-1}{\kappa }^{\mathfrak{j}-1}\left[{\hslash }_1\left(\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(1-\kappa \right){\hslash }_2\left(1-\kappa \right) \\ +{\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(1-\kappa \right){\hslash }_2\left(1-\kappa \right) \\ +{\hslash }_1\left(\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(\kappa \right) \\ +{\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(\kappa \right)\right]d\upsilon d\kappa \\ +\mathcal{N}\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right){\int }_0^1{\int }_0^1{\upsilon }^{\mathfrak{i}-1}{\kappa }^{\mathfrak{j}-1}\left[{\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(1-\kappa \right) \\ +{\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(1-\kappa \right){\hslash }_2\left(\kappa \right) \\ +{\hslash }_1\left(\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(1-\kappa \right){\hslash }_2\left(\kappa \right) \\ +{\hslash }_1\left(\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(1-\kappa \right)\right]d\upsilon d\kappa \\ +Q\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right){\int }_0^1{\int }_0^1{\upsilon }^{\mathfrak{i}-1}{\kappa }^{\mathfrak{j}-1}\left[{\hslash }_1\left(\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(1-\kappa \right) \\ +{\hslash }_1\left(\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(1-\kappa \right){\hslash }_2\left(\kappa \right) \\ +{\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(1-\kappa \right) \\ +{\hslash }_1\left(\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(1-\kappa \right)\right]d\upsilon d\kappa . \end{gathered}$$

Hence, the required result. □

Remark 4.

Assuming that $\mathcal{G}$is left-$\mathcal{C}$·$\mathsf{L}$·$\mathsf{R}$-$\mathit{\hslash }$-pre-invex with$\hslash \left(\upsilon \right)=\upsilon , \hslash \left(\kappa \right)=\kappa$, $\mathfrak{i}=1$, and $\mathfrak{j}=1$, then from (59), the following inequality emerges, see [54]:

$$\begin{gathered} \frac{1}{\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{\int }_{\mathrm{ȵ}}^{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{\int }_{ⱬ}^{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}\mathcal{G}\left(x,y\right)\times \mathcal{J}\left(x,y\right)dydx \\ \supseteq \frac{1}{9}\mathcal{M}\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right)+\frac{1}{18}\left[P\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right)+\mathcal{N}\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right)\right]+\frac{1}{36}Q\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right). \end{gathered}$$

(61)

If $\mathcal{G}$ is $\mathcal{C}$·$\mathsf{L}$·$\mathsf{R}$-$\mathbf{\hslash }$- pre-invex with $\hslash \left(\upsilon \right)=\upsilon , \hslash \left(\kappa \right)=\kappa$ and one assumes that $\mathfrak{i}=1$ and $\mathfrak{j}=1$, then from (59), the following inequality emerges, see [58]:

$$\begin{gathered} \frac{1}{\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{\int }_{\mathrm{ȵ}}^{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{\int }_{ⱬ}^{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}\mathcal{G}\left(x,y\right)\times \mathcal{J}\left(x,y\right)dydx \\ {\le }_p\frac{1}{9}\mathcal{M}\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right)+\frac{1}{18}\left[P\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right)+\mathcal{N}\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right)\right]+ \\ \frac{1}{36}Q\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right). \end{gathered}$$

(62)

If ${\mathcal{G}}_{\ast }\left(x,y\right)\ne {\mathcal{G}}^{\ast }\left(x,y\right)$ and $\hslash \left(\upsilon \right)=\upsilon , \hslash \left(\kappa \right)=\kappa$, then we can derive the following inequality from (57); for further information, see [53]:

$$\begin{gathered} \frac{\Gamma \left(\mathfrak{i}+1\right)\Gamma \left(\mathfrak{j}+1\right)}{{4\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}{\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[\begin{array}{c}{\mathcal{I}}_{{\mathrm{ȵ}}^{+},{ⱬ}^{+}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\times J\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),\theta \right)\\ +{\mathcal{I}}_{{\mathrm{ȵ}}^{+},{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right)\times J\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right)\end{array}\right] \\ +\frac{\Gamma \left(\mathfrak{i}+1\right)\Gamma \left(\mathfrak{j}+1\right)}{{4\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}{\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[\begin{array}{c}{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-},{ⱬ}^{+}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\times J\left(\mathrm{ȵ},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\\ +{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-},{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)\times J\left(\mathrm{ȵ},ⱬ\right)\end{array}\right] \\ {\le }_p\left(\frac{1}{2}-\frac{\mathfrak{i}}{\left(\mathfrak{i}+1\right)\left(\mathfrak{i}+2\right)}\right)\left(\frac{1}{2}-\frac{\mathfrak{j}}{\left(\mathfrak{j}+1\right)\left(\mathfrak{j}+2\right)}\right)\mathcal{M}\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right) \\ +\frac{\mathfrak{i}}{(\mathfrak{i}+1)(\mathfrak{i}+2)}\left(\frac{1}{2}-\frac{\mathfrak{j}}{(\mathfrak{j}+1)(\mathfrak{j}+2)}\right)P\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right) \\ +\left(\frac{1}{2}-\frac{\mathfrak{i}}{(\mathfrak{i}+1)(\mathfrak{i}+2)}\right)\frac{\mathfrak{j}}{(\mathfrak{j}+1)(\mathfrak{j}+2)}\mathcal{N}\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right)+\frac{\mathfrak{j}}{(\mathfrak{j}+1)(\mathfrak{j}+2)}\frac{\mathfrak{i}}{(\mathfrak{i}+1)(\mathfrak{i}+2)}Q\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right). \end{gathered}$$

(63)

If $\hslash \left(\upsilon \right)=\upsilon , \hslash \left(\kappa \right)=\kappa$, and ${\mathcal{G}}_{\ast }\left(x,y\right)\ne {\mathcal{G}}^{\ast }\left(x,y\right)$, then we can derive the following inequality from (57); for further information, see [46]:

$$\begin{gathered} \frac{1}{\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{\int }_{\mathrm{ȵ}}^{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{\int }_{ⱬ}^{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}\mathcal{G}\left(x,y\right)\times \mathcal{J}\left(x,y\right)dydx \\ {\le }_p\frac{1}{9}\mathcal{M}\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right)+\frac{1}{18}\left[P\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right)+\mathcal{N}\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right)\right]+\frac{1}{36}Q\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right). \end{gathered}$$

(64)

If ${\mathcal{G}}_{\ast }\left(x,y\right)={\mathcal{G}}^{\ast }\left(x,y\right)$ and ${\mathcal{J}}_{\ast }\left(x,y\right)={\mathcal{J}}^{\ast }\left(x,y\right)$ and $\hslash \left(\upsilon \right)=\upsilon , \hslash \left(\kappa \right)=\kappa$, then we can derive the following classical inequality from (66):

$$\begin{gathered} \frac{\Gamma \left(\mathfrak{i}+1\right)\Gamma \left(\mathfrak{j}+1\right)}{{4\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}{\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[\begin{array}{c}{\mathcal{I}}_{{\mathrm{ȵ}}^{+},{ⱬ}^{+}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\times J\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\\ +{\mathcal{I}}_{{\mathrm{ȵ}}^{+},{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right)\times J\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right)\end{array}\right] \\ +\frac{\Gamma \left(\mathfrak{i}+1\right)\Gamma \left(\mathfrak{j}+1\right)}{{4\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}{\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}} \left[\begin{array}{c}+{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-},{ⱬ}^{+}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\times J\left(\mathrm{ȵ},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\\ +{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-},{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)\times J\left(\mathrm{ȵ},ⱬ\right)\end{array}\right] \\ \le \left(\frac{1}{2}-\frac{\mathfrak{i}}{\left(\mathfrak{i}+1\right)\left(\mathfrak{i}+2\right)}\right)\left(\frac{1}{2}-\frac{\mathfrak{j}}{\left(\mathfrak{j}+1\right)\left(\mathfrak{j}+2\right)}\right)\mathcal{M}\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right) \\ +\frac{\mathfrak{i}}{(\mathfrak{i}+1)(\mathfrak{i}+2)}\left(\frac{1}{2}-\frac{\mathfrak{j}}{(\mathfrak{j}+1)(\mathfrak{j}+2)}\right)P\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right) \\ +\left(\frac{1}{2}-\frac{\mathfrak{i}}{(\mathfrak{i}+1)(\mathfrak{i}+2)}\right)\frac{\mathfrak{j}}{(\mathfrak{j}+1)(\mathfrak{j}+2)}\mathcal{N}\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right)+\frac{\mathfrak{j}}{(\mathfrak{j}+1)(\mathfrak{j}+2)}\frac{\mathfrak{i}}{(\mathfrak{i}+1)(\mathfrak{i}+2)}Q\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right). \end{gathered}$$

(65)

Theorem 7.

Let $\mathcal{G}, \mathcal{J}:\Omega \to {\mathbb{R}}_I^{+}$ be two $\mathcal{C}$·$\mathsf{L}$·$\mathsf{R}$-$\hslash$-pre-invex ${\rm I}$·$\mathsf{V}$-$\mathsf{M}$s on $\Omega$, given by $\mathcal{G}\left(x,y\right)=\left[{\mathcal{G}}_{\ast }\left(x,y\right), {\mathcal{G}}^{\ast }\left(x,y\right)\right]$ and $\mathcal{J}\left(x,y\right)=\left[{\mathcal{J}}_{\ast }\left(x,y\right), {\mathcal{J}}^{\ast }\left(x,y\right)\right]$ for all $\left(x,y\right)\in \Omega$ and let $\hslash :\left[0, 1\right]\to {\mathbb{R}}^{+}$. If ${\mathcal{G}\times \mathcal{J}\in \mathcal{F}\mathfrak{O}}_{\Omega }$, then the following inequalities holds:

$$\begin{gathered} \frac{1}{2\mathfrak{i}\mathfrak{j}{{\hslash }_1}^2\left(\frac{1}{2}\right){{\hslash }_2}^2\left(\frac{1}{2}\right)}\mathcal{G}\left(\frac{2\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2}, \frac{2ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right)\times \mathcal{J}\left(\frac{2\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2}, \frac{2ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right) \\ {\le }_p\frac{\Gamma \left(\mathfrak{i}\right)\Gamma \left(\mathfrak{j}\right)}{{2\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}{\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[{\mathcal{I}}_{{\mathrm{ȵ}}^{+},{ⱬ}^{+}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\times \mathcal{J}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+ \\ {\mathcal{I}}_{{\mathrm{ȵ}}^{+},{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right)\times \mathcal{J}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right)\right] \\ +\frac{\Gamma \left(\mathfrak{i}\right)\Gamma \left(\mathfrak{j}\right)}{2{\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}{\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-},{ⱬ}^{+}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\times \mathcal{J}\left(\mathrm{ȵ},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right) \\ +{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-},{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)\times \mathcal{J}\left(\mathrm{ȵ},ⱬ\right)\right] \\ +\mathcal{M}\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right){\int }_0^1{\upsilon }^{\mathfrak{i}-1}{\kappa }^{\mathfrak{j}-1}\left[{\hslash }_1\left(\upsilon \right){\hslash }_1\left(\kappa \right)\left[{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right) \\ +{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)\right] \\ +{\hslash }_1\left(\upsilon \right){\hslash }_1\left(1-\kappa \right)\left[{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)\right]\right]d\upsilon d\kappa \\ +P\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right){\int }_0^1{\upsilon }^{\mathfrak{i}-1}{\kappa }^{\mathfrak{j}-1}\left[{\hslash }_1\left(\upsilon \right){\hslash }_1\left(\kappa \right)\left[{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)\right] \\ +{\hslash }_1\left(\upsilon \right){\hslash }_1\left(1-\kappa \right)\left[{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)\right]\right]d\upsilon d\kappa \\ +\mathcal{N}\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right){\int }_0^1{\upsilon }^{\mathfrak{i}-1}{\kappa }^{\mathfrak{j}-1}\left[{\hslash }_1\left(\upsilon \right){\hslash }_1\left(\kappa \right)\left[{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)\right] \\ +{\hslash }_1\left(\upsilon \right){\hslash }_1\left(1-\kappa \right)\left[{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)\right]\right]d\upsilon d\kappa \\ +Q\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right){\int }_0^1{\upsilon }^{\mathfrak{i}-1}{\kappa }^{\mathfrak{j}-1}\left[{\hslash }_1\left(\upsilon \right){\hslash }_1\left(\kappa \right)\left[{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)\right]+ \\ {\hslash }_1\left(\upsilon \right){\hslash }_1\left(1-\kappa \right)\left[{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)\right]\right]d\upsilon d\kappa . \end{gathered}$$

(66)

If $\mathcal{G}$ is $\mathcal{C}$·$\mathsf{L}$·$\mathsf{R}$-$\mathbf{\hslash }$-pre-concave, then the inequality (66) is reversed such that

$$\begin{gathered} \frac{1}{2\mathfrak{i}\mathfrak{j}{{\hslash }_1}^2\left(\frac{1}{2}\right){{\hslash }_2}^2\left(\frac{1}{2}\right)}\mathcal{G}\left(\frac{2\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2}, \frac{2ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right)\times \mathcal{J}\left(\frac{2\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2}, \frac{2ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right) \\ {\ge }_p\frac{\Gamma \left(\mathfrak{i}\right)\Gamma \left(\mathfrak{j}\right)}{{2\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}{\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[{\mathcal{I}}_{{\mathrm{ȵ}}^{+},{ⱬ}^{+}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\times \mathcal{J}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ+ \\ {\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+{\mathcal{I}}_{{\mathrm{ȵ}}^{+},{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right)\times \mathcal{J}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right)\right] \\ +\frac{\Gamma \left(\mathfrak{i}\right)\Gamma \left(\mathfrak{j}\right)}{2{\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}{\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-},{ⱬ}^{+}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\times \mathcal{J}\left(\mathrm{ȵ},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right) \\ +{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-},{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)\times \mathcal{J}\left(\mathrm{ȵ},ⱬ\right)\right] \\ +\mathcal{M}\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right){\int }_0^1{\upsilon }^{\mathfrak{i}-1}{\kappa }^{\mathfrak{j}-1}\left[{\hslash }_1\left(\upsilon \right){\hslash }_1\left(\kappa \right)\left[{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right) \\ +{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)\right] \\ +{\hslash }_1\left(\upsilon \right){\hslash }_1\left(1-\kappa \right)\left[{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right) \\ +{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)\right]\right]d\upsilon d\kappa \\ +P\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right){\int }_0^1{\upsilon }^{\mathfrak{i}-1}{\kappa }^{\mathfrak{j}-1}\left[{\hslash }_1\left(\upsilon \right){\hslash }_1\left(\kappa \right)\left[{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)+ \\ {\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)\right]+{\hslash }_1\left(\upsilon \right){\hslash }_1\left(1-\kappa \right)\left[{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)+ \\ {\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)\right]\right]d\upsilon d\kappa \\ +\mathcal{N}\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right){\int }_0^1{\upsilon }^{\mathfrak{i}-1}{\kappa }^{\mathfrak{j}-1}\left[{\hslash }_1\left(\upsilon \right){\hslash }_1\left(\kappa \right)\left[{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right) \\ +{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)\right] \\ +{\hslash }_1\left(\upsilon \right){\hslash }_1\left(1-\kappa \right)\left[{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right) \\ +{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)\right]\right]d\upsilon d\kappa \\ +Q\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right){\int }_0^1{\upsilon }^{\mathfrak{i}-1}{\kappa }^{\mathfrak{j}-1}\left[{\hslash }_1\left(\upsilon \right){\hslash }_1\left(\kappa \right)\left[{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)+ \\ {\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)\right]+{\hslash }_1\left(\upsilon \right){\hslash }_1\left(1-\kappa \right)\left[{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)+ \\ {\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)\right]\right]d\upsilon d\kappa . \end{gathered}$$

(67)

where $\mathcal{M}\left(\mathrm{ȵ},\mathrm{ȶ},\mathrm{ⱬ},\theta \right)$, $P\left(\mathrm{ȵ},\mathrm{ȶ},\mathrm{ⱬ},\theta \right)$, $\mathcal{N}\left(\mathrm{ȵ},\mathrm{ȶ},\mathrm{ⱬ},\theta \right)$, and $Q\left(\mathrm{ȵ},\mathrm{ȶ},\mathrm{ⱬ},\theta \right)$ are given in Theorem 6.

Proof. 

Since $\mathcal{G},\mathcal{J}:\Omega \to {\mathbb{R}}_I^{+}$ be two $\mathsf{L}\mathsf{R}$-$\hslash$- pre-invex ${\rm I}$·$\mathsf{V}$-$\mathsf{M}$s, then from the inequality (17)$,$ we have

$$\begin{gathered} \mathcal{G}\left(\frac{2\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2},\frac{2ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right)\times \mathcal{J}\left(\frac{2\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2},\frac{2ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right) \\ =\mathcal{G}\left(\frac{\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2}+\frac{\mathrm{ȵ}+\upsilon {\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2},\frac{ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}+\frac{ⱬ+\kappa {\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right) \\ \times \mathcal{J}\left(\frac{\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2}+\frac{\mathrm{ȵ}+\upsilon {\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2},\frac{ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2} \\ +\frac{ⱬ+\kappa {\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right) \\ {\le }_p{{\hslash }_1}^2\left(\frac{1}{2}\right){{\hslash }_2}^2\left(\frac{1}{2}\right) \\ \times \left[\begin{array}{c}\mathcal{G}\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+\mathcal{G}\left(\mathrm{ȵ}+\upsilon {\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\\ +\mathcal{G}\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\kappa {\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+\mathcal{G}\left(\mathrm{ȵ}+\upsilon {\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\kappa {\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\end{array}\right] \\ \times \left[\begin{array}{c}J\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+J\left(\mathrm{ȵ}+\upsilon {\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\\ +J\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\kappa {\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+J\left(\mathrm{ȵ}+\upsilon {\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\kappa {\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\end{array}\right] \\ {\le }_p{{\hslash }_1}^2\left(\frac{1}{2}\right){{\hslash }_2}^2\left(\frac{1}{2}\right) \\ \times \left[\begin{array}{c}\mathcal{G}\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\times J\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\\ +\mathcal{G}\left(\mathrm{ȵ}+\upsilon {\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\times J\left(\mathrm{ȵ}+\upsilon {\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\\ +\mathcal{G}\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\kappa {\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\times J\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\kappa {\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\\ +\mathcal{G}\left(\mathrm{ȵ}+\upsilon {\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\kappa {\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\times J\left(\mathrm{ȵ}+\upsilon {\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\kappa {\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\end{array}\right] \\ +{{\hslash }_1}^2\left(\frac{1}{2}\right){{\hslash }_2}^2\left(\frac{1}{2}\right) \\ \times \left[\begin{array}{c}{\hslash }_1\left(\upsilon \right){\hslash }_1\left(\kappa \right)\left[{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)\right]\\ +{\hslash }_1\left(\upsilon \right){\hslash }_1\left(1-\kappa \right)\left[{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)\right]\\ \begin{array}{c}+{\hslash }_1\left(1-\upsilon \right){\hslash }_1\left(\kappa \right)\left[{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)\right]\\ +{\hslash }_1\left(1-\upsilon \right){\hslash }_1\left(1-\kappa \right)\left[{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)\right]\end{array}\end{array}\right]\mathcal{M}\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right) \\ +{{\hslash }_1}^2\left(\frac{1}{2}\right){{\hslash }_2}^2\left(\frac{1}{2}\right) \\ \times \left[\begin{array}{c}{\hslash }_1\left(\upsilon \right){\hslash }_1\left(\kappa \right)\left[{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)\right]\\ +{\hslash }_1\left(\upsilon \right){\hslash }_1\left(1-\kappa \right)\left[{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)\right]\\ +{\hslash }_1\left(1-\upsilon \right){\hslash }_1\left(\kappa \right)\left[{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)\right]\\ +{\hslash }_1\left(1-\upsilon \right){\hslash }_1\left(1-\kappa \right)\left[{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)\right]\end{array}\right]P\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right) \\ +{{\hslash }_1}^2\left(\frac{1}{2}\right){{\hslash }_2}^2\left(\frac{1}{2}\right) \\ \times \left[\begin{array}{c}{\hslash }_1\left(\upsilon \right){\hslash }_1\left(\kappa \right)\left[{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)\right]\\ +{\hslash }_1\left(\upsilon \right){\hslash }_1\left(1-\kappa \right)\left[{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)\right]\\ +{\hslash }_1\left(1-\upsilon \right){\hslash }_1\left(\kappa \right)\left[{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)\right]\\ +{\hslash }_1\left(1-\upsilon \right){\hslash }_1\left(1-\kappa \right)\left[{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)\right]\end{array}\right]\mathcal{N}\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right) \\ +{{\hslash }_1}^2\left(\frac{1}{2}\right){{\hslash }_2}^2\left(\frac{1}{2}\right) \\ \times \left[\begin{array}{c}{\hslash }_1\left(\upsilon \right){\hslash }_1\left(\kappa \right)\left[{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)\right]\\ +{\hslash }_1\left(\upsilon \right){\hslash }_1\left(1-\kappa \right)\left[{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)\right]\\ \begin{array}{c}+{\hslash }_1\left(1-\upsilon \right){\hslash }_1\left(\kappa \right)\left[{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)\right]\\ +{\hslash }_1\left(\upsilon \right){\hslash }_1\left(1-\kappa \right)\left[{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)\right]\end{array}\end{array}\right]Q\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right). \end{gathered}$$

Taking the multiplication of the above fuzzy inclusion with ${\upsilon }^{\mathfrak{i}-1}{\kappa }^{\mathfrak{j}-1}$ and then taking the double integration of the resultant over $\left[0, 1\right]\times \left[0, 1\right]$ with respect to ($\upsilon , \kappa$), we have

$$\begin{gathered} {\int }_0^1{\int }_0^1{\upsilon }^{\mathfrak{i}-1}{\kappa }^{\mathfrak{j}-1}\mathcal{G}\left(\frac{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2},\frac{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right)\times \mathcal{J}\left(\frac{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2},\frac{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right)d\upsilon d\kappa \\ {\le }_p{{\hslash }_1}^2\left(\frac{1}{2}\right){{\hslash }_2}^2\left(\frac{1}{2}\right) \\ \times {\int }_0^1{\int }_0^1{\upsilon }^{\mathfrak{i}-1}{\kappa }^{\mathfrak{j}-1}\left[\begin{array}{c}\mathcal{G}\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\times J\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\\ +\mathcal{G}\left(\mathrm{ȵ}+\upsilon {\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\times J\left(\mathrm{ȵ}+\upsilon {\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\left(1-\kappa \right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\\ +\mathcal{G}\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\kappa {\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\times J\left(\mathrm{ȵ}+\left(1-\upsilon \right){\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\kappa {\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\\ +\mathcal{G}\left(\mathrm{ȵ}+\upsilon {\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\kappa {\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\times J\left(\mathrm{ȵ}+\upsilon {\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right), ⱬ+\kappa {\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\end{array}\right]d\upsilon d\kappa \\ +{{\hslash }_1}^2\left(\frac{1}{2}\right){{\hslash }_2}^2\left(\frac{1}{2}\right)\mathcal{M}\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right) \\ \times {\int }_0^1{\int }_0^1{\upsilon }^{\mathfrak{i}-1}{\kappa }^{\mathfrak{j}-1}\left[\begin{array}{c}{\hslash }_1\left(\upsilon \right){\hslash }_1\left(\kappa \right)\left[{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)\right]\\ +{\hslash }_1\left(\upsilon \right){\hslash }_1\left(1-\kappa \right)\left[{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)\right]\\ \begin{array}{c}+{\hslash }_1\left(1-\upsilon \right){\hslash }_1\left(\kappa \right)\left[{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)\right]\\ +{\hslash }_1\left(1-\upsilon \right){\hslash }_1\left(1-\kappa \right)\left[{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)\right]\end{array}\end{array}\right]d\upsilon d\kappa \\ +{{\hslash }_1}^2\left(\frac{1}{2}\right){{\hslash }_2}^2\left(\frac{1}{2}\right)P\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right) \\ \times {\int }_0^1{\int }_0^1{\upsilon }^{\mathfrak{i}-1}{\kappa }^{\mathfrak{j}-1}\left[\begin{array}{c}{\hslash }_1\left(\upsilon \right){\hslash }_1\left(\kappa \right)\left[{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)\right]\\ +{\hslash }_1\left(\upsilon \right){\hslash }_1\left(1-\kappa \right)\left[{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)\right]\\ +{\hslash }_1\left(1-\upsilon \right){\hslash }_1\left(\kappa \right)\left[{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)\right]\\ +{\hslash }_1\left(1-\upsilon \right){\hslash }_1\left(1-\kappa \right)\left[{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)\right]\end{array}\right]d\upsilon d\kappa \\ +{{\hslash }_1}^2\left(\frac{1}{2}\right){{\hslash }_2}^2\left(\frac{1}{2}\right)\mathcal{N}\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right) \\ \times {\int }_0^1{\int }_0^1{\upsilon }^{\mathfrak{i}-1}{\kappa }^{\mathfrak{j}-1}\left[\begin{array}{c}{\hslash }_1\left(\upsilon \right){\hslash }_1\left(\kappa \right)\left[{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)\right]\\ +{\hslash }_1\left(\upsilon \right){\hslash }_1\left(1-\kappa \right)\left[{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)\right]\\ +{\hslash }_1\left(1-\upsilon \right){\hslash }_1\left(\kappa \right)\left[{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)\right]\\ +{\hslash }_1\left(1-\upsilon \right){\hslash }_1\left(1-\kappa \right)\left[{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)\right]\end{array}\right]d\upsilon d\kappa \\ +{{\hslash }_1}^2\left(\frac{1}{2}\right){{\hslash }_2}^2\left(\frac{1}{2}\right)Q\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right) \\ \times {\int }_0^1{\int }_0^1{\upsilon }^{\mathfrak{i}-1}{\kappa }^{\mathfrak{j}-1}\left[\begin{array}{c}{\hslash }_1\left(\upsilon \right){\hslash }_1\left(\kappa \right)\left[{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)\right]\\ +{\hslash }_1\left(\upsilon \right){\hslash }_1\left(1-\kappa \right)\left[{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)\right]\\ \begin{array}{c}+{\hslash }_1\left(1-\upsilon \right){\hslash }_1\left(\kappa \right)\left[{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)\right]\\ +{\hslash }_1\left(\upsilon \right){\hslash }_1\left(1-\kappa \right)\left[{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)\right]\end{array}\end{array}\right]d\upsilon d\kappa , \end{gathered}$$

which implies that

$$\begin{gathered} \frac{1}{\mathfrak{i}\mathfrak{j}}\mathcal{G}\left(\frac{2\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2}, \frac{2ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right)\times \mathcal{J}\left(\frac{2\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2}, \frac{2ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right) \\ {\le }_p\frac{\Gamma \left(\mathfrak{i}\right)\Gamma \left(\mathfrak{j}\right){{\hslash }_1}^2\left(\frac{1}{2}\right){{\hslash }_2}^2\left(\frac{1}{2}\right)}{{\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}{\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[{\mathcal{I}}_{{\mathrm{ȵ}}^{+},{ⱬ}^{+}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\times \mathcal{J}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)+ \\ {\mathcal{I}}_{{\mathrm{ȵ}}^{+},{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right)\times \mathcal{J}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right)\right] \\ +\frac{\Gamma \left(\mathfrak{i}\right)\Gamma \left(\mathfrak{j}\right){{\hslash }_1}^2\left(\frac{1}{2}\right){{\hslash }_2}^2\left(\frac{1}{2}\right)}{{\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}{\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-},{ⱬ}^{+}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\times \mathcal{J}\left(\mathrm{ȵ},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right) \\ +{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-},{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)\times \mathcal{J}\left(\mathrm{ȵ},ⱬ\right)\right] \\ +2{{\hslash }_1}^2\left(\frac{1}{2}\right){{\hslash }_2}^2\left(\frac{1}{2}\right)\mathcal{M}\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right){\int }_0^1{\upsilon }^{\mathfrak{i}-1}{\kappa }^{\mathfrak{j}-1}\left[{\hslash }_1\left(\upsilon \right){\hslash }_1\left(\kappa \right)\left[{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right) \\ +{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)\right] \\ +{\hslash }_1\left(\upsilon \right){\hslash }_1\left(1-\kappa \right)\left[{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right) \\ +{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)\right]\right]d\upsilon d\kappa \\ +2{{\hslash }_1}^2\left(\frac{1}{2}\right){{\hslash }_2}^2\left(\frac{1}{2}\right)P\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right){\int }_0^1{\upsilon }^{\mathfrak{i}-1}{\kappa }^{\mathfrak{j}-1}\left[{\hslash }_1\left(\upsilon \right){\hslash }_1\left(\kappa \right)\left[{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right) \\ +{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)\right] \\ +{\hslash }_1\left(\upsilon \right){\hslash }_1\left(1-\kappa \right)\left[{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right) \\ +{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)\right]\right]d\upsilon d\kappa \\ +2{{\hslash }_1}^2\left(\frac{1}{2}\right){{\hslash }_2}^2\left(\frac{1}{2}\right)\mathcal{N}\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right){\int }_0^1{\upsilon }^{\mathfrak{i}-1}{\kappa }^{\mathfrak{j}-1}\left[{\hslash }_1\left(\upsilon \right){\hslash }_1\left(\kappa \right)\left[{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right) \\ +{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)\right] \\ +{\hslash }_1\left(\upsilon \right){\hslash }_1\left(1-\kappa \right)\left[{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right) \\ +{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)\right]\right]d\upsilon d\kappa \\ +2{{\hslash }_1}^2\left(\frac{1}{2}\right){{\hslash }_2}^2\left(\frac{1}{2}\right)Q\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right){\int }_0^1{\upsilon }^{\mathfrak{i}-1}{\kappa }^{\mathfrak{j}-1}\left[{\hslash }_1\left(\upsilon \right){\hslash }_1\left(\kappa \right)\left[{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)+ \\ {\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)\right]+{\hslash }_1\left(\upsilon \right){\hslash }_1\left(1-\kappa \right)\left[{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)+ \\ {\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)\right]\right]d\upsilon d\kappa , \end{gathered}$$

hence, the required result. □

Remark 5.

Assuming that $\mathcal{G}$ is left-$\mathcal{C}$·$\mathsf{L}$·$\mathsf{R}$-$\mathit{\hslash }$- pre-invex with $\hslash \left(\upsilon \right)=\upsilon , \hslash \left(\kappa \right)=\kappa$ and $\mathfrak{i}=1$ and $\mathfrak{j}=1$, then from (66), the following inequality emerges, see [35]:

$$\begin{gathered} 4\mathcal{G}\left(\frac{2\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2},\frac{2ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right)\times \mathcal{J}\left(\frac{2\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2},\frac{2ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right) \\ \supseteq \frac{1}{\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{\int }_{\mathrm{ȵ}}^{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{\int }_{ⱬ}^{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}\mathcal{G}\left(x,y\right)\times \mathcal{J}\left(x,y\right)dydx \\ +\frac{5}{36}\mathcal{M}\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right) \\ +\frac{7}{36}\left[P\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right)+\mathcal{N}\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right)\right]+\frac{2}{9}Q\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right). \end{gathered}$$

(68)

If $\mathcal{G}$ is $\mathcal{C}$·$\mathsf{L}$·$\mathsf{R}$-$\mathbf{\hslash }$-pre-invex with $\hslash \left(\upsilon \right)=\upsilon , \hslash \left(\kappa \right)=\kappa$ and one assumes that $\mathfrak{i}=1$ and $\mathfrak{j}=1$, then from (66), the following inequality emerges, see [46]:

$$\begin{gathered} 4\mathcal{G}\left(\frac{2\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2},\frac{2ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right)\times \mathcal{J}\left(\frac{2\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2},\frac{2ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right) \\ {\le }_p\frac{1}{\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right){\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{\int }_{\mathrm{ȵ}}^{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{\int }_{ⱬ}^{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}\mathcal{G}\left(x,y\right)\times \mathcal{J}\left(x,y\right)dydx \\ +\frac{5}{36}\mathcal{M}\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right) \\ +\frac{7}{36}\left[P\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right)+\mathcal{N}\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right)\right]+\frac{2}{9}Q\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right). \end{gathered}$$

(69)

If $\mathcal{G}$ is left-$\mathcal{C}$·$\mathsf{L}$·$\mathsf{R}$-$\mathbf{\hslash }$-pre-invex and ${\mathcal{G}}_{\ast }\left(x,y\right)\ne {\mathcal{G}}^{\ast }\left(x,y\right)$ with $\hslash \left(\upsilon \right)=\upsilon , \hslash \left(\kappa \right)=\kappa$, then from (66), we derive the following inequality, see [53]:

$$\begin{gathered} 4\mathcal{G}\left(\frac{2\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2}, \frac{2ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right)\times \mathcal{J}\left(\frac{2\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2}, \frac{2ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right) \\ \supseteq \frac{\Gamma \left(\mathfrak{i}+1\right)\Gamma \left(\mathfrak{j}+1\right)}{{4\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}{\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[\begin{array}{c}{\mathcal{I}}_{{\mathrm{ȵ}}^{+},{ⱬ}^{+}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\times J\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\\ +{\mathcal{I}}_{{\mathrm{ȵ}}^{+},{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right)\times J\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right)\\ +{\mathcal{I}}_{{\mathrm{ȶ}}^{-},{ⱬ}^{+}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\times J\left(\mathrm{ȵ},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\\ +{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-},{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)\times J\left(\mathrm{ȵ},ⱬ\right)\end{array}\right]+\left[\frac{\mathfrak{i}}{2\left(\mathfrak{i}+1\right)\left(\mathfrak{i}+2\right)}+ \\ \frac{\mathfrak{j}}{\left(\mathfrak{j}+1\right)\left(\mathfrak{j}+2\right)}\left(\frac{1}{2}-\frac{\mathfrak{i}}{\left(\mathfrak{i}+1\right)\left(\mathfrak{i}+2\right)}\right)\right]\mathcal{M}\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right) \\ +\left[\frac{1}{2}\left(\frac{1}{2}-\frac{\mathfrak{i}}{\left(\mathfrak{i}+1\right)\left(\mathfrak{i}+2\right)}\right)+\frac{\mathfrak{i}}{(\mathfrak{i}+1)(\mathfrak{i}+2)}\frac{\mathfrak{j}}{\left(\mathfrak{j}+1\right)\left(\mathfrak{j}+2\right)}\right]P\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right) \\ +\left[\frac{1}{2}\left(\frac{1}{2}-\frac{\mathfrak{j}}{\left(\mathfrak{j}+1\right)\left(\mathfrak{j}+2\right)}\right)+\frac{\mathfrak{i}}{(\mathfrak{i}+1)(\mathfrak{i}+2)}\frac{\mathfrak{j}}{\left(\mathfrak{j}+1\right)\left(\mathfrak{j}+2\right)}\right]\mathcal{N}\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right) \\ +\left[\frac{1}{4}-\frac{\mathfrak{i}}{(\mathfrak{i}+1)(\mathfrak{i}+2)}\frac{\mathfrak{j}}{\left(\mathfrak{j}+1\right)\left(\mathfrak{j}+2\right)}\right]Q\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right). \end{gathered}$$

(70)

If ${\mathcal{G}}_{\ast }\left(x,y\right)\ne {\mathcal{G}}^{\ast }\left(x,y\right)$ and $\hslash \left(\upsilon \right)=\upsilon , \hslash \left(\kappa \right)=\kappa$, then we can derive the following inequality from (66); for further information, see [53]:

$$\begin{gathered} 4\mathcal{G}\left(\frac{2\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2}, \frac{2ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right)\times \mathcal{J}\left(\frac{2\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2}, \frac{2ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right) \\ {\le }_p\frac{\Gamma \left(\mathfrak{i}+1\right)\Gamma \left(\mathfrak{j}+1\right)}{{4\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}{\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[\begin{array}{c}{\mathcal{I}}_{{\mathrm{ȵ}}^{+},{ⱬ}^{+}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\times J\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\\ +{\mathcal{I}}_{{\mathrm{ȵ}}^{+},{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right)\times J\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right)\\ +{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-},{ⱬ}^{+} }^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\times J\left(\mathrm{ȵ},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\\ +{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-},{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)\times J\left(\mathrm{ȵ},ⱬ\right)\end{array}\right] \\ +\left[\frac{\mathfrak{i}}{2\left(\mathfrak{i}+1\right)\left(\mathfrak{i}+2\right)}+\frac{\mathfrak{j}}{\left(\mathfrak{j}+1\right)\left(\mathfrak{j}+2\right)}\left(\frac{1}{2}-\frac{\mathfrak{i}}{\left(\mathfrak{i}+1\right)\left(\mathfrak{i}+2\right)}\right)\right]\mathcal{M}\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right) \\ +\left[\frac{1}{2}\left(\frac{1}{2}-\frac{\mathfrak{i}}{\left(\mathfrak{i}+1\right)\left(\mathfrak{i}+2\right)}\right)+\frac{\mathfrak{i}}{(\mathfrak{i}+1)(\mathfrak{i}+2)}\frac{\mathfrak{j}}{\left(\mathfrak{j}+1\right)\left(\mathfrak{j}+2\right)}\right]P\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right) \\ +\left[\frac{1}{2}\left(\frac{1}{2}-\frac{\mathfrak{j}}{\left(\mathfrak{j}+1\right)\left(\mathfrak{j}+2\right)}\right)+\frac{\mathfrak{i}}{(\mathfrak{i}+1)(\mathfrak{i}+2)}\frac{\mathfrak{j}}{\left(\mathfrak{j}+1\right)\left(\mathfrak{j}+2\right)}\right]\mathcal{N}\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right) \\ +\left[\frac{1}{4}-\frac{\mathfrak{i}}{(\mathfrak{i}+1)(\mathfrak{i}+2)}\frac{\mathfrak{j}}{\left(\mathfrak{j}+1\right)\left(\mathfrak{j}+2\right)}\right]Q\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right). \end{gathered}$$

(71)

If ${\mathcal{G}}_{\ast }\left(x,y\right)={\mathcal{G}}^{\ast }\left(x,y\right)$ and ${\mathcal{J}}_{\ast }\left(x,y\right)={\mathcal{J}}^{\ast }\left(x,y\right)$ and $\hslash \left(\upsilon \right)=\upsilon , \hslash \left(\kappa \right)=\kappa$, then we can derive the following inequality from (66) such that

$$\begin{gathered} 4\mathcal{G}\left(\frac{2\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2}, \frac{2ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right)\times \mathcal{J}\left(\frac{2\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}{2}, \frac{2ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}{2}\right) \\ \le \\ \frac{\Gamma \left(\mathfrak{i}+1\right)\Gamma \left(\mathfrak{j}+1\right)}{{4\left({\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)\right)}^{\mathfrak{i}}{\left({\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)}^{\mathfrak{j}}}\left[\begin{array}{c}{\mathcal{I}}_{{\mathrm{ȵ}}^{+},{ⱬ}^{+}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\times J\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\\ +{\mathcal{I}}_{{\mathrm{ȵ}}^{+},{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right)\times J\left(\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right),ⱬ\right)\\ +{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-},{ⱬ}^{+}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\times J\left(\mathrm{ȵ},ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)\right)\\ +{\mathcal{I}}_{{\mathrm{ȵ}+{\mathrm{ʎ}}_1\left(\mathrm{ȶ},\mathrm{ȵ}\right)}^{-},{ⱬ+{\mathrm{ʎ}}_2\left(\theta ,ⱬ\right)}^{-}}^{\mathfrak{i},\mathfrak{j}}\mathcal{G}\left(\mathrm{ȵ},ⱬ\right)\times J\left(\mathrm{ȵ},ⱬ\right)\end{array}\right] \\ +\left[\frac{\mathfrak{i}}{2\left(\mathfrak{i}+1\right)\left(\mathfrak{i}+2\right)}+\frac{\mathfrak{j}}{\left(\mathfrak{j}+1\right)\left(\mathfrak{j}+2\right)}\left(\frac{1}{2}-\frac{\mathfrak{i}}{\left(\mathfrak{i}+1\right)\left(\mathfrak{i}+2\right)}\right)\right]\mathcal{M}\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right) \\ +\left[\frac{1}{2}\left(\frac{1}{2}-\frac{\mathfrak{i}}{\left(\mathfrak{i}+1\right)\left(\mathfrak{i}+2\right)}\right)+\frac{\mathfrak{i}}{\left(\mathfrak{i}+1\right)\left(\mathfrak{i}+2\right)}\frac{\mathfrak{j}}{\left(\mathfrak{j}+1\right)\left(\mathfrak{j}+2\right)}\right]P\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right) \\ +\left[\frac{1}{2}\left(\frac{1}{2}-\frac{\mathfrak{j}}{\left(\mathfrak{j}+1\right)\left(\mathfrak{j}+2\right)}\right)+\frac{\mathfrak{i}}{\left(\mathfrak{i}+1\right)\left(\mathfrak{i}+2\right)}\frac{\mathfrak{j}}{\left(\mathfrak{j}+1\right)\left(\mathfrak{j}+2\right)}\right]\mathcal{N}\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right) \\ +\left[\frac{1}{4}-\frac{\mathfrak{i}}{\left(\mathfrak{i}+1\right)\left(\mathfrak{i}+2\right)}\frac{\mathfrak{j}}{\left(\mathfrak{j}+1\right)\left(\mathfrak{j}+2\right)}\right]Q\left(\mathrm{ȵ},\mathrm{ȶ},ⱬ,\theta \right). \end{gathered}$$

(72)

4. Conclusions and Future Plans

This article aims to establish Hermite–Hadamard and Pachpatte’s type integral inequalities for the newly defined nonconvex class which is known as $\mathcal{C}$·$L$·$R$-$\hslash$-pre-invex ${\rm I}$·$V$-$M$s using the interval fractional operator. First, we present integral equalities that are crucial for deriving the main results of the paper by employing the fractional integral operators, where integrable mappings are ${\rm I}$·$V$-$M$s. Additionally, the obtained results generalize many previous findings [53,54]. We also provide some exceptional cases as applications for the main results. In future work, we plan to expand and generalize our results using different classes of convex functions and other generalized fractional integral operators. We believe that our results and techniques will be useful to researchers working with various classes of convex functions and other generalized fractional integral operators.