New Version of Fractional Pachpatte-Type Integral Inequalities via Coordinated ℏ-Convexity via Left and Right Order Relation

Abstract

In particular, the fractional forms of Hermite–Hadamard inequalities for the newly defined class of convex mappings proposed that are known as coordinated left and right $\hslash$-convexity ($LR$-$\hslash$-convexity) over interval-valued codomain. We exploit the use of double Riemann–Liouville fractional integral to derive the major results of the research. We also examine the key results’ numerical validations that examples are nontrivial. By taking the product of two left and right coordinated $\hslash$-convexity, some new versions of fractional integral inequalities are also obtained. Moreover, some new and classical exceptional cases are also discussed by taking some restrictions on endpoint functions of interval-valued functions that can be seen as applications of these new outcomes.

1. Introduction

There are many uses for the concepts of convex sets and convex functions in the realms of applied and pure sciences. Furthermore, because of its many applications and tight relationship to the theory of inequalities, convexity has advanced quickly in recent years. When determining exact values for a mathematical problem proves to be challenging, inequalities can be used to approximate the solution. Since many inequalities can be directly derived from convex functions, there is a close relationship between convexity and the theory of inequalities.

The Hermite–Hadamard inequality is one of the most well-known findings in the category of classical convex functions, according to Dragomir and Pearce [1]. This inequality has several applications and a straightforward intrinsic geometric explanation. The result was mainly credited to Hermite (1822–1901), even though Hadamard (1865–1963) was the one who first identified it [2,3]. The following is how this inequality is stated:

Theorem 1. 

Assume that the convex mapping $\mathsf{Ԓ}:\left[\mathcal{o},\mathfrak{i}\right]\to \mathfrak{R}$. Then, the following double inequality holds: 

$$\mathsf{Ԓ}\left(\frac{\mathcal{o}+\mathfrak{i}}{2}\right)\le \frac{1}{\mathfrak{i}-\mathcal{o}}{\int }_{\mathcal{o}}^{\mathfrak{i}}\mathsf{Ԓ}\left(x\right)dx\le \frac{\mathsf{Ԓ}\left(\mathfrak{i}\right)+\mathsf{Ԓ}\left(\mathcal{o}\right)}{2},$$

(1)

where $\mathfrak{R}$ is a set of real numbers. One can check the concavity of the mappings by replacing the symbol  “≤” with “≥” in double inequality (1).

The midpoint and trapezoidal-type inequalities, which are the two sides of the Hermite–Hadamard inequality, are used to estimate error boundaries for specific quadrature rules. The original derivation of these inequalities was in [4,5].

New extended versions of Simpson-type inequalities were derived by Awan et al. [6] using differentiable, strongly $\left(s,m\right)$-convex maps. Simpson’s integral inequality has been further expanded upon, refined, and generalized in [7,8,9,10].

In the course of research, various variations of the Hermite–Hadamard inequality have been derived by expanding the definition of convex functions. Conversely, the notion of 𝔰}-convexity [11,12] is divided into two halves, as follows, with the fundamental requirement that $1\ge s>0$. The following references [13,14,15,16,17] contain more generalizations and expansions of classical convex functions.

Fractional calculus is the study of integrals and derivatives of any real order. Fractional integrals are used to solve a wide range of problems involving mathematical science’s special functions, as well as their generalizations and extensions to one or more variables. Furthermore, compared to traditional derivatives, fractional-order derivatives describe the memory and hereditary characteristics of distinct processes far better. Actually, current applications in fluid mechanics, mathematical biology, electrochemistry, physics, differential and integral equations, signal processing, and fluid mechanics have been the driving forces behind the recent developments in fractional calculus. Without a doubt, fractional calculus can be used to solve a wide range of diverse problems in science, engineering, and mathematics [18,19,20]. The reference [21] provides a thorough history of fractional calculus.

Creating different kinds of integral inequalities is a modern issue. Utilizing a range of integrals, including the Sugeno integral [22,23], the pseudo integral [24], the Choquet integral [25], and others, a significant amount of important work has been accomplished in recent years. As a notion of generalization of functions and a significant mathematical subject, interval-valued functions [26] have grown in importance as a tool for resolving real-world problems, especially in mathematical economics [27]. Certain classical integral inequalities have been expanded to the domain of interval-valued functions through recent studies. New interval variations of Minkowski and Beckenbach’s integral inequalities were introduced by Costa et al. [28]. This generalization established Jensen, Ostrowski, and Hermite–Hadamard-type inequalities [29]. Fractional integrals of Riemann–Liouville with interval values were also used to solve Hermite–Hadamard and Hermite–Hadamard-type inequalities [30]. Zhao and colleagues [31,32,33] employed the gH -differentiable or h-convex notion to study Jensen and Hermite–Hadamard-type inequalities, Opial-type integral inequalities, and Chebyshev-type inequalities for interval-valued functions. Budaka et al. [34] used the definitions of gH-derivatives to develop new fractional inequalities of the Ostrowski type for interval-valued functions. Log-h-convex fuzzy-interval-valued functions are a new class of convex fuzzy-interval-valued functions that were introduced by Khan et al. [35] using a fuzzy order relation. The Jensen and Hermite–Hadamard inequalities were established in this class. To include the Ostrowski-type inequality in the domain of fuzzy-valued functions, the Hukuhara derivative had to be applied, as Anastassiou [36] showed. Anastassiou’s research focused heavily on fuzzy-valued functions, commonly referred to as functions with an interval value. An interesting finding is that Anastassiou’s fuzzy Ostrowski-type inequalities could also be used for interval-valued functions. Bede and Gal’s [37] and Chalco-Cano et al.’s [38] publications should be studied in order to gain a thorough grasp of the limitations placed on interval-valued functions by the idea of the H-derivative. Significantly, Chalco-Cano et al.’s recent work [39] has produced an Ostrowski-type inequality that is tailored to generalized Hukuhara differentiable interval-valued functions. On the other hand, Lupulescu [40] introduced the concept of left-fractional integral in interval-valued calculus. Then, right fractional integrals are proposed by Budak et al. [41], as well as providing the fractional Hermite–Hadamard-type inequalities for interval-valued mappings over coordinates. Zhao et al. [42] generalized the Riemann integral inequalities for coordinated convex interval-valued mappings and produced the inequalities for the product of coordinated convexity. After that, Khan et al. [43] provide a new direction in interval-valued calculus by introducing new versions of coordinated integral inequalities via double Riemann integrals and left and right relations. Budak and Sarıkaya [44] and Khan et al. [45] defined Pachpatte’s inequalities for the product of coordinated and left and right coordinated convex mappings via fractional integrals, respectively. By using this approach, Khan et al. [46] obtained the left- and right-coordinated interval-valued functions and acquired some integral inequalities in interval fractional calculus for left- and right-coordinated interval-valued functions. Zhang et al. [47] first defined the up and down relations and then discussed some of the properties of these relations. Moreover, he defined the new versions of Jensen-type inequalities using up and down relations.

The structure of this research article is as follows. Some classical notions, definitions, and results are recalled and a new definition of convexity over interval-valued mapping is also introduced, which is known as coordinated $LR$-$\hslash$-convexity in Section 2. We also report several other findings that follow from this definition of convexity. Using coordinated $LR$-$\hslash$-convexity defined in Section 2, some new and classical exceptional cases are also obtained in Section 2. In Section 3, involving interval fractional integrals for $LR$-$\hslash$-convexity, some well-known inequalities have been generalized, as well as nontrivial examples have also been provided to validate the main outcomes of this paper. Section 4 concludes this study and discusses future work.

2. Preliminaries

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

$$\mathcal{i}=\left[{\mathcal{i}}_{*}, {\mathcal{i}}^{*}\right]=\left\{\mathsf{y}\in \mathbb{R}|{\mathcal{i}}_{*}\le \mathsf{y}\le {\mathcal{i}}^{*}\right\},\left({\mathcal{i}}_{*}, {\mathcal{i}}^{*}\in \mathbb{R}\right).$$

(2)

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

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

• Scaler multiplication:

$$\varrho .\mathcal{i}=\left\{\begin{array}{c}\left[\varrho {\mathcal{i}}_{*}, {\varrho \mathcal{i}}^{*}\right] if \varrho >0,\\ \left\{0\right\} if \varrho =0,\\ \left[\varrho {\mathcal{i}}^{*}{,\varrho \mathcal{i}}_{*}\right] if \varrho <0. \end{array}\right.$$

(3)

• Addition:

$$\begin{array}{c}\left[{\mathcal{j}}_{*}, {\mathcal{j}}^{*}\right]+\left[{\mathcal{i}}_{*}, {\mathcal{i}}^{*}\right]=\left[{\mathcal{j}}_{*}+{\mathcal{i}}_{*}, {\mathcal{j}}^{*}+{\mathcal{i}}^{*}\right]\end{array}.$$

(4)

• Multiplication:

$$\left[{\mathcal{j}}_{*}, {\mathcal{j}}^{*}\right]\times \left[{\mathcal{i}}_{*}, {\mathcal{i}}^{*}\right]=\left[min\left\{{\mathcal{j}}_{*}{\mathcal{i}}_{*}, {\mathcal{j}}^{*}{\mathcal{i}}_{*}, {\mathcal{j}}_{*}{\mathcal{i}}^{*}, {\mathcal{j}}^{*}{\mathcal{i}}^{*}\right\}, max\left\{{\mathcal{j}}_{*}{\mathcal{i}}_{*}, {\mathcal{j}}^{*}{\mathcal{i}}_{*}, {\mathcal{j}}_{*}{\mathcal{i}}^{*}, {\mathcal{j}}^{*}{\mathcal{i}}^{*}\right\}\right].$$

(5)

The inclusion $“\supseteq ”$ means that $\mathcal{i}\supseteq \mathcal{j}$ if and only if, $\left[{\mathcal{i}}_{\mathrm{*}},{\mathcal{i}}^{\mathrm{*}}\right]\supseteq \left[{\mathcal{j}}_{\mathrm{*}},{\mathcal{j}}^{\mathrm{*}}\right]$, and if and only if ${\mathcal{i}}_{\mathrm{*}}\le {\mathcal{j}}_{\mathrm{*}}$, ${\mathcal{j}}^{\mathrm{*}}\le {\mathcal{i}}^{\mathrm{*}}.$

Remark 1 

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

$$\left[{\mathcal{j}}_{*}, {\mathcal{j}}^{*}\right]{\le }_p\left[{\mathcal{i}}_{*}, {\mathcal{i}}^{*}\right] \mathrm{if} \mathrm{and} \mathrm{only} \mathrm{if} {\mathcal{j}}_{*}{\le \mathcal{i}}_{*}, {\mathcal{j}}^{*}{\le \mathcal{i}}^{*},$$

(6)

for all $\left[{\mathcal{j}}_{*}, {\mathcal{j}}^{*}\right], \left[{\mathcal{i}}_{*}, {\mathcal{i}}^{*}\right]\in {\mathbb{R}}_I,$ and it is a pseudo-order relation. The relation $\left[{\mathcal{j}}_{*}, {\mathcal{j}}^{*}\right]{\le }_p\left[{\mathcal{i}}_{*}, {\mathcal{i}}^{*}\right]$ coincident to $\left[{\mathcal{j}}_{*}, {\mathcal{j}}^{*}\right]\le \left[{\mathcal{i}}_{*}, {\mathcal{i}}^{*}\right]$ 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 $“{\le }_p”$ “left and right” (or “LR” order, in short).

The Hausdorff–Pompeiu distance between intervals $\left[{\mathcal{j}}_{*}, {\mathcal{j}}^{*}\right]$, $\left[{\mathcal{i}}_{*}, {\mathcal{i}}^{*}\right]\in {\mathbb{R}}_I$ is given by

$$d\left(\left[{\mathcal{j}}_{*}, {\mathcal{j}}^{*}\right], \left[{\mathcal{i}}_{*}, {\mathcal{i}}^{*}\right]\right)=max\left\{\left|{\mathcal{j}}_{*}-{\mathcal{i}}_{*}\right|, \left|{\mathcal{j}}^{*}-{\mathcal{i}}^{*}\right|\right\}.$$

(7)

It is a familiar fact that $\left({\mathbb{R}}_I,d\right)$ is a complete metric space.

Definition 1 

([30,40]). Let $\mathsf{Ԓ}:\left[\mathcal{o},\mathfrak{i}\right]\to {\mathbb{R}}_I^{+}$ be an interval-valued mapping ($IVM$) and $\mathsf{Ԓ}\in {\mathcal{I}\mathcal{R}}_{\left[\mathcal{o},\mathfrak{i}\right]}$. Then, interval Riemann–Liouville-type integrals of $\mathsf{Ԓ}$ are defined as 

$${\mathcal{I}}_{{\mathcal{o}}^{+}}^{\alpha }\mathsf{Ԓ}\left(y\right)=\frac{1}{\Gamma \left(\alpha \right)}{\int }_{\mathcal{o}}^y{\left(y-\mathtt{t}\right)}^{\alpha -1}\mathsf{Ԓ}\left(\mathtt{t}\right)d\mathtt{t}\hspace{1em}\hspace{1em}\left(y>\mathcal{o}\right),$$

(8)

$${\mathcal{I}}_{{\mathfrak{i}}^{-}}^{\alpha }\mathsf{Ԓ}\left(y\right)=\frac{1}{\Gamma \left(\alpha \right)}{\int }_y^{\mathfrak{i}}{\left(\mathtt{t}-y\right)}^{\alpha -1}\mathsf{Ԓ}\left(\mathtt{t}\right)d\mathtt{t}\hspace{1em}\hspace{1em}(y<\mathfrak{i}),$$

(9)

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

Interval and fuzzy Riemann-type integrals are defined as follows for coordinated $VM$ $\mathsf{Ԓ}\left(x,y\right)$.

Theorem 2 

([42]). Let $\mathsf{Ԓ}:\left[\mathcal{o},\mathfrak{i}\right]\subset \mathbb{R}\to {\mathbb{R}}_I^{ }$ be an ${\rm I}VM$, given by $\mathsf{Ԓ}\left(x\right)=\left[{\mathsf{Ԓ}}_{*}\left(x\right), {\mathsf{Ԓ}}^{*}\left(x\right)\right]$ for all $x\in \left[\mathcal{o},\mathfrak{i}\right]$. Then, $\mathsf{Ԓ}$ is Riemann integrable ($IR$-integrable) over $\left[\mathcal{o},\mathfrak{i}\right]$ if and only if ${\mathsf{Ԓ}}_{*}\left(x\right)$ and ${\mathsf{Ԓ}}^{*}\left(x\right)$ both are Riemann integrable ($R$-integrable) over $\left[\mathcal{o},\mathfrak{i}\right].$ Moreover, if $\mathsf{Ԓ}$ is $IR$-integrable over $\left[\mathcal{o},\mathfrak{i}\right],$ then 

$$\left(IR\right){\int }_{\mathcal{o}}^{\mathfrak{i}}\mathsf{Ԓ}\left(x\right)dx=\left[\left(R\right){\int }_{\mathcal{o}}^{\mathfrak{i}}{\mathsf{Ԓ}}_{*}\left(x\right)dx, \left(R\right){\int }_{\mathcal{o}}^{\mathfrak{i}}{\mathsf{Ԓ}}^{*}\left(x\right)dx\right].$$

(10)

The family of all $IR$-integrable of ${\rm I}VM$s over coordinates and $R$-integrable functions over $\left[\mathcal{o},\mathfrak{i}\right]$ are denoted by ${IR}_{\left[\mathcal{o},\mathfrak{i}\right]}$ and $R_{\left[\mathcal{o},\mathfrak{i}\right]},$ respectively.

Theorem 3 

([31]). Let $\mathsf{Ԓ}:\Omega =\left[\mathcal{o},\mathfrak{i}\right]\times \left[\epsilon , \mathfrak{v}\right]\subset {\mathbb{R}}^2\to {\mathbb{R}}_I^{ }$ be an ${\rm I}VM$ on coordinates, given by $\mathsf{Ԓ}\left(x,y\right)=\left[{\mathsf{Ԓ}}_{*}\left(x,y\right), {\mathsf{Ԓ}}^{*}\left(x,y\right)\right]$ for all $\left(x,y\right)\in \Omega =\left[\mathcal{o},\mathfrak{i}\right]\times \left[\epsilon , \mathfrak{v}\right]$. Then, $\mathsf{Ԓ}$ is double integrable ($ID$-integrable) over $\Omega$ if and only if ${\mathsf{Ԓ}}_{*}\left(x,y\right)$ and ${\mathsf{Ԓ}}^{*}\left(x,y\right)$ both are $D$-integrable over $\Omega .$ Moreover, if $\mathsf{Ԓ}$ is $FD$-integrable over $\Omega ,$ then 

$$\left(ID\right){\int }_{\mathcal{o}}^{\mathfrak{i}}{\int }_{\epsilon }^{\mathfrak{v}}\mathsf{Ԓ}\left(x,y\right)dydx=\left(IR\right){\int }_{\mathcal{o}}^{\mathfrak{i}}\left(IR\right){\int }_{\epsilon }^{\mathfrak{v}}\mathsf{Ԓ}\left(x,y\right)dydx.$$

(11)

The family of all $ID$-integrable of ${\rm I}VM$s over coordinates over coordinates is denoted by ${\mathfrak{T}\mathfrak{O}}_{\Omega }.$

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

Definition 2 

([41]). Let $\mathsf{Ԓ}:\Omega \to {\mathbb{R}}_I^{ }$ and $\mathsf{Ԓ}\in {\mathfrak{T}\mathfrak{O}}_{\Omega }$. The double fuzzy interval Rieman–-Liouville-type integrals ${\mathcal{I}}_{{\mathcal{o}}^{+},{\epsilon }^{+}}^{\alpha , \beta }, {\mathcal{I}}_{{\mathcal{o}}^{+},{\mathfrak{v}}^{-}}^{\alpha , \beta }, {\mathcal{I}}_{{\mathfrak{i}}^{-},{\epsilon }^{+}}^{\alpha , \beta }, {\mathcal{I}}_{{\mathfrak{i}}^{-},{\mathfrak{v}}^{-} }^{\alpha , \beta }$ of $\mathsf{Ԓ}$ order $\alpha , \beta >0$ are defined by: 

$${\mathcal{I}}_{{\mathcal{o}}^{+},{\epsilon }^{+}}^{\alpha , \beta }\mathsf{Ԓ}\left(x,y\right)=\frac{1}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{\int }_{\mathcal{o}}^x{\int }_{\epsilon }^y{{\left(x-\mathtt{t}\right)}^{\alpha -1}\left(y-\mathtt{s}\right)}^{\beta -1}\mathsf{Ԓ}\left(\mathtt{t},\mathtt{s}\right)d\mathtt{s}d\mathtt{t}, \left(x>\mathcal{o},y>\epsilon \right),$$

(12)

$${\mathcal{I}}_{{\mathcal{o}}^{+},{\mathfrak{v}}^{-}}^{\alpha , \beta }\mathsf{Ԓ}\left(x,y\right)=\frac{1}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{\int }_{\mathcal{o}}^x{\int }_y^{\mathfrak{v}}{{\left(x-\mathtt{t}\right)}^{\alpha -1}\left(\mathtt{s}-y\right)}^{\beta -1}\mathsf{Ԓ}\left(\mathtt{t},\mathtt{s}\right)d\mathtt{s}d\mathtt{t}, (x>\mathcal{o},y<\mathfrak{v}),$$

(13)

$${\mathcal{I}}_{{\mathfrak{i}}^{-},{\epsilon }^{+}}^{\alpha , \beta }\mathsf{Ԓ}\left(x,y\right)=\frac{1}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{\int }_x^{\mathfrak{i}}{\int }_{\epsilon }^y{{\left(\mathtt{t}-x\right)}^{\alpha -1}\left(y-\mathtt{s}\right)}^{\beta -1}\mathsf{Ԓ}\left(\mathtt{t},\mathtt{s}\right)d\mathtt{s}d\mathtt{t}, \left(x<\mathfrak{i},y>\epsilon \right),$$

(14)

$${\mathcal{I}}_{{\mathfrak{i}}^{-},{\mathfrak{v}}^{-}}^{\alpha , \beta }\mathsf{Ԓ}\left(x,y\right)=\frac{1}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{\int }_x^{\mathfrak{i}}{\int }_y^{\mathfrak{v}}{{\left(\mathtt{t}-x\right)}^{\alpha -1}\left(\mathtt{s}-y\right)}^{\beta -1}\mathsf{Ԓ}\left(\mathtt{t},\mathtt{s}\right)d\mathtt{s}d\mathtt{t}, (x<\mathfrak{i},y<\mathfrak{v}).$$

(15)

Here is the classical and newly defined concept of coordinated $LR$-$\hslash$-convexity over fuzzy number space in the codomain via fuzzy relation given by:

Definition 3 

([45]). The ${\rm I}VM$ $\mathsf{Ԓ}:\left[\epsilon , \mathfrak{v}\right]\to {\mathbb{R}}_I^{+}$ is referred to be $LR$-$\hslash$-convex ${\rm I}VM$ on $\left[\epsilon , \mathfrak{v}\right]$ if 

$$\mathsf{Ԓ}\left(\kappa \epsilon +\left(1-\kappa \right)\mathfrak{v}\right){\le }_p\hslash \left(\kappa \right)\mathsf{Ԓ}\left(\epsilon \right)+\hslash \left(1-\kappa \right)\mathsf{Ԓ}\left(\mathfrak{v}\right),$$

(16)

where $\hslash :\left[0, 1\right]\to {\mathbb{R}}_{ }^{+}$. If inequality (16) is reversed, then $\mathsf{Ԓ}$ is referred to be $\mathit{\hslash }$-concave ${\rm I}VM$ on $\left[\epsilon , \mathfrak{v}\right]$.

Theorem 4 

([45]). Let $\hslash :\left[0, 1\right]\to {\mathbb{R}}^{+}$ and $\mathsf{Ԓ}:\left[\epsilon , \mathfrak{v}\right]\to {\mathbb{R}}_I^{+}$ be a $LR$-$\hslash$-convex ${\rm I}VM$ on $\left[\epsilon , \mathfrak{v}\right],$ given by $\mathsf{Ԓ}\left(y\right)=\left[{\mathsf{Ԓ}}_{*}\left(y\right), {\mathsf{Ԓ}}^{*}\left(y\right)\right]$ for all $y\in \left[\epsilon , \mathfrak{v}\right]$. If $\mathsf{Ԓ}\in L\left(\left[\epsilon , \mathfrak{v}\right],{\mathbb{R}}_I^{+}\right)$, then 

$$\frac{1}{\alpha \hslash \left(\frac{1}{2}\right)}\mathsf{Ԓ}\left(\frac{\epsilon +\mathfrak{v}}{2}\right){\le }_p\frac{\Gamma \left(\alpha \right)}{{\left(\mathfrak{v}-\epsilon \right)}^{\alpha }}\left[{\mathcal{I}}_{{\epsilon }^{+}}^{\alpha }\mathsf{Ԓ}\left(\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\alpha }\mathsf{Ԓ}\left(\epsilon \right)\right]{\le }_p\left[\mathsf{Ԓ}\left(\epsilon \right)+\mathsf{Ԓ}\left(\mathfrak{v}\right)\right]{\int }_0^1{\upsilon }^{\alpha -1}\left[\hslash \left(\upsilon \right)+\hslash \left(1-\upsilon \right)\right]d\upsilon .$$

(17)

Definition 4. 

The ${\rm I}VM$ $\mathsf{Ԓ}:\Omega \to {\mathbb{R}}_I^{+}$ is referred to be coordinated $LR$-$\hslash$-convex ${\rm I}VM$ on $\Omega$ if 

$$\begin{array}{c}\mathsf{Ԓ}\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},\kappa \epsilon +\left(1-\kappa \right)\mathfrak{v}\right)\hfill \\ {\le }_p\hslash \left(\upsilon \right)\hslash \left(\kappa \right)\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)+\hslash \left(\upsilon \right)\hslash \left(1-\kappa \right)\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)+\hslash \left(1-\upsilon \right)\hslash \left(\kappa \right)\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)+\hslash \left(1-\upsilon \right)\hslash \left(1-\kappa \right)\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)\hfill \end{array}$$

(18)

for all $\left(\mathcal{o},\mathfrak{i}\right), \left(\epsilon ,\mathfrak{v}\right)\in \Omega ,$ and $\upsilon ,\kappa \in \left[0, 1\right],$where $\mathsf{Ԓ}\left(x\right){\ge }_{p}0.$ If inequality (18) is reversed, then $\mathsf{Ԓ}$ is referred to be coordinate $\mathit{\hslash }$-concave ${\rm I}VM$ on $\Omega$. If $\mathit{\hslash }$ is the identity function, we recover the $LR$-convex ${\rm I}VM$ on $\Omega$ given in [46].

Lemma 1. 

Let $\mathsf{Ԓ}:\Omega \to {\mathbb{R}}_I^{+}$ be a coordinated ${\rm I}VM$ on $\Omega$. Then, $\mathsf{Ԓ}$ is coordinated $LR$-$\hslash$-convex ${\rm I}VM$ on $\Omega$ if and only if there exist two coordinated $LR$-$\hslash$-convex ${\rm I}VM$s ${\mathsf{Ԓ}}_x:\left[\epsilon ,\mathfrak{v}\right]\to {\mathbb{R}}_I^{+}$, ${\mathsf{Ԓ}}_x\left(w\right)=\mathsf{Ԓ}\left(x,w\right)$ and ${\mathsf{Ԓ}}_y:\left[\mathcal{o},\mathfrak{i}\right]\to {\mathbb{R}}_I^{+}$, ${\mathsf{Ԓ}}_y\left(z\right)=\mathsf{Ԓ}\left(z,y\right)$.

Theorem 5. 

Let $\mathsf{Ԓ}:\Omega \to {\mathbb{R}}_I^{+}$ be a ${\rm I}VM$ on $\Omega$, given by 

$$\mathsf{Ԓ}\left(x,y\right)=\left[{\mathsf{Ԓ}}_{*}\left(x,y\right), {\mathsf{Ԓ}}^{*}\left(x,y\right)\right],$$

(19)

for all $\left(x,y\right)\in \Omega$. Then, $\mathsf{Ԓ}$ is coordinated $LR$-$\hslash$-convex ${\rm I}VM$ on $\Omega ,$ if and only if both ${\mathsf{Ԓ}}_{*}\left(x,y\right)$ and ${\mathsf{Ԓ}}^{*}\left(x,y\right)$ are coordinated $LR$-$\hslash$-convex.

Proof. 

Assume that ${\mathsf{Ԓ}}_{*}\left(\mathcal{x}\right)$ and ${\mathsf{Ԓ}}^{*}\left(\mathcal{x}\right)$ are coordinated $LR$-$\hslash$-convex and $\mathbf{\hslash }$-concave on $\Omega$, respectively. Then, from Equation (19), for all $\left(\mathcal{o},\mathfrak{i}\right), \left(\epsilon ,\mathfrak{v}\right)\in \Omega , \upsilon$ and $\kappa \in \left[0, 1\right]$, we have

$$\begin{array}{c}{\mathsf{Ԓ}}_{*}\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},\kappa \epsilon +\left(1-\kappa \right)\mathfrak{v}\right)\hfill \\ \le \hslash \left(\upsilon \right){\hslash \left(\kappa \right)\mathsf{Ԓ}}_{*}\left(\mathcal{o},\epsilon \right)+\hslash \left(\upsilon \right)\hslash \left(1-\kappa \right){\mathsf{Ԓ}}_{*}\left(\mathcal{o},\mathfrak{v}\right)+{\hslash \left(\kappa \right)\hslash \left(1-\upsilon \right)\mathsf{Ԓ}}_{*}\left(\mathfrak{i},\epsilon \right)+\hslash \left(1-\upsilon \right)\hslash \left(1-\kappa \right){\mathsf{Ԓ}}_{*}\left(\mathfrak{i},\mathfrak{v}\right)\hfill \end{array}$$

(20)

and

$$\begin{array}{c}{\mathsf{Ԓ}}^{*}\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},\kappa \epsilon +\left(1-\kappa \right)\mathfrak{v}\right)\hfill \\ \le \hslash \left(\upsilon \right)\hslash \left(\kappa \right){\mathsf{Ԓ}}^{*}\left(\mathcal{o},\epsilon \right)+\hslash \left(\upsilon \right)\hslash \left(1-\kappa \right){\mathsf{Ԓ}}^{*}\left(\mathcal{o},\mathfrak{v}\right)+\hslash \left(\kappa \right)\hslash \left(1-\upsilon \right){\mathsf{Ԓ}}^{*}\left(\mathfrak{i},\epsilon \right)+\hslash \left(1-\upsilon \right)\hslash \left(1-\kappa \right){\mathsf{Ԓ}}^{*}\left(\mathfrak{i},\mathfrak{v}\right).\hfill \end{array}$$

(21)

Then, by Equations (19), (3) and (4), we obtain

$$\begin{array}{cc}\hfill \mathsf{Ԓ}(\upsilon \mathcal{o}& +\left(1-\upsilon \right)\mathfrak{i},\kappa \epsilon +\left(1-\kappa \right)\mathfrak{v})\hfill \\ & =\left[{\mathsf{Ԓ}}_{*}\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},\kappa \epsilon +\left(1-\kappa \right)\mathfrak{v}\right), {\mathsf{Ԓ}}^{*}\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},\kappa \epsilon +\left(1-\kappa \right)\mathfrak{v}\right)\right]\hfill \\ & {\le }_p\hslash \left(\upsilon \right)\hslash \left(\kappa \right)\left[{\mathsf{Ԓ}}_{*}\left(\mathcal{o},\epsilon \right), {\mathsf{Ԓ}}^{*}\left(\mathcal{o},\epsilon \right)\right]+\hslash \left(\upsilon \right)\hslash \left(1-\kappa \right)[{\mathsf{Ԓ}}_{*}\left(\mathcal{o},\mathfrak{v}\right), {\mathsf{Ԓ}}^{*}\left(\mathcal{o},\mathfrak{v}\right)]\hfill \\ & +\hslash \left(\kappa \right)\hslash \left(1-\upsilon \right)\left[{\mathsf{Ԓ}}_{*}\left(\mathfrak{i},\epsilon \right), {\mathsf{Ԓ}}^{*}\left(\mathfrak{i},\epsilon \right)\right]+\hslash \left(1-\upsilon \right)\hslash \left(1-\kappa \right)\left[{\mathsf{Ԓ}}_{*}\left(\mathfrak{i},\mathfrak{v}\right), {\mathsf{Ԓ}}^{*}\left(\mathfrak{i},\mathfrak{v}\right)\right]\hfill \end{array}$$

From (20) and (21), we have

$$\begin{array}{cc}\hfill \mathsf{Ԓ}(\upsilon \mathcal{o}& +\left(1-\upsilon \right)\mathfrak{i},\kappa \epsilon +\left(1-\kappa \right)\mathfrak{v})\hfill \\ & {\le }_p\hslash \left(\upsilon \right)\hslash \left(\kappa \right)\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)+\hslash \left(\upsilon \right)\hslash \left(1-\kappa \right)\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)+\hslash \left(1-\upsilon \right)\hslash \left(1-\kappa \right)\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)+\hslash \left(1-\upsilon \right)\hslash \left(1-\kappa \right)\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right),\hfill \end{array}$$

hence, $\mathsf{Ԓ}$ is coordinated $LR$-$\hslash$-convex ${\rm I}VM$ on $\Omega$.

Conversely, let $\mathsf{Ԓ}$ be coordinated $LR$-$\hslash$-convex ${\rm I}VM$ on $\Omega .$ Then, for all $\left(\mathcal{o},\mathfrak{i}\right), \left(\epsilon ,\mathfrak{v}\right)\in \Omega , \upsilon$ and $\kappa \in \left[0, 1\right],$ we have

$$\begin{array}{c}\mathsf{Ԓ}\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},\kappa \epsilon +\left(1-\kappa \right)\mathfrak{v}\right)\hfill \\ \hspace{1em}\hspace{1em}{\le }_p\hslash \left(\upsilon \right)\hslash \left(\kappa \right)\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)+\hslash \left(\upsilon \right)\hslash \left(1-\kappa \right)\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)+\hslash \left(1-\upsilon \right)\hslash \left(\kappa \right)\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)+\hslash \left(1-\upsilon \right)\hslash \left(1-\kappa \right)\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right).\hfill \end{array}$$

Therefore, again from Equation (20), we have

$$\begin{array}{c}\mathsf{Ԓ}\left(\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},\kappa \epsilon +\left(1-\kappa \right)\mathfrak{v}\right)\right)\hfill \\ =\left[{\mathsf{Ԓ}}_{*}\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},\kappa \epsilon +\left(1-\kappa \right)\mathfrak{v}\right), {\mathsf{Ԓ}}^{*}\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},\kappa \epsilon +\left(1-\kappa \right)\mathfrak{v}\right)\right].\hfill \end{array}$$

(22)

Again, Equations (3) and (4), we obtain

$$\begin{array}{c}\hslash \left(\upsilon \right)\hslash \left(\kappa \right)\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)+\hslash \left(\upsilon \right)\hslash \left(1-\kappa \right)\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)+\hslash \left(1-\upsilon \right)\hslash \left(\kappa \right)\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)+\hslash \left(1-\upsilon \right)\hslash \left(1-\kappa \right)\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)\hfill \\ \hspace{1em}=\hslash \left(\upsilon \right)\hslash \left(\kappa \right)\left[{\mathsf{Ԓ}}_{*}\left(\mathcal{o},\epsilon \right), {\mathsf{Ԓ}}^{*}\left(\mathcal{o},\epsilon \right)\right]+\hslash \left(\upsilon \right)\hslash \left(1-\kappa \right)\left[{\mathsf{Ԓ}}_{*}\left(\mathcal{o},\mathfrak{v}\right), {\mathsf{Ԓ}}^{*}\left(\mathcal{o},\mathfrak{v}\right)\right]\hfill \\ \hspace{1em}+\hslash \left(\kappa \right)\hslash \left(1-\upsilon \right)\left[{\mathsf{Ԓ}}_{*}\left(\mathcal{o},\epsilon \right), {\mathsf{Ԓ}}^{*}\left(\mathcal{o},\epsilon \right)\right]+\hslash \left(1-\upsilon \right)\hslash \left(1-\kappa \right)\left[{\mathsf{Ԓ}}_{*}\left(\mathcal{o},\mathfrak{v}\right), {\mathsf{Ԓ}}^{*}\left(\mathcal{o},\mathfrak{v}\right)\right],\hfill \end{array}$$

(23)

for all $\mathcal{x},\omega \in \Omega$ and $\upsilon \in \left[0, 1\right].$ Then, from (22) and (23), we have for all $\mathcal{x},\omega \in \Omega$ and $\upsilon \in \left[0, 1\right],$ such that

$$\begin{array}{c}{\mathsf{Ԓ}}_{*}\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},\kappa \epsilon +\left(1-\kappa \right)\mathfrak{v}\right)\hfill \\ \le \hslash \left(\upsilon \right)\hslash \left(\kappa \right){\mathsf{Ԓ}}_{*}\left(\mathcal{o},\epsilon \right)+\hslash \left(\upsilon \right)\hslash \left(1-\kappa \right){\mathsf{Ԓ}}_{*}\left(\mathcal{o},\mathfrak{v}\right)+\hslash \left(1-\upsilon \right)\hslash \left(\kappa \right){\mathsf{Ԓ}}_{*}\left(\mathfrak{i},\epsilon \right)+\hslash \left(1-\upsilon \right)\hslash (1-\hfill \\ \kappa ){\mathsf{Ԓ}}_{*}\left(\mathfrak{i},\mathfrak{v}\right),\hfill \end{array}$$

and

$$\begin{array}{c}{\mathsf{Ԓ}}^{*}\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},\kappa \epsilon +\left(1-\kappa \right)\mathfrak{v}\right)\hfill \\ \le \hslash \left(\upsilon \right)\hslash \left(\kappa \right){\mathsf{Ԓ}}^{*}\left(\mathcal{o},\epsilon \right)+\hslash \left(\upsilon \right)\hslash \left(1-\kappa \right){\mathsf{Ԓ}}^{*}\left(\mathcal{o},\mathfrak{v}\right)+\hslash \left(1-\upsilon \right)\hslash \left(\kappa \right){\mathsf{Ԓ}}^{*}\left(\mathfrak{i},\epsilon \right)+\hfill \\ \hslash \left(1-\upsilon \right)\hslash \left(1-\kappa \right){\mathsf{Ԓ}}^{*}\left(\mathfrak{i},\mathfrak{v}\right),\hfill \end{array}$$

hence, the result follows. □

Example 1. 

We consider the  ${\rm I}VM$ $\mathsf{Ԓ}:\left[0, 1\right]\times \left[0, 1\right]\to {\mathbb{R}}_I^{+}$ defined by, 

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

Endpoint functions ${\mathsf{Ԓ}}_{*}\left(x,y\right),$ ${\mathsf{Ԓ}}^{*}\left(x,y\right)$ are coordinate $\mathit{\hslash }$-concave functions. Hence, $\mathsf{Ԓ}\left(x,y\right)$ is coordinated $LR$-$\hslash$-convex ${\rm I}VM$.

From Lemma 1 and Example 1, we can easily note that each $LR$-$\hslash$-convex ${\rm I}VM$ is coordinated $LR$-$\hslash$-convex ${\rm I}VM$. But the inverse is not true.

Remark 2. 

If one assumes that ${\mathsf{Ԓ}}_{*}\left(x,y\right)={\mathsf{Ԓ}}^{*}\left(x,y\right)$, then $\mathsf{Ԓ}$ is referred to as a classical coordinated $LR$-$\hslash$-convex function if $\mathsf{Ԓ}$ meets the stated inequality here:

$$\begin{array}{c}\mathsf{Ԓ}\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},\kappa \epsilon +\left(1-\kappa \right)\mathfrak{v}\right)\hfill \\ \le \hslash \left(\upsilon \right)\hslash \left(\kappa \right)\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)+\hslash \left(\upsilon \right)\hslash \left(1-\kappa \right)\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)+\hslash \left(\kappa \right)\hslash \left(1-\upsilon \right)\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)+\hslash \left(1-\upsilon \right)\hslash \left(1-\kappa \right)\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right).\hfill \end{array}$$

(24)

If one assumes that$\hslash \left(\upsilon \right)=\upsilon , \hslash \left(\kappa \right)=\kappa$ and ${\mathsf{Ԓ}}_{*}\left(x,y\right)={\mathsf{Ԓ}}^{*}\left(x,y\right)$, then $\mathsf{Ԓ}$ is referred to as a classical coordinated convex function if $\mathsf{Ԓ}$ meets the stated inequality here:

$$\begin{array}{c}\mathsf{Ԓ}\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},\kappa \epsilon +\left(1-\kappa \right)\mathfrak{v}\right)\hfill \\ \le \upsilon \kappa \mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)+\upsilon \left(1-\kappa \right)\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)+\left(1-\upsilon \right)\kappa \mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)+\left(1-\upsilon \right)\left(1-\kappa \right)\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right).\hfill \end{array}$$

(25)

Let one assume that $\hslash \left(\upsilon \right)=\upsilon , \hslash \left(\kappa \right)=\kappa$ and ${\mathsf{Ԓ}}_{*}\left(x,y\right)\ne {\mathsf{Ԓ}}^{*}\left(x,y\right)$, and ${\mathsf{Ԓ}}_{*}\left(x,y\right)$ is an affine function and ${\mathsf{Ԓ}}^{*}\left(x,y\right)$ is a concave function for the stated inequality here (see [42]) 

$$\begin{array}{c}\mathsf{Ԓ}\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},\kappa \epsilon +\left(1-\kappa \right)\mathfrak{v}\right)\hfill \\ \supseteq \upsilon \kappa \mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)+\upsilon \left(1-\kappa \right)\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)+\left(1-\upsilon \right)\kappa \mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)+\left(1-\upsilon \right)\left(1-\kappa \right)\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right),\hfill \end{array}$$

(26)

 is true.

Definition 5. 

Let $\mathsf{Ԓ}:\Omega \to {\mathbb{R}}_I^{+}$ be a ${\rm I}VM$ on $\Omega$. Then, we have 

$$\mathsf{Ԓ}\left(x,y\right)=\left[{\mathsf{Ԓ}}_{*}\left(x,y\right), {\mathsf{Ԓ}}^{*}\left(x,y\right)\right],$$

for all $\left(x,y\right)\in \Omega$. Then, $\mathsf{Ԓ}$ is coordinated left-$LR$-$\hslash$-convex (concave) ${\rm I}VM$ on $\Omega ,$ if and only if, ${\mathsf{Ԓ}}_{*}\left(x,y\right)$ and ${\mathsf{Ԓ}}^{*}\left(x,y\right)$ are coordinated $LR$-$\hslash$-convex (concave) and affine functions on $\Omega$, respectively.

Definition 6. 

Let $\mathsf{Ԓ}:\Omega \to {\mathbb{R}}_I^{+}$ be a ${\rm I}VM$ on $\Omega$. Then, we have 

$$\mathsf{Ԓ}\left(x,y\right)=\left[{\mathsf{Ԓ}}_{*}\left(x,y\right), {\mathsf{Ԓ}}^{*}\left(x,y\right)\right],$$

for all $\left(x,y\right)\in \Omega$. Then, $\mathsf{Ԓ}$ is coordinated right-$LR$-$\hslash$-convex (concave) ${\rm I}VM$ on $\Omega ,$ if and only if ${\mathsf{Ԓ}}_{*}\left(x,y\right)$ and ${\mathsf{Ԓ}}^{*}\left(x,y\right)$ are coordinated $\mathit{\hslash }$-affine and $\mathit{\hslash }$-(concave) functions on $\Omega$, respectively.

Theorem 6. 

Let $\Omega$ be a coordinated convex set, and let $\mathsf{Ԓ}:\Omega \to {\mathbb{R}}_I^{+}$ be a ${\rm I}VM$, defined by 

$$\mathsf{Ԓ}\left(x,y\right)=\left[{\mathsf{Ԓ}}_{*}\left(x,y\right), {\mathsf{Ԓ}}^{*}\left(x,y\right)\right],$$

for all $\left(x,y\right)\in \Omega$. Then, $\mathsf{Ԓ}$ is coordinated $\mathit{\hslash }$-concave ${\rm I}VM$ on $\Omega ,$ if and only if ${\mathsf{Ԓ}}_{*}\left(x,y\right)$ and ${\mathsf{Ԓ}}^{*}\left(x,y\right)$ are coordinated $\mathit{\hslash }$-concave and $LR$-$\hslash$-convex functions, respectively.

Proof. 

The demonstration of proof of Theorem 6 is similar to the demonstration proof of Theorem 5. □

Example 2. 

We consider the ${\rm I}VM$s $\mathsf{Ԓ}:\left[0, 1\right]\times \left[0, 1\right]\to {\mathbb{R}}_I^{+}$ defined by, 

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

Then, we have endpoint functions ${\mathsf{Ԓ}}_{*}\left(x,y\right),$ ${\mathsf{Ԓ}}^{*}\left(x,y\right)$, which are both coordinated $\mathit{\hslash }$-concave functions. Hence, $\mathsf{Ԓ}\left(x,y\right)$ is coordinated $LR$-$\mathit{\hslash }$-concave ${\rm I}VM$.

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 coordinated $LR$-$\mathbf{\hslash }$-concave ${\rm I}VM$s.

Theorem 7. 

Let $\mathsf{Ԓ}:\Omega \to {\mathbb{R}}_I^{+}$ be a coordinate $LR$-$\hslash$-convex ${\rm I}VM$ on $\Omega$, where $\mathsf{Ԓ}\left(x,y\right)=\left[{\mathsf{Ԓ}}_{*}\left(x,y\right), {\mathsf{Ԓ}}^{*}\left(x,y\right)\right]$ for all $\left(x,y\right)\in \Omega$ and let $\hslash :\left[0, 1\right]\to {\mathbb{R}}^{+}$. If ${\mathsf{Ԓ}\in \mathfrak{T}\mathfrak{O}}_{\Omega }$, then the following inequalities hold: 

$$\begin{array}{c}\hspace{1em}\frac{1}{{\hslash }^2\left(\frac{1}{2}\right)}\mathsf{Ԓ}\left(\frac{\mathcal{o}+\mathfrak{i}}{2}, \frac{\epsilon +\mathfrak{v}}{2}\right)\hfill \\ \hspace{1em}{\le }_p\frac{\Gamma \left(\alpha +1\right)}{{2\hslash \left(\frac{1}{2}\right)\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathcal{o}}^{+}}^{\alpha }\mathsf{Ԓ}\left(\mathfrak{i},\frac{\epsilon +\mathfrak{v}}{2}\right)+{\mathcal{I}}_{{\mathfrak{i}}^{-}}^{\alpha }\mathsf{Ԓ}\left(\mathcal{o},\frac{\epsilon +\mathfrak{v}}{2}\right)\right]+\frac{\Gamma \left(\beta +1\right)}{{2\hslash \left(\frac{1}{2}\right)\left(\mathfrak{v}-\epsilon \right)}^{\beta }}\left[{\mathcal{I}}_{{\epsilon }^{+}}^{\beta }\mathsf{Ԓ}\left(\frac{\mathcal{o}+\mathfrak{i}}{2},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta }\mathsf{Ԓ}\left(\frac{\mathcal{o}+\mathfrak{i}}{2},\epsilon \right)\right]\hfill \\ \hspace{1em}{\le }_p\frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }{\left(\mathfrak{v}-\epsilon \right)}^{\beta }}\left[{\mathcal{I}}_{{\mathcal{o}}^{+},{\epsilon }^{+}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathcal{o}}^{+},{\mathfrak{v}}^{-}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)+{\mathcal{I}}_{{\mathfrak{i}}^{-},{\epsilon }^{+}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{i}}^{-},{\mathfrak{v}}^{-}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)\right]\hfill \\ \hspace{1em}{\le }_p\frac{\beta \Gamma \left(\alpha +1\right)}{{\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathcal{o}}^{+}}^{\alpha }\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)+{\mathcal{I}}_{{\mathcal{o}}^{+}}^{\alpha }\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{i}}^{-}}^{\alpha }\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)+{\mathcal{I}}_{{\mathfrak{i}}^{-}}^{\alpha }\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)\right]\times {\int }_0^1{\kappa }^{\beta -1}\left[\hslash \left(\kappa \right)+\hslash \left(1-\kappa \right)\right]d\kappa \hfill \\ \hspace{1em}\hspace{1em}+\frac{\alpha \Gamma \left(\beta +1\right)}{{\left(\mathfrak{v}-\epsilon \right)}^{\beta }}\left[{\mathcal{I}}_{{\epsilon }^{+}}^{ \beta }\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta }\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)+{\mathcal{I}}_{{\epsilon }^{+}}^{\beta }\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta }\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)\right]\times {\int }_0^1{\upsilon }^{\alpha -1}\hslash \left(\upsilon \right)+\hslash \left(1-\upsilon \right)d\upsilon \hfill \\ {\le }_p\alpha \beta \left[\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)+\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)+\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)+\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)\right]\times {\int }_0^1{\kappa }^{\beta -1}\left[\hslash \left(\kappa \right)+\hslash \left(1-\kappa \right)\right]d\kappa {\int }_0^1{\upsilon }^{\alpha -1}\left[\hslash \left(\upsilon \right)+\hslash \left(1-\upsilon \right)\right]d\upsilon .\hfill \end{array}$$

(27)

If $\mathsf{Ԓ}\left(x,y\right)$ coordinated $\mathit{\hslash }$-concave ${\rm I}VM$, then, 

$$\begin{array}{c}\frac{1}{{\hslash }^2\left(\frac{1}{2}\right)}\mathsf{Ԓ}\left(\frac{\mathcal{o}+\mathfrak{i}}{2}, \frac{\epsilon +\mathfrak{v}}{2}\right)\hfill \\ {\ge }_p\frac{\Gamma \left(\alpha +1\right)}{{2\hslash \left(\frac{1}{2}\right)\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathcal{o}}^{+}}^{\alpha }\mathsf{Ԓ}\left(\mathfrak{i},\frac{\epsilon +\mathfrak{v}}{2}\right)+{\mathcal{I}}_{{\mathfrak{i}}^{-}}^{\alpha }\mathsf{Ԓ}\left(\mathcal{o},\frac{\epsilon +\mathfrak{v}}{2}\right)\right]+\frac{\Gamma \left(\beta +1\right)}{{2\hslash \left(\frac{1}{2}\right)\left(\mathfrak{v}-\epsilon \right)}^{\beta }}\left[{\mathcal{I}}_{{\epsilon }^{+}}^{\beta }\mathsf{Ԓ}\left(\frac{\mathcal{o}+\mathfrak{i}}{2},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta }\mathsf{Ԓ}\left(\frac{\mathcal{o}+\mathfrak{i}}{2},\epsilon \right)\right]\hfill \\ {\ge }_p\frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }{\left(\mathfrak{v}-\epsilon \right)}^{\beta }}\left[{\mathcal{I}}_{{\mathcal{o}}^{+},{\epsilon }^{+}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathcal{o}}^{+},{\mathfrak{v}}^{-}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)+{\mathcal{I}}_{{\mathfrak{i}}^{-},{\epsilon }^{+}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{i}}^{-},{\mathfrak{v}}^{-}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)\right]\hfill \\ {\ge }_p\frac{\beta \Gamma \left(\alpha +1\right)}{{\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathcal{o}}^{+}}^{\alpha }\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)+{\mathcal{I}}_{{\mathcal{o}}^{+}}^{\alpha }\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{i}}^{-}}^{\alpha }\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)+{\mathcal{I}}_{{\mathfrak{i}}^{-}}^{\alpha }\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)\right]\times {\int }_0^1{\kappa }^{\beta -1}\left[\hslash \left(\kappa \right)+\hslash \left(1-\kappa \right)\right]d\kappa \hfill \\ +\frac{\alpha \Gamma \left(\beta +1\right)}{{\left(\mathfrak{v}-\epsilon \right)}^{\beta }}\left[{\mathcal{I}}_{{\epsilon }^{+}}^{ \beta }\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta }\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)+{\mathcal{I}}_{{\epsilon }^{+}}^{\beta }\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta }\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)\right]\times {\int }_0^1{\upsilon }^{\alpha -1}\left[\hslash \left(\upsilon \right)+\hslash \left(1-\upsilon \right)\right]d\upsilon \hfill \\ {\ge }_p\alpha \beta \left[\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)+\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)+\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)+\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)\right]\times {\int }_0^1{\kappa }^{\beta -1}\left[\hslash \left(\kappa \right)+\hslash \left(1-\kappa \right)\right]d\kappa {\int }_0^1{\upsilon }^{\alpha -1}\left[\hslash \left(\upsilon \right)+\hslash \left(1-\upsilon \right)\right]d\upsilon .\hfill \end{array}$$

(28)

Proof. 

Let $\mathsf{Ԓ}:\left[\mathcal{o},\mathfrak{i}\right]\to {\mathbb{R}}_I^{+}$ be a coordinated $LR$-$\hslash$-convex ${\rm I}VM$. Then, by hypothesis, we have

$$\frac{1}{{\hslash }^2\left(\frac{1}{2}\right)}\mathsf{Ԓ}\left(\frac{\mathcal{o}+\mathfrak{i}}{2},\frac{\epsilon +\mathfrak{v}}{2}\right){\le }_p\mathsf{Ԓ}\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},\upsilon \epsilon +\left(1-\upsilon \right)\mathfrak{v}\right)+\mathsf{Ԓ}\left(\left(1-\upsilon \right)\mathcal{o}+\upsilon \mathfrak{i},\left(1-\upsilon \right)\epsilon +\upsilon \mathfrak{v}\right)$$

By using Theorem 5, we have

$$\begin{array}{c}\frac{1}{{\hslash }^2\left(\frac{1}{2}\right)}{\mathsf{Ԓ}}_{*}\left(\frac{\mathcal{o}+\mathfrak{i}}{2},\frac{\epsilon +\mathfrak{v}}{2}\right) \\ \le {\mathsf{Ԓ}}_{*}\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},\upsilon \epsilon +\left(1-\upsilon \right)\mathfrak{v}\right)+{\mathsf{Ԓ}}_{*}\left(\left(1-\upsilon \right)\mathcal{o}+\upsilon \mathfrak{i},\left(1-\upsilon \right)\epsilon +\upsilon \mathfrak{v}\right), \\ \frac{1}{{\hslash }^2\left(\frac{1}{2}\right)}{\mathsf{Ԓ}}^{*}\left(\frac{\mathcal{o}+\mathfrak{i}}{2},\frac{\epsilon +\mathfrak{v}}{2}\right) \\ \le {\mathsf{Ԓ}}^{*}\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},\upsilon \epsilon +\left(1-\upsilon \right)\mathfrak{v}\right)+{\mathsf{Ԓ}}^{*}\left(\left(1-\upsilon \right)\mathcal{o}+\upsilon \mathfrak{i},\left(1-\upsilon \right)\epsilon +\upsilon \mathfrak{v}\right).\end{array}$$

By using Lemma 1, we have

$$\begin{array}{c}\frac{1}{\hslash \left(\frac{1}{2}\right)}{\mathsf{Ԓ}}_{*}\left(x,\frac{\epsilon +\mathfrak{v}}{2}\right)\le {\mathsf{Ԓ}}_{*}\left(x, \upsilon \epsilon +\left(1-\upsilon \right)\mathfrak{v}\right)+{\mathsf{Ԓ}}_{*}\left(x, \left(1-\upsilon \right)\epsilon +\upsilon \mathfrak{v}\right), \\ \frac{1}{\hslash \left(\frac{1}{2}\right)}{\mathsf{Ԓ}}^{*}\left(x,\frac{\epsilon +\mathfrak{v}}{2}\right)\le {\mathsf{Ԓ}}^{*}\left(x, \upsilon \epsilon +\left(1-\upsilon \right)\mathfrak{v}\right)+{\mathsf{Ԓ}}^{*}\left(x, \left(1-\upsilon \right)\epsilon +\upsilon \mathfrak{v}\right),\end{array}$$

(29)

and

$$\begin{array}{c}\frac{1}{\hslash \left(\frac{1}{2}\right)}{\mathsf{Ԓ}}_{*}\left(\frac{\mathcal{o}+\mathfrak{i}}{2},y\right)\le {\mathsf{Ԓ}}_{*}\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},y\right)+{\mathsf{Ԓ}}_{*}\left(\left(1-\upsilon \right)\mathcal{o}+\upsilon \mathfrak{i},y\right), \\ \frac{1}{\hslash \left(\frac{1}{2}\right)}{\mathsf{Ԓ}}^{*}\left(\frac{\mathcal{o}+\mathfrak{i}}{2},y\right)\le {\mathsf{Ԓ}}^{*}\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},y\right)+{\mathsf{Ԓ}}^{*}\left(\left(1-\upsilon \right)\mathcal{o}+\upsilon \mathfrak{i},y\right).\end{array}$$

(30)

From (29) and (30), we have

$$\begin{gathered} \frac{1}{\hslash \left(\frac{1}{2}\right)}\left[{\mathsf{Ԓ}}_{*}\left(x,\frac{\epsilon +\mathfrak{v}}{2}\right),{\mathsf{Ԓ}}^{*}\left(x,\frac{\epsilon +\mathfrak{v}}{2}\right)\right] \\ {\le }_p\left[{\mathsf{Ԓ}}_{*}\left(x, \upsilon \epsilon +\left(1-\upsilon \right)\mathfrak{v}\right),{\mathsf{Ԓ}}^{*}\left(x, \upsilon \epsilon +\left(1-\upsilon \right)\mathfrak{v}\right)\right] \\ +\left[{\mathsf{Ԓ}}_{*}\left(x, \left(1-\upsilon \right)\epsilon +\upsilon \mathfrak{v}\right),{\mathsf{Ԓ}}^{*}\left(x, \left(1-\upsilon \right)\epsilon +\upsilon \mathfrak{v}\right)\right], \end{gathered}$$

and

$$\begin{array}{c}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\frac{1}{\hslash \left(\frac{1}{2}\right)}\left[{\mathsf{Ԓ}}_{*}\left(\frac{\mathcal{o}+\mathfrak{i}}{2},y\right),{\mathsf{Ԓ}}^{*}\left(\frac{\mathcal{o}+\mathfrak{i}}{2},y\right)\right]\hfill \\ {\le }_p\left[{\mathsf{Ԓ}}_{*}\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},y\right),{\mathsf{Ԓ}}^{*}\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},y\right)\right]\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\left[{\mathsf{Ԓ}}_{*}\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},y\right),{\mathsf{Ԓ}}^{*}\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},y\right)\right],\hfill \end{array}$$

It follows that

$$\frac{1}{\hslash \left(\frac{1}{2}\right)}\mathsf{Ԓ}\left(x,\frac{\epsilon +\mathfrak{v}}{2}\right){\le }_p\mathsf{Ԓ}\left(x, \upsilon \epsilon +\left(1-\upsilon \right)\mathfrak{v}\right)+\mathsf{Ԓ}\left(x, \left(1-\upsilon \right)\epsilon +\upsilon \mathfrak{v}\right),$$

(31)

and

$$\frac{1}{\hslash \left(\frac{1}{2}\right)}\mathsf{Ԓ}\left(\frac{\mathcal{o}+\mathfrak{i}}{2},y\right){\le }_p\mathsf{Ԓ}\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},y\right)+\mathsf{Ԓ}\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},y\right).$$

(32)

Since $\mathsf{Ԓ}\left(x,.\right)$ and $\mathsf{Ԓ}\left(.,y\right)$ are both coordinated $LR$-$\hslash$-convex-${\rm I}VM$s; then, from (17), (31), and (32), we have

$$\frac{1}{\beta \hslash \left(\frac{1}{2}\right)}{\mathsf{Ԓ}}_x\left(\frac{\epsilon +\mathfrak{v}}{2}\right){\le }_p\frac{\Gamma \left(\beta \right)}{{\left(\mathfrak{v}-\epsilon \right)}^{\beta }}\left[{\mathcal{I}}_{{\epsilon }^{+}}^{\beta }{\mathsf{Ԓ}}_x\left(\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta } {\mathsf{Ԓ}}_x\left(\epsilon \right)\right]{\le }_p\left[{\mathsf{Ԓ}}_x\left(\epsilon \right)+{\mathsf{Ԓ}}_x\left(\mathfrak{v}\right)\right]{\int }_0^1{\kappa }^{\beta -1}\left[\hslash \left(\kappa \right)+\hslash \left(1-\kappa \right)\right]d\kappa$$

(33)

and

$$\frac{1}{\alpha \hslash \left(\frac{1}{2}\right)}{\mathsf{Ԓ}}_y\left(\frac{\mathcal{o}+\mathfrak{i}}{2}\right){\le }_p\frac{\Gamma \left(\alpha \right)}{{\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathcal{o}}^{+}}^{\alpha }{\mathsf{Ԓ}}_y\left(\mathfrak{i}\right)+{\mathcal{I}}_{{\mathfrak{i}}^{-}}^{\alpha }{\mathsf{Ԓ}}_y\left(\mathcal{o}\right)\right]{\le }_p\left[{\mathsf{Ԓ}}_y\left(\mathcal{o}\right)+{\mathsf{Ԓ}}_y\left(\mathfrak{i}\right)\right]{\int }_0^1{\upsilon }^{\alpha -1}\hslash \left(\upsilon \right)+\hslash \left(1-\upsilon \right)d\upsilon$$

(34)

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

$$\frac{1}{\beta \hslash \left(\frac{1}{2}\right)}\mathsf{Ԓ}\left(x,\frac{\epsilon +\mathfrak{v}}{2}\right){\le }_p\frac{\Gamma \left(\beta \right)}{{\left(\mathfrak{v}-\epsilon \right)}^{\beta }}\left[{\mathcal{I}}_{{\epsilon }^{+}}^{\alpha }\mathsf{Ԓ}\left(x,\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\alpha }\mathsf{Ԓ}\left(x,\epsilon \right)\right]{\le }_p\left[\mathsf{Ԓ}\left(x,\epsilon \right)+\mathsf{Ԓ}\left(x,\mathfrak{v}\right)\right]{\int }_0^1{\kappa }^{\beta -1}\left[\hslash \left(\kappa \right)+\hslash \left(1-\kappa \right)\right]d\kappa .$$

(35)

That is,

$$\begin{array}{c}\frac{1}{\beta \hslash \left(\frac{1}{2}\right)}\mathsf{Ԓ}\left(x,\frac{\epsilon +\mathfrak{v}}{2}\right){\le }_p\frac{1}{{\left(\mathfrak{v}-\epsilon \right)}^{\beta }}\left[ {\int }_{\epsilon }^{\mathfrak{v}}{\left(\mathfrak{v}-\kappa \right)}^{\beta -1}\mathsf{Ԓ}\left(x,\kappa \right)d\kappa +{\int }_{\epsilon }^{\mathfrak{v}}{\left(\kappa -\epsilon \right)}^{\beta -1}\mathsf{Ԓ}\left(x,\kappa \right)d\kappa \right]\hfill \\ {\le }_p\left[\mathsf{Ԓ}\left(x,\epsilon \right)+\mathsf{Ԓ}\left(x,\mathfrak{v}\right)\right]{\int }_0^1{\kappa }^{\beta -1}\left[\hslash \left(\kappa \right)+\hslash \left(1-\kappa \right)\right]d\kappa .\hfill \end{array}$$

(36)

Multiplying double inequality (36) by $\frac{{\left(\mathfrak{i}-x\right)}^{\alpha -1}}{{\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }}$ and integrating with respect to $x$ over $\left[\mathcal{o},\mathfrak{i}\right],$ we have

$$\begin{array}{c}\frac{1}{{\beta \left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }\hslash \left(\frac{1}{2}\right)}{\int }_{\mathcal{o}}^{\mathfrak{i}}\mathsf{Ԓ}\left(x,\frac{\epsilon +\mathfrak{v}}{2}\right){\left(\mathfrak{i}-x\right)}^{\alpha -1}dx\hfill \\ {\le }_p\frac{1}{{{\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }\left(\mathfrak{v}-\epsilon \right)}^{\beta }}{\int }_{\mathcal{o}}^{\mathfrak{i}}{\int }_{\epsilon }^{\mathfrak{v}}{\left(\mathfrak{i}-x\right)}^{\alpha -1}{\left(\mathfrak{v}-\kappa \right)}^{\beta -1}\mathsf{Ԓ}\left(x,\kappa \right)d\kappa dx+{\int }_{\mathcal{o}}^{\mathfrak{i}}{\int }_{\epsilon }^{\mathfrak{v}}{\left(\mathfrak{i}-x\right)}^{\alpha -1}{\left(\kappa -\epsilon \right)}^{\beta -1}\mathsf{Ԓ}\left(x,\kappa \right)d\kappa dx\hfill \\ {\le }_p\frac{1}{{\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }}\left[{\int }_{\mathcal{o}}^{\mathfrak{i}}{\left(\mathfrak{i}-x\right)}^{\alpha -1}\mathsf{Ԓ}\left(x,\epsilon \right)dx+{\int }_{\mathcal{o}}^{\mathfrak{i}}{\left(\mathfrak{i}-x\right)}^{\alpha -1}\mathsf{Ԓ}\left(x,\mathfrak{v}\right)dx\right]{\int }_0^1{\kappa }^{\beta -1}\left[\hslash \left(\kappa \right)+\hslash \left(1-\kappa \right)\right]d\kappa .\hfill \end{array}$$

(37)

Again, multiplying double inequality (36) by $\frac{{\left(x-\mathcal{o}\right)}^{\alpha -1}}{{\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }}$ and integrating with respect to $x$ over $\left[\mathcal{o},\mathfrak{i}\right],$ we have

$$\begin{array}{c}\frac{1}{{\beta \left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }\hslash \left(\frac{1}{2}\right)}{\int }_{\mathcal{o}}^{\mathfrak{i}}\mathsf{Ԓ}\left(x,\frac{\epsilon +\mathfrak{v}}{2}\right){\left(x-\mathcal{o}\right)}^{\alpha -1}dx\hfill \\ {\le }_p\frac{1}{{\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }{\left(\mathfrak{v}-\epsilon \right)}^{\beta }}{\int }_{\mathcal{o}}^{\mathfrak{i}}{\int }_{\epsilon }^{\mathfrak{v}}{\left(x-\mathcal{o}\right)}^{\alpha -1}{\left(\mathfrak{v}-\kappa \right)}^{\beta -1}\mathsf{Ԓ}\left(x,\kappa \right)d\kappa dx\hfill \\ +\frac{1}{{\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }{\left(\mathfrak{v}-\epsilon \right)}^{\beta }}{\int }_{\mathcal{o}}^{\mathfrak{i}}{\int }_{\epsilon }^{\mathfrak{v}}{\left(x-\mathcal{o}\right)}^{\alpha -1}{\left(\kappa -\epsilon \right)}^{\beta -1}\mathsf{Ԓ}\left(x,\kappa \right)d\kappa dx\hfill \\ {\le }_p\frac{1}{{\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }}\left[{\int }_{\mathcal{o}}^{\mathfrak{i}}{\left(x-\mathcal{o}\right)}^{\alpha -1}\mathsf{Ԓ}\left(x,\epsilon \right)dx+{\int }_{\mathcal{o}}^{\mathfrak{i}}{\left(x-\mathcal{o}\right)}^{\alpha -1}\mathsf{Ԓ}\left(x,\mathfrak{v}\right)dx\right]{\int }_0^1{\kappa }^{\beta -1}\left[\hslash \left(\kappa \right)+\hslash \left(1-\kappa \right)\right]d\kappa .\hfill \end{array}$$

(38)

From (37), we have

$$\begin{array}{c}\frac{\Gamma \left(\alpha +1\right)}{{2\hslash \left(\frac{1}{2}\right)\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathcal{o}}^{+}}^{\alpha }\mathsf{Ԓ}\left(\mathfrak{i},\frac{\epsilon +\mathfrak{v}}{2}\right)\right]\hfill \\ {\le }_p\frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }{\left(\mathfrak{v}-\epsilon \right)}^{\beta }}\left[{\mathcal{I}}_{{\mathcal{o}}^{+},{\epsilon }^{+}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{i}}^{-},{\epsilon }^{+}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)\right]\hfill \\ {\le }_p\frac{\beta \Gamma \left(\alpha +1\right)}{{\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathcal{o}}^{+}}^{\alpha }\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)+{\mathcal{I}}_{{\mathcal{o}}^{+}}^{\alpha }\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)\right]{\int }_0^1{\kappa }^{\beta -1}\left[\hslash \left(\kappa \right)+\hslash \left(1-\kappa \right)\right]d\kappa .\hfill \end{array}$$

(39)

From (38), we have

$$\begin{array}{c}\frac{\Gamma \left(\alpha +1\right)}{{2\hslash \left(\frac{1}{2}\right)\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathfrak{i}}^{-}}^{\alpha }\mathsf{Ԓ}\left(\mathcal{o},\frac{\epsilon +\mathfrak{v}}{2}\right)\right]\hfill \\ {\le }_p\frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }{\left(\mathfrak{v}-\epsilon \right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{i}}^{-},{\epsilon }^{+}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{i}}^{-},{\mathfrak{v}}^{-}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)\right]\hfill \\ {\le }_p\frac{\beta \Gamma \left(\alpha +1\right)}{{\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathfrak{i}}^{-}}^{\alpha }\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)+{\mathcal{I}}_{{\mathfrak{i}}^{-}}^{\alpha }\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)\right]{\int }_0^1{\kappa }^{\beta -1}\left[\hslash \left(\kappa \right)+\hslash \left(1-\kappa \right)\right]d\kappa .\hfill \end{array}$$

(40)

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

$$\begin{array}{c}\frac{\Gamma \left(\beta +1\right)}{{2\hslash \left(\frac{1}{2}\right)\left(\mathfrak{v}-\epsilon \right)}^{\beta }}\left[{\mathcal{I}}_{{\epsilon }^{+}}^{\beta }\mathsf{Ԓ}\left(\frac{\mathcal{o}+\mathfrak{i}}{2},\mathfrak{v}\right)\right]\hfill \\ {\le }_p\frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }{\left(\mathfrak{v}-\epsilon \right)}^{\beta }}\left[{\mathcal{I}}_{{\mathcal{o}}^{+},{\epsilon }^{+}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{i}}^{-},{\epsilon }^{+}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)\right]\hfill \\ {\le }_p\frac{\alpha \Gamma \left(\beta +1\right)}{{\left(\mathfrak{v}-\epsilon \right)}^{\beta }}\left[{\mathcal{I}}_{{\epsilon }^{+} }^{\beta }\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)+{\mathcal{I}}_{{\epsilon }^{+}}^{\beta }\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)\right].\hfill \end{array}$$

(41)

and

$$\begin{array}{c}\frac{\Gamma \left(\beta +1\right)}{{2\hslash \left(\frac{1}{2}\right)\left(\mathfrak{v}-\epsilon \right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta }\mathsf{Ԓ}\left(\frac{\mathcal{o}+\mathfrak{i}}{2},\epsilon \right)\right]\hfill \\ {\le }_p\frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }{\left(\mathfrak{v}-\epsilon \right)}^{\beta }}\left[{\mathcal{I}}_{{\mathcal{o}}^{+},{\mathfrak{v}}^{-}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)+{\mathcal{I}}_{{\mathfrak{i}}^{-},{\mathfrak{v}}^{-}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)\right]\hfill \\ {\le }_p\frac{\alpha \Gamma \left(\beta +1\right)}{{\left(\mathfrak{v}-\epsilon \right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta }\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)+{\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta }\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)\right].\hfill \end{array}$$

(42)

The second, third, and fourth inequalities of (27) will be the consequence of adding the inequalities (41) and (42).

Now, we have inequality (17)’s left portion.

$$\frac{1}{{\hslash }^2\left(\frac{1}{2}\right)}\mathsf{Ԓ}\left(\frac{\mathcal{o}+\mathfrak{i}}{2},\frac{\epsilon +\mathfrak{v}}{2}\right){\le }_p\frac{\Gamma \left(\beta +1\right)}{{\hslash \left(\frac{1}{2}\right)\left(\mathfrak{v}-\epsilon \right)}^{\beta }}\left[{\mathcal{I}}_{{\epsilon }^{+}}^{\beta }\mathsf{Ԓ}\left(\frac{\mathcal{o}+\mathfrak{i}}{2},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta }\mathsf{Ԓ}\left(\frac{\mathcal{o}+\mathfrak{i}}{2},\epsilon \right)\right]$$

(43)

and

$$\frac{1}{{\hslash }^2\left(\frac{1}{2}\right)}\mathsf{Ԓ}\left(\frac{\mathcal{o}+\mathfrak{i}}{2},\frac{\epsilon +\mathfrak{v}}{2}\right){\le }_p\frac{\Gamma \left(\alpha +1\right)}{{\hslash \left(\frac{1}{2}\right)\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathcal{o}}^{+}}^{\alpha }\mathsf{Ԓ}\left(\mathfrak{i},\frac{\epsilon +\mathfrak{v}}{2}\right)+{\mathcal{I}}_{{\mathfrak{i}}^{-}}^{\alpha }\mathsf{Ԓ}\left(\mathcal{o},\frac{\epsilon +\mathfrak{v}}{2}\right)\right]$$

(44)

The following inequality is created by adding the two inequalities (43) and (44):

$$\begin{gathered} \frac{1}{{\hslash }^2\left(\frac{1}{2}\right)}\mathsf{Ԓ}\left(\frac{\mathcal{o}+\mathfrak{i}}{2},\frac{\epsilon +\mathfrak{v}}{2}\right){\le }_p\frac{\Gamma \left(\alpha +1\right)}{{\hslash \left(\frac{1}{2}\right)\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathcal{o}}^{+}}^{\alpha }\mathsf{Ԓ}\left(\mathfrak{i},\frac{\epsilon +\mathfrak{v}}{2}\right)+{\mathcal{I}}_{{\mathfrak{i}}^{-}}^{\alpha }\mathsf{Ԓ}\left(\mathcal{o},\frac{\epsilon +\mathfrak{v}}{2}\right)\right] \\ +\frac{\Gamma \left(\beta +1\right)}{{\hslash \left(\frac{1}{2}\right)\left(\mathfrak{v}-\epsilon \right)}^{\beta }}\left[{\mathcal{I}}_{{\epsilon }^{+}}^{\beta }\mathsf{Ԓ}\left(\frac{\mathcal{o}+\mathfrak{i}}{2},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta }\mathsf{Ԓ}\left(\frac{\mathcal{o}+\mathfrak{i}}{2},\epsilon \right)\right]. \end{gathered}$$

Similarly, since we obtain the set of ${\rm I}VM$s $\mathsf{Ԓ}:\Omega \to {\mathbb{R}}_I^{+}$, the inequality can be expressed as follows:

$$\begin{array}{c}\frac{1}{{\hslash }^2\left(\frac{1}{2}\right)}\mathsf{Ԓ}\left(\frac{\mathcal{o}+\mathfrak{i}}{2},\frac{\epsilon +\mathfrak{v}}{2}\right)\hfill \\ {\le }_p\frac{\Gamma \left(\alpha +1\right)}{{\hslash \left(\frac{1}{2}\right)\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathcal{o}}^{+}}^{\alpha }\mathsf{Ԓ}\left(\mathfrak{i},\frac{\epsilon +\mathfrak{v}}{2}\right)+{\mathcal{I}}_{{\mathfrak{i}}^{-}}^{\alpha }\mathsf{Ԓ}\left(\mathcal{o},\frac{\epsilon +\mathfrak{v}}{2}\right)\right]+\frac{\Gamma \left(\beta +1\right)}{{\hslash \left(\frac{1}{2}\right)\left(\mathfrak{v}-\epsilon \right)}^{\beta }}\left[{\mathcal{I}}_{{\epsilon }^{+}}^{\beta }\mathsf{Ԓ}\left(\frac{\mathcal{o}+\mathfrak{i}}{2},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta }\mathsf{Ԓ}\left(\frac{\mathcal{o}+\mathfrak{i}}{2},\epsilon \right)\right].\hfill \end{array}$$

(45)

The first inequality of (27) is this one.

Now, we have inequality (17)’s right portion:

$$\frac{\Gamma \left(\beta \right)}{{\left(\mathfrak{v}-\epsilon \right)}^{\beta }}\left[{\mathcal{I}}_{{\epsilon }^{+}}^{\beta }\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta }\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)\right]{\le }_p\left[\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)+\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)\right]\times {\int }_0^1{\kappa }^{\beta -1}\left[\hslash \left(\kappa \right)+\hslash \left(1-\kappa \right)\right]d\kappa$$

(46)

$$\frac{\Gamma \left(\beta \right)}{{\left(\mathfrak{v}-\epsilon \right)}^{\beta }}\left[{\mathcal{I}}_{{\epsilon }^{+}}^{\beta }\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta }\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)\right]{\le }_p\left[\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)+\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)\right]\times {\int }_0^1{\kappa }^{\beta -1}\left[\hslash \left(\kappa \right)+\hslash \left(1-\kappa \right)\right]d\kappa$$

(47)

$$\frac{\Gamma \left(\alpha \right)}{{\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathcal{o}}^{+}}^{\alpha }\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)+{\mathcal{I}}_{{\mathfrak{i}}^{-}}^{\alpha }\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)\right]{\le }_p\left[\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)+\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)\right]\times {\int }_0^1{\upsilon }^{\alpha -1}\hslash \left(\upsilon \right)+\hslash \left(1-\upsilon \right)d\upsilon$$

(48)

$$\frac{\Gamma \left(\alpha \right)}{{\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathcal{o}}^{+}}^{\alpha }\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{i}}^{-}}^{\alpha }\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)\right]{\le }_p\left[\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)+\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)\right]\times {\int }_0^1{\upsilon }^{\alpha -1}\hslash \left(\upsilon \right)+\hslash \left(1-\upsilon \right)d\upsilon$$

(49)

Summing inequalities (46), (47), (48), and (49), and then taking the multiplication of the resultant with $\alpha \beta$, we have

$$\begin{array}{c}\frac{\beta \Gamma \left(\alpha +1\right)}{{\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathcal{o}}^{+}}^{\alpha }\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)+{\mathcal{I}}_{{\mathfrak{i}}^{-}}^{\alpha }\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)+{\mathcal{I}}_{{\mathcal{o}}^{+}}^{\alpha }\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{i}}^{-}}^{\alpha }\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)\right]\hfill \\ +\frac{\alpha \Gamma \left(\beta +1\right)}{{\left(\mathfrak{v}-\epsilon \right)}^{\beta }}\left[{\mathcal{I}}_{{\epsilon }^{+}}^{\beta }\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta }\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)+{\mathcal{I}}_{{\epsilon }^{+}}^{\beta }\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta }\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)\right]\hfill \\ {\le }_p\left[\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)+\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)+\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)+\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)\right]\times {\int }_0^1{\kappa }^{\beta -1}\left[\hslash \left(\kappa \right)+\hslash \left(1-\kappa \right)\right]d\kappa {\int }_0^1{\upsilon }^{\alpha -1}\hslash \left(\upsilon \right)+\hslash \left(1-\upsilon \right)d\upsilon .\hfill \end{array}$$

(50)

This is the final inequality of (27) and the conclusion has been established. □

Example 3. 

We assume the ${\rm I}VM$s $\mathsf{Ԓ}:\left[0, 2\right]\times \left[0, 2\right]\to {\mathbb{R}}_I^{+}$ defined by 

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

(51)

 then, for each $\gamma \in \left[0, 1\right],$ we have. Endpoint functions ${\mathsf{Ԓ}}_{*}\left(x,y\right),$ ${\mathsf{Ԓ}}^{*}\left(x,y\right)$ are coordinate $LR$-$\hslash$-convex and $\mathit{\hslash }$-concave functions. Hence, $\widetilde{\mathsf{Ԓ}}\left(x,y\right)$ is $\mathit{\hslash }$-coordinated $LR$-$\hslash$-convex ${\rm I}VM$.

$$\begin{array}{c}\mathsf{Ԓ}\left(\frac{\mathcal{o}+\mathfrak{i}}{2}, \frac{\epsilon +\mathfrak{v}}{2}\right)=\left[1,2\right],\hfill \\ \frac{\Gamma \left(\alpha +1\right)}{{4\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathcal{o}}^{+}}^{\alpha }\mathsf{Ԓ}\left(\mathfrak{i},\frac{\epsilon +\mathfrak{v}}{2}\right)+{\mathcal{I}}_{{\mathfrak{i}}^{-}}^{\alpha }\mathsf{Ԓ}\left(\mathcal{o},\frac{\epsilon +\mathfrak{v}}{2}\right)\right]+\frac{\Gamma \left(\beta +1\right)}{{4\left(\mathfrak{v}-\epsilon \right)}^{\beta }}\left[{\mathcal{I}}_{{\epsilon }^{+}}^{\beta }\mathsf{Ԓ}\left(\frac{\mathcal{o}+\mathfrak{i}}{2},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta }\mathsf{Ԓ}\left(\frac{\mathcal{o}+\mathfrak{i}}{2},\epsilon \right)\right]\hfill \\ =2-\frac{\sqrt{2}}{4}-\frac{\sqrt{2}}{8}\pi ·\left[1,2\right]\hfill \\ \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{4\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }{\left(\mathfrak{v}-\epsilon \right)}^{\beta }}\left[{\mathcal{I}}_{{\mathcal{o}}^{+},{\epsilon }^{+}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathcal{o}}^{+},{\mathfrak{v}}^{-}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)+{\mathcal{I}}_{{\mathfrak{i}}^{-},{\epsilon }^{+}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{i}}^{-},{\mathfrak{v}}^{-}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)\right]\hfill \\ =\frac{33}{8}-\sqrt{2}-\frac{\sqrt{2}}{2}\pi +\frac{\pi }{8}+\frac{{\pi }^2}{32}·\left[1,2\right]\hfill \\ \frac{\Gamma \left(\alpha +1\right)}{{8\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathcal{o}}^{+}}^{\alpha }\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)+{\mathcal{I}}_{{\mathcal{o}}^{+}}^{\alpha }\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{i}}^{-}}^{\alpha }\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)+{\mathcal{I}}_{{\mathfrak{i}}^{-}}^{\alpha }\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)\right]\hfill \\ +\frac{\Gamma \left(\beta +1\right)}{{8\left(\mathfrak{v}-\epsilon \right)}^{\beta }}\left[{\mathcal{I}}_{{\epsilon }^{+}}^{\beta }\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)+{\mathcal{I}}_{{\epsilon }^{+}}^{\beta }\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta }\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)+{\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta }\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)\right]\hfill \\ =\frac{34\sqrt{2}+\left(\sqrt{2}-4\right)\pi -24}{8\sqrt{2}}·\left[1,2\right]\hfill \\ \frac{\mathsf{Ԓ}\left(\epsilon ,\mathfrak{i}\right)+\mathsf{Ԓ}\left(\sigma ,\mathfrak{i}\right)+\mathsf{Ԓ}\left(\epsilon ,\mathfrak{v}\right)+\mathsf{Ԓ}\left(\sigma ,\mathfrak{v}\right)}{4}=\left(\frac{9}{2}-2\sqrt{2}\right)·\left[1,2\right].\hfill \end{array}$$

That is 

$$\begin{array}{c}\left[1,2\right]{\le }_{p}2-\frac{\sqrt{2}}{4}-\frac{\sqrt{2}}{8}\pi ·\left[1,2\right]{\le }_p\frac{33}{8}-\sqrt{2}-\frac{\sqrt{2}}{2}\pi +\frac{\pi }{8}+\frac{{\pi }^2}{32}·\left[1,2\right]{\le }_p\frac{34\sqrt{2}+\left(\sqrt{2}-4\right)\pi -24}{8\sqrt{2}}·\left[1,2\right]{\le }_p\left(\frac{9}{2}-2\sqrt{2}\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}·\left[1,2\right]\hfill \end{array}$$

 Hence, Theorem 6 has been verified.

Remark 3. 

If one assumes that $\alpha =1$ and $\beta =1$, and $\hslash \left(\upsilon \right)=\upsilon , \hslash \left(\kappa \right)=\kappa$, then from (27), as a result, there will be inequality (see [43]): 

$$\begin{array}{c}\mathsf{Ԓ}\left(\frac{\mathcal{o}+\mathfrak{i}}{2}, \frac{\epsilon +\mathfrak{v}}{2}\right)\hfill \\ {\le }_p\frac{\begin{array}{c} \\ 1\end{array}}{2}\left[\frac{1}{\mathfrak{i}-\mathcal{o}}{\int }_{\mathcal{o}}^{\mathfrak{i}}\mathsf{Ԓ}\left(x,\frac{\epsilon +\mathfrak{v}}{2}\right)dx+\frac{1}{\mathfrak{v}-\epsilon }{\int }_{\epsilon }^{\mathfrak{v}}\mathsf{Ԓ}\left(\frac{\mathcal{o}+\mathfrak{i}}{2},y\right)dy\right]{\le }_p\frac{1}{\left(\mathfrak{i}-\mathcal{o}\right)\left(\mathfrak{v}-\epsilon \right)}{\int }_{\mathcal{o}}^{\mathfrak{i}}{\int }_{\epsilon }^{\mathfrak{v}}\mathsf{Ԓ}\left(x,y\right)dydx\hfill \\ {\le }_p\frac{\begin{array}{c} \\ 1\end{array}}{4\left(\mathfrak{i}-\mathcal{o}\right)}\left[{\int }_{\mathcal{o}}^{\mathfrak{i}}\mathsf{Ԓ}\left(x,\epsilon \right)dx+{\int }_{\mathcal{o}}^{\mathfrak{i}}\mathsf{Ԓ}\left(x,\mathfrak{v}\right)dx\right]+\frac{\begin{array}{c} \\ 1\end{array}}{4\left(\mathfrak{v}-\epsilon \right)}\left[{\int }_{\epsilon }^{\mathfrak{v}}\mathsf{Ԓ}\left(\mathcal{o},y\right)dy+{\int }_{\epsilon }^{\mathfrak{v}}\mathsf{Ԓ}\left(\mathfrak{i},y\right)dy\right]\hfill \\ {\le }_p\frac{\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)+\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)+\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)+\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)}{4}.\hfill \end{array}$$

(52)

If one assumes that $\alpha =1$ and $\beta =1$, $\hslash \left(\upsilon \right)=\upsilon , \hslash \left(\kappa \right)=\kappa$ and $\mathsf{Ԓ}$ is coordinated left-$LR$-$\mathit{\hslash }$-convex, then from (27), as a result, there will be inequality (see [42]): 

$$\begin{array}{c}\mathsf{Ԓ}\left(\frac{\mathcal{o}+\mathfrak{i}}{2}, \frac{\epsilon +\mathfrak{v}}{2}\right)\hfill \\ \supseteq \frac{\begin{array}{c} \\ 1\end{array}}{2}\left[\frac{1}{\mathfrak{i}-\mathcal{o}}{\int }_{\mathcal{o}}^{\mathfrak{i}}\mathsf{Ԓ}\left(x,\frac{\epsilon +\mathfrak{v}}{2}\right)dx+\frac{1}{\mathfrak{v}-\epsilon }{\int }_{\epsilon }^{\mathfrak{v}}\mathsf{Ԓ}\left(\frac{\mathcal{o}+\mathfrak{i}}{2},y\right)dy\right]\supseteq \frac{1}{\left(\mathfrak{i}-\mathcal{o}\right)\left(\mathfrak{v}-\epsilon \right)}{\int }_{\mathcal{o}}^{\mathfrak{i}}{\int }_{\epsilon }^{\mathfrak{v}}\mathsf{Ԓ}\left(x,y\right)dydx\hfill \\ \supseteq \frac{\begin{array}{c} \\ 1\end{array}}{4\left(\mathfrak{i}-\mathcal{o}\right)}\left[{\int }_{\mathcal{o}}^{\mathfrak{i}}\mathsf{Ԓ}\left(x,\epsilon \right)dx+{\int }_{\mathcal{o}}^{\mathfrak{i}}\mathsf{Ԓ}\left(x,\mathfrak{v}\right)dx\right]+\frac{\begin{array}{c} \\ 1\end{array}}{4\left(\mathfrak{v}-\epsilon \right)}\left[{\int }_{\epsilon }^{\mathfrak{v}}\mathsf{Ԓ}\left(\mathcal{o},y\right)dy+{\int }_{\epsilon }^{\mathfrak{v}}\mathsf{Ԓ}\left(\mathfrak{i},y\right)dy\right]\hfill \\ \supseteq \frac{\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)+\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)+\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)+\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)}{4}.\hfill \end{array}$$

(53)

If $\hslash \left(\upsilon \right)=\upsilon , \hslash \left(\kappa \right)=\kappa$ and ${\mathsf{Ԓ}}_{*}\left(x,y\right)\ne {\mathsf{Ԓ}}^{*}\left(x,y\right)$, then from (27), we succeed in bringing about the upcoming inequality (see [46]): 

$$\begin{array}{c}\hfill \hspace{1em}\mathsf{Ԓ}\left(\frac{\mathcal{o}+\mathfrak{i}}{2}, \frac{\epsilon +\mathfrak{v}}{2}\right)\hfill \\ \hspace{1em}{\le }_p\frac{\Gamma \left(\alpha +1\right)}{{4\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathcal{o}}^{+}}^{\alpha }\mathsf{Ԓ}\left(\mathfrak{i},\frac{\epsilon +\mathfrak{v}}{2}\right)+{\mathcal{I}}_{{\mathfrak{i}}^{-}}^{\alpha }\mathsf{Ԓ}\left(\mathcal{o},\frac{\epsilon +\mathfrak{v}}{2}\right)\right]+\frac{\Gamma \left(\beta +1\right)}{{4\left(\mathfrak{v}-\epsilon \right)}^{\beta }}\left[{\mathcal{I}}_{{\epsilon }^{+}}^{\beta }\mathsf{Ԓ}\left(\frac{\mathcal{o}+\mathfrak{i}}{2},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta }\mathsf{Ԓ}\left(\frac{\mathcal{o}+\mathfrak{i}}{2},\epsilon \right)\right]\hfill \\ {\le }_p\frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{4\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }{\left(\mathfrak{v}-\epsilon \right)}^{\beta }} \left[{\mathcal{I}}_{{\mathcal{o}}^{+},{\epsilon }^{+}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathcal{o}}^{+},{\mathfrak{v}}^{-}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)+{\mathcal{I}}_{{\mathfrak{i}}^{-},{\epsilon }^{+}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{i}}^{-},{\mathfrak{v}}^{-}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)\right]\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}{\le }_p\frac{\Gamma \left(\alpha +1\right)}{{8\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathcal{o}}^{+} }^{\alpha }\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)+{\mathcal{I}}_{{\mathcal{o}}^{+}}^{\alpha }\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{i}}^{-}}^{\alpha }\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)+{\mathcal{I}}_{{\mathfrak{i}}^{-}}^{\alpha }\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)\right].\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}+\frac{\Gamma \left(\beta +1\right)}{{8\left(\mathfrak{v}-\epsilon \right)}^{\beta }}\left[{\mathcal{I}}_{{\epsilon }^{+} }^{ \beta }\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta }\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)+{\mathcal{I}}_{{\epsilon }^{+}}^{\beta }\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta }\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)\right]\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\le }_p\frac{\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)+\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)+\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)+\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)}{4}.\hfill \end{array}$$

(54)

If $\hslash \left(\upsilon \right)=\upsilon , \hslash \left(\kappa \right)=\kappa$ and ${\mathsf{Ԓ}}_{*}\left(x,y\right)\ne {\mathsf{Ԓ}}^{*}\left(x,y\right)$, then by (27), we succeed in bringing about the upcoming inequality (see [43]): 

$$\begin{gathered} \mathsf{Ԓ}\left(\frac{\mathcal{o}+\mathfrak{i}}{2}, \frac{\epsilon +\mathfrak{v}}{2}\right) \\ {\le }_p\frac{\begin{array}{c} \\ 1\end{array}}{2}\left[\frac{1}{\mathfrak{i}-\mathcal{o}}{\int }_{\mathcal{o}}^{\mathfrak{i}}\mathsf{Ԓ}\left(x,\frac{\epsilon +\mathfrak{v}}{2}\right)dx+\frac{1}{\mathfrak{v}-\epsilon }{\int }_{\epsilon }^{\mathfrak{v}}\mathsf{Ԓ}\left(\frac{\mathcal{o}+\mathfrak{i}}{2},y\right)dy\right]{\le }_p\frac{1}{\left(\mathfrak{i}-\mathcal{o}\right)\left(\mathfrak{v}-\epsilon \right)} {\int }_{\mathcal{o}}^{\mathfrak{i}}{\int }_{\epsilon }^{\mathfrak{v}}\mathsf{Ԓ}\left(x,y\right)dydx \\ {\le }_p\frac{\begin{array}{c} \\ 1\end{array}}{4\left(\mathfrak{i}-\mathcal{o}\right)}\left[{\int }_{\mathcal{o}}^{\mathfrak{i}}\mathsf{Ԓ}\left(x,\epsilon \right)dx+{\int }_{\mathcal{o}}^{\mathfrak{i}}\mathsf{Ԓ}\left(x,\mathfrak{v}\right)dx\right]+\frac{\begin{array}{c} \\ 1\end{array}}{4\left(\mathfrak{v}-\epsilon \right)}\left[{\int }_{\epsilon }^{\mathfrak{v}}\mathsf{Ԓ}\left(\mathcal{o},y\right)dy+{\int }_{\epsilon }^{\mathfrak{v}}\mathsf{Ԓ}\left(\mathfrak{i},y\right)dy\right] \\ {\le }_p\frac{\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)+\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)+\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)+\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)}{4}. \end{gathered}$$

(55)

If $\mathsf{Ԓ}$ is coordinated $LR$-$\mathit{\hslash }$-convex with $\hslash \left(\upsilon \right)=\upsilon , \hslash \left(\kappa \right)=\kappa$ and ${\mathsf{Ԓ}}_{*}\left(x,y\right)={\mathsf{Ԓ}}^{*}\left(x,y\right)$, then from (28), we succeed in bringing about the upcoming classical inequality: 

$$\begin{array}{c}\hfill \mathsf{Ԓ}\left(\frac{\mathcal{o}+\mathfrak{i}}{2}, \frac{\epsilon +\mathfrak{v}}{2}\right)\hfill \\ \le \frac{\Gamma \left(\alpha +1\right)}{{4\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathcal{o}}^{+}}^{\alpha } \mathsf{Ԓ}\left(\mathfrak{i},\frac{\epsilon +\mathfrak{v}}{2}\right)+{\mathcal{I}}_{{\mathfrak{i}}^{-}}^{\alpha }\mathsf{Ԓ}\left(\mathcal{o},\frac{\epsilon +\mathfrak{v}}{2}\right)\right]+\frac{\Gamma \left(\beta +1\right)}{{4\left(\mathfrak{v}-\epsilon \right)}^{\beta }}\left[{\mathcal{I}}_{{\epsilon }^{+}}^{\beta } \mathsf{Ԓ}\left(\frac{\mathcal{o}+\mathfrak{i}}{2},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta } \mathsf{Ԓ}\left(\frac{\mathcal{o}+\mathfrak{i}}{2},\epsilon \right)\right]\hfill \\ \hfill \le \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{4\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }{\left(\mathfrak{v}-\epsilon \right)}^{\beta }} \left[{\mathcal{I}}_{{\mathcal{o}}^{+},{\epsilon }^{+} }^{\alpha , \beta }\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathcal{o}}^{+},{\mathfrak{v}}^{-}}^{\alpha , \beta } \mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)+{\mathcal{I}}_{{\mathfrak{i}}^{-},{\epsilon }^{+} }^{\alpha , \beta }\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{i}}^{-},{\mathfrak{v}}^{-}}^{\alpha , \beta } \mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)\right]\hfill \\ \le \frac{\Gamma \left(\alpha +1\right)}{{8\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathcal{o}}^{+} }^{\alpha }\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)\mathsf{Ԓ}{\mathcal{I}}_{{\mathcal{o}}^{+}}^{\alpha } \mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{i}}^{-} }^{\alpha }\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)+{\mathcal{I}}_{{\mathfrak{i}}^{-}}^{\alpha } \mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)\right].\hfill \\ \hspace{1em}+\frac{\Gamma \left(\beta +1\right)}{{8\left(\mathfrak{v}-\epsilon \right)}^{\beta }}\left[{\mathcal{I}}_{{\epsilon }^{+} }^{ \beta }\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)\widetilde{+}{\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta } \mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)+{\mathcal{I}}_{{\epsilon }^{+} }^{\beta }\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta } \mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)\right]\hfill \\ \hfill \le \frac{\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)+\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)+\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)+\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)}{4}.\hfill \end{array}$$

(56)

In the next outcomes, we are going to find very interesting outcomes that will be obtained over the product of two coordinated $LR$-$\hslash$-convex ${\rm I}VM$s. These inequalities are known as Pachpatte’s inequalities.

Theorem 7. 

Let $\mathsf{Ԓ}, \mathcal{J}:\Omega \to {\mathbb{R}}_I^{+}$ be two coordinated $LR$-$\hslash$-convex ${\rm I}VM$s on $\Omega$, given by $\mathsf{Ԓ}\left(x,y\right)=\left[{\mathsf{Ԓ}}_{*}\left(x,y\right), {\mathsf{Ԓ}}^{*}\left(x,y\right)\right]$ and $\mathcal{J}\left(x,y\right)=\left[{\mathcal{J}}_{*}\left(x,y\right), {\mathcal{J}}^{*}\left(x,y\right)\right]$ for all $\left(x,y\right)\in \Omega$ and let ${\hslash }_1,{\hslash }_2 :\left[0, 1\right]\to {\mathbb{R}}^{+}$. If ${\mathsf{Ԓ}\times \mathcal{J}\in \mathfrak{T}\mathfrak{O}}_{\Omega }$, then the following inequalities hold: 

$$\begin{array}{c}\hspace{1em}\frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }{\left(\mathfrak{v}-\epsilon \right)}^{\beta }}\left[{\mathcal{I}}_{{\mathcal{o}}^{+},{\epsilon }^{+}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)\times \mathcal{J}\left(\mathfrak{i},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathcal{o}}^{+},{\mathfrak{v}}^{-}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)\times \mathcal{J}\left(\mathfrak{i},\epsilon \right)\right]\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }{\left(\mathfrak{v}-\epsilon \right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{i}}^{-},{\epsilon }^{+}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)\times \mathcal{J}\left(\mathcal{o},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{i}}^{-},{\mathfrak{v}}^{-}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)\times \mathcal{J}\left(\mathcal{o},\epsilon \right)\right]\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\le }_p\mathcal{M}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right){\int }_0^1{\upsilon }^{\alpha -1}{\kappa }^{\beta -1}[{\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(1-\kappa \right){\hslash }_2\left(1-\kappa \right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\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)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\hslash }_1\left(\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(\kappa \right)]d\upsilon d\kappa \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+P\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right){\int }_0^1{\upsilon }^{\alpha -1}{\kappa }^{\beta -1}[{\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(1-\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\upsilon ){\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(1-\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\upsilon \left){\hslash }_2\left(\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(\kappa \right)\right]d\upsilon d\kappa \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em} +\mathcal{N}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right){\int }_0^1{\upsilon }^{\alpha -1}{\kappa }^{\beta -1}[{\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(1-\kappa \right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\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)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\hslash }_1\left(\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(1-\kappa \right)]d\upsilon d\kappa \hfill \\ +Q\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right){\int }_0^1{\upsilon }^{\alpha -1}{\kappa }^{\beta -1}[{\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)+\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\hslash }_1(1-\upsilon ){\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)]d\upsilon d\kappa .\hfill \end{array}$$

(57)

If $\mathsf{Ԓ}$ and $\mathcal{J}$ are both coordinated $\mathit{\hslash }$-concave ${\rm I}VM$s on $\Omega$, then the inequality above can be expressed as follows: 

$$\begin{array}{c}\frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }{\left(\mathfrak{v}-\epsilon \right)}^{\beta }}\left[{\mathcal{I}}_{{\mathcal{o}}^{+},{\epsilon }^{+}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)\times \mathcal{J}\left(\mathfrak{i},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathcal{o}}^{+},{\mathfrak{v}}^{-}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)\times \mathcal{J}\left(\mathfrak{i},\epsilon \right)\right]\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }{\left(\mathfrak{v}-\epsilon \right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{i}}^{-},{\epsilon }^{+}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)\times \mathcal{J}\left(\mathcal{o},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{i}}^{-},{\mathfrak{v}}^{-}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)\times \mathcal{J}\left(\mathcal{o},\epsilon \right)\right]\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\ge }_p\mathcal{M}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right){\int }_0^1{\upsilon }^{\alpha -1}{\kappa }^{\beta -1}[{\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(1-\kappa \right){\hslash }_2\left(1-\kappa \right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\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)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\hslash }_1\left(\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(\kappa \right)]d\upsilon d\kappa \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+P\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right){\int }_0^1{\upsilon }^{\alpha -1}{\kappa }^{\beta -1}[{\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(1-\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\upsilon ){\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(1-\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\upsilon \left){\hslash }_2\left(\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(\kappa \right)\right]d\upsilon d\kappa \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em} +\mathcal{N}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right){\int }_0^1{\upsilon }^{\alpha -1}{\kappa }^{\beta -1}[{\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(1-\kappa \right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\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)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\hslash }_1\left(\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(1-\kappa \right)]d\upsilon d\kappa \hfill \\ +Q\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right){\int }_0^1{\upsilon }^{\alpha -1}{\kappa }^{\beta -1}[{\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)+\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\hslash }_1(1-\upsilon ){\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)]d\upsilon d\kappa \hfill \end{array}$$

(58)

where 

$$\begin{array}{c}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathcal{M}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)=\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)\times \mathcal{J}\left(\mathcal{o},\epsilon \right)+\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)\times \mathcal{J}\left(\mathfrak{i},\epsilon \right)+\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)\times \mathcal{J}\left(\mathcal{o},\mathfrak{v}\right)+\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)\times \mathcal{J}\left(\mathfrak{i},\mathfrak{v}\right),\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}P\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)=\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)\times \mathcal{J}\left(\mathfrak{i},\epsilon \right)+\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)\times \mathcal{J}\left(\mathcal{o},\epsilon \right)+\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)\times \mathcal{J}\left(\mathfrak{i},\mathfrak{v}\right)+\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)\times \mathcal{J}\left(\mathcal{o},\mathfrak{v}\right),\hfill \\ \hspace{1em}\hspace{1em}\mathcal{N}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)=\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)\times \mathcal{J}\left(\mathcal{o},\mathfrak{v}\right)+\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)\times \mathcal{J}\left(\mathfrak{i},\mathfrak{v}\right)+\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)\times \mathcal{J}\left(\mathcal{o},\epsilon \right)+\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)\times \mathcal{J}\left(\mathfrak{i},\epsilon \right),\hfill \\ Q\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)=\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)\times \mathcal{J}\left(\mathfrak{i},\mathfrak{v}\right)+\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)\times \mathcal{J}\left(\mathcal{o},\mathfrak{v}\right)+\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)\times \mathcal{J}\left(\mathfrak{i},\epsilon \right)+\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)\times \mathcal{J}\left(\mathcal{o},\epsilon \right) ,\hfill \end{array}$$

 and $\mathcal{M}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)$, $P\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)$, $\mathcal{N}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)$ and $Q\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)$ are defined as follows: 

$$\mathcal{M}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)=\left[{\mathcal{M}}_{*}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right), {\mathcal{M}}^{*}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)\right],$$

$$P\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)=\left[P_{*}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right), P^{*}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)\right],$$

$$\mathcal{N}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)=\left[{\mathcal{N}}_{*}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right), {\mathcal{N}}^{*}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)\right],$$

$$Q\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)=\left[Q_{*}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right), Q^{*}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)\right].$$

Proof. 

Let $\mathsf{Ԓ}$ and $\mathcal{J}$ be two coordinated $LR$-${\hslash }_1$ and $LR$-${\hslash }_2$-convex ${\rm I}VM$s on $\left[\mathcal{o},\mathfrak{i}\right]\times \left[\epsilon , \mathfrak{v}\right]$, respectively. Then,

$$\begin{array}{c}\hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathsf{Ԓ}\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},\kappa \epsilon +\left(1-\kappa \right)\mathfrak{v}\right)\hfill \\ \hfill \hspace{1em}\hspace{1em}{\le }_p{\hslash }_1\left(\upsilon \right){\hslash }_1\left(\kappa \right)\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)+{\hslash }_1\left(\upsilon \right){\hslash }_1\left(1-\kappa \right)\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)+{\hslash }_1\left(1-\upsilon \right){\hslash }_1\left(\kappa \right)\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)\hfill \\ \hfill \hspace{1em}\hspace{1em}+{\hslash }_1\left(1-\upsilon \right){\hslash }_1\left(1-\kappa \right)\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right),\hfill \\ \hfill \mathsf{Ԓ}\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},\left(1-\kappa \right)\epsilon +\kappa \mathfrak{v}\right)\hfill \\ \hfill {\le }_p{\hslash }_1\left(\upsilon \right){\hslash }_1\left(1-\kappa \right)\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)+{\hslash }_1\left(\upsilon \right){\hslash }_1\left(\kappa \right)\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)+{\hslash }_1\left(1-\upsilon \right){\hslash }_1\left(1-\kappa \right)\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)+{\hslash }_1\left(1-\upsilon \right){\hslash }_1\left(\kappa \right)\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right),\hfill \end{array}$$

$$\begin{array}{c}\hfill \mathsf{Ԓ}\left(\left(1-\upsilon \right)\mathcal{o}+\upsilon \mathfrak{i},\kappa \epsilon +\left(1-\kappa \right)\mathfrak{v}\right)\hfill \\ \hfill {\le }_p{\hslash }_1\left(1-\upsilon \right){\hslash }_1\left(\kappa \right)\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)+{\hslash }_1\left(1-\upsilon \right){\hslash }_1\left(1-\kappa \right)\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)+{\hslash }_1\left(\upsilon \right){\hslash }_1\left(\kappa \right)\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\hslash }_1\left(\upsilon \right){\hslash }_1\left(1-\kappa \right)\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right),\hfill \\ \hfill \mathsf{Ԓ}\left(\left(1-\upsilon \right)\mathcal{o}+\upsilon \mathfrak{i},\left(1-\kappa \right)\epsilon +\kappa \mathfrak{v}\right)\hfill \\ \hfill {\le }_p{\hslash }_1\left(1-\upsilon \right){\hslash }_1\left(1-\kappa \right)\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)+{\hslash }_1\left(1-\upsilon \right){\hslash }_1\left(\kappa \right)\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)+{\hslash }_1\left(\upsilon \right){\hslash }_1\left(1-\kappa \right)\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\hslash }_1\left(\upsilon \right){\hslash }_1\left(\kappa \right)\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right),\hfill \end{array}$$

and

$$\begin{array}{c}\hfill \hspace{1em}\hspace{1em}\mathcal{J}\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},\kappa \epsilon +\left(1-\kappa \right)\mathfrak{v}\right)\hfill \\ \hfill \hspace{1em}\hspace{1em}{\le }_p{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)\mathcal{J}\left(\mathcal{o},\epsilon \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)\mathcal{J}\left(\mathcal{o},\mathfrak{v}\right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)\mathcal{J}\left(\mathfrak{i},\epsilon \right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)\mathcal{J}\left(\mathfrak{i},\mathfrak{v}\right),\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathcal{J}\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},\left(1-\kappa \right)\epsilon +\kappa \mathfrak{v}\right)\hfill \\ \hfill {\le }_p{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)\mathcal{J}\left(\mathcal{o},\epsilon \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)\mathcal{J}\left(\mathcal{o},\mathfrak{v}\right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)\mathcal{J}\left(\mathfrak{i},\epsilon \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)\mathcal{J}\left(\mathfrak{i},\mathfrak{v}\right),\hfill \end{array}$$

$$\begin{array}{c}\hfill \left(\left(1-\upsilon \right)\mathcal{o}+\upsilon \mathfrak{i},\kappa \epsilon +\left(1-\kappa \right)\mathfrak{v}\right)\hfill \\ \hfill {\le }_p{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)\mathcal{J}\left(\mathcal{o},\epsilon \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)\mathcal{J}\left(\mathcal{o},\mathfrak{v}\right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)\mathcal{J}\left(\mathfrak{i},\epsilon \right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)\mathcal{J}\left(\mathfrak{i},\mathfrak{v}\right),\hfill \\ \hfill \mathcal{J}\left(\left(1-\upsilon \right)\mathcal{o}+\upsilon \mathfrak{i},\left(1-\kappa \right)\epsilon +\kappa \mathfrak{v}\right)\hfill \\ \hfill {\le }_p{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)\mathcal{J}\left(\mathcal{o},\epsilon \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)\mathcal{J}\left(\mathcal{o},\mathfrak{v}\right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)\mathcal{J}\left(\mathfrak{i},\epsilon \right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)\mathcal{J}\left(\mathfrak{i},\mathfrak{v}\right),\hfill \end{array}$$

Since $\mathsf{Ԓ}$ and $\mathcal{J}$ both are coordinated $LR$-${\hslash }_1$ and $LR$-${\hslash }_2$-convex ${\rm I}VM$s on $\left[\mathcal{o},\mathfrak{i}\right]\times \left[\epsilon , \mathfrak{v}\right]$, respectively, we have

$$\begin{array}{c}\mathsf{Ԓ}\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},\kappa \epsilon +\left(1-\kappa \right)\mathfrak{v}\right)\times \mathcal{J}\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},\kappa \epsilon +\left(1-\kappa \right)\mathfrak{v}\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\mathsf{Ԓ}\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},\left(1-\kappa \right)\epsilon +\kappa \mathfrak{v}\right)\times \mathcal{J}\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},\left(1-\kappa \right)\epsilon +\kappa \mathfrak{v}\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\mathsf{Ԓ}\left(\left(1-\upsilon \right)\mathcal{o}+\upsilon \mathfrak{i},\kappa \epsilon +\left(1-\kappa \right)\mathfrak{v}\right)\times \mathcal{J}\left(\left(1-\upsilon \right)\mathcal{o}+\upsilon \mathfrak{i},\kappa \epsilon +\left(1-\kappa \right)\mathfrak{v}\right)\hfill \\ \hspace{1em}+\mathsf{Ԓ}\left(\left(1-\upsilon \right)\mathcal{o}+\upsilon \mathfrak{i},\left(1-\kappa \right)\epsilon +\kappa \mathfrak{v}\right)\times \mathcal{J}\left(\left(1-\upsilon \right)\mathcal{o}+\upsilon \mathfrak{i},\left(1-\kappa \right)\epsilon +\kappa \mathfrak{v}\right)\hfill \\ \hspace{1em}\hspace{1em}{\le }_p\mathcal{M}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)[{\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(1-\kappa \right){\hslash }_2\left(1-\kappa \right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\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)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\hslash }_1\left(\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(\kappa \right)]\hfill \\ \hspace{1em}\hspace{1em}+P\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)[{\hslash }_1\left(\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(1-\kappa \right){\hslash }_2\left(1-\kappa \right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\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)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(\kappa \right)]\hfill \\ \hspace{1em}\hspace{1em}+\mathcal{N}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)[{\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(1-\kappa \right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\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)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\hslash }_1\left(\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(1-\kappa \right)]\hfill \\ \hspace{1em}+Q\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)[{\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)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\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)].\hfill \end{array}$$

Taking the multiplication of the above fuzzy inclusion with ${\upsilon }^{\alpha -1}{\kappa }^{\beta -1}$ 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{array}{c}{\int }_0^1{\int }_0^1{\upsilon }^{\alpha -1}{\kappa }^{\beta -1}\mathsf{Ԓ}\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},\kappa \epsilon +\left(1-\kappa \right)\mathfrak{v}\right)\times \mathcal{J}\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},\kappa \epsilon +\left(1-\kappa \right)\mathfrak{v}\right)d\upsilon d\kappa \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\int }_0^1{\int }_0^1{\upsilon }^{\alpha -1}{\kappa }^{\beta -1}\mathsf{Ԓ}\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},\left(1-\kappa \right)\epsilon +\kappa \mathfrak{v}\right)\times \mathcal{J}(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},\left(1-\kappa \right)\epsilon +\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\kappa \mathfrak{v})d\upsilon d\kappa \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\int }_0^1{\int }_0^1{\upsilon }^{\alpha -1}{\kappa }^{\beta -1}\mathsf{Ԓ}\left(\left(1-\upsilon \right)\mathcal{o}+\upsilon \mathfrak{i},\kappa \epsilon +\left(1-\kappa \right)\mathfrak{v}\right)\times \mathcal{J}\left((|1-\upsilon \right)\mathcal{o}+\upsilon \mathfrak{i},\kappa \epsilon +(1-\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\kappa \left)\mathfrak{v}\right)d\upsilon d\kappa \hfill \\ \hspace{1em}+{\int }_0^1{\int }_0^1{\upsilon }^{\alpha -1}{\kappa }^{\beta -1}\mathsf{Ԓ}\left(\left(1-\upsilon \right)\mathcal{o}+\upsilon \mathfrak{i},\left(1-\kappa \right)\epsilon +\kappa \mathfrak{v}\right)\times \mathcal{J}\left(\left(1-\upsilon \right)\mathcal{o}+\upsilon \mathfrak{i},\left(1-\kappa \right)\epsilon +\kappa \mathfrak{v}\right)d\upsilon d\kappa \hfill \\ \hspace{1em}\hspace{1em}{\le }_p\mathcal{M}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right){\int }_0^1{\int }_0^1{\upsilon }^{\alpha -1}{\kappa }^{\beta -1}[{\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(1-\kappa \right){\hslash }_2\left(1-\kappa \right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\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)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\hslash }_1\left(\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(\kappa \right)]d\upsilon d\kappa \hfill \\ \hspace{1em}\hspace{1em}+P\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right){\int }_0^1{\int }_0^1{\upsilon }^{\alpha -1}{\kappa }^{\beta -1}[{\hslash }_1\left(\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(1-\kappa \right){\hslash }_2\left(1-\kappa \right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\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)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(\kappa \right)]d\upsilon d\kappa \hfill \\ \hspace{1em}\hspace{1em}+\mathcal{N}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right){\int }_0^1{\int }_0^1{\upsilon }^{\alpha -1}{\kappa }^{\beta -1}[{\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(1-\kappa \right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\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)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\hslash }_1\left(\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(1-\kappa \right)]d\upsilon d\kappa \hfill \\ \hspace{1em}+Q\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right){\int }_0^1{\int }_0^1{\upsilon }^{\alpha -1}{\kappa }^{\beta -1}[{\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)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\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)]d\upsilon d\kappa \hfill \end{array}$$

(59)

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

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\upsilon }^{\alpha -1}{\kappa }^{\beta -1}\mathsf{Ԓ}\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},\kappa \epsilon +\left(1-\kappa \right)\mathfrak{v}\right)\times \mathcal{J}\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},\kappa \epsilon +\left(1-\kappa \right)\mathfrak{v}\right)d\upsilon d\kappa \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\int }_0^1{\int }_0^1{\upsilon }^{\alpha -1}{\kappa }^{\beta -1}\mathsf{Ԓ}\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},\left(1-\kappa \right)\epsilon +\kappa \mathfrak{v}\right)\times \mathcal{J}(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},\left(1-\kappa \right)\epsilon +\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\kappa \mathfrak{v})d\upsilon d\kappa \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\int }_0^1{\int }_0^1{\upsilon }^{\alpha -1}{\kappa }^{\beta -1}\mathsf{Ԓ}\left(\left(1-\upsilon \right)\mathcal{o}+\upsilon \mathfrak{i},\kappa \epsilon +\left(1-\kappa \right)\mathfrak{v}\right)\times \mathcal{J}(\left(1-\upsilon \right)\mathcal{o}+\upsilon \mathfrak{i},\kappa \epsilon +(1-\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\kappa \left)\mathfrak{v}\right)d\upsilon d\kappa \hfill \\ \hspace{1em}+{\int }_0^1{\int }_0^1{\upsilon }^{\alpha -1}{\kappa }^{\beta -1}\mathsf{Ԓ}\left(\left(1-\upsilon \right)\mathcal{o}+\upsilon \mathfrak{i},\left(1-\kappa \right)\epsilon +\kappa \mathfrak{v}\right)\times \mathcal{J}\left(\left(1-\upsilon \right)\mathcal{o}+\upsilon \mathfrak{i},\left(1-\kappa \right)\epsilon +\kappa \mathfrak{v}\right)d\upsilon d\kappa \hfill \\ \hspace{1em}\hspace{1em}=\frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }{\left(\mathfrak{v}-\epsilon \right)}^{\beta }}\left[{\mathcal{I}}_{{\mathcal{o}}^{+},{\epsilon }^{+}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)\times \mathcal{J}\left(\mathfrak{i},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathcal{o}}^{+},{\mathfrak{v}}^{-}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)\times \mathcal{J}\left(\mathfrak{i},\epsilon \right)\right]\hfill \end{array}$$

(60)

Combining (59) and (60), we have

$$\begin{array}{c}\hfill \frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }{\left(\mathfrak{v}-\epsilon \right)}^{\beta }}\left[{\mathcal{I}}_{{\mathcal{o}}^{+},{\epsilon }^{+}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)\times \mathcal{J}\left(\mathfrak{i},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathcal{o}}^{+},{\mathfrak{v}}^{-}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)\times \mathcal{J}\left(\mathfrak{i},\epsilon \right)\right]\hfill \\ \hspace{1em}{\le }_p\mathcal{M}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right){\int }_0^1{\int }_0^1{\upsilon }^{\alpha -1}{\kappa }^{\beta -1}[{\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(1-\kappa \right){\hslash }_2\left(1-\kappa \right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\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)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\hslash }_1\left(\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(\kappa \right)]d\upsilon d\kappa \hfill \\ \hspace{1em}+P\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right){\int }_0^1{\int }_0^1{\upsilon }^{\alpha -1}{\kappa }^{\beta -1}[{\hslash }_1\left(\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(1-\kappa \right){\hslash }_2\left(1-\kappa \right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\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)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(\kappa \right)]d\upsilon d\kappa \hfill \\ \hspace{1em}+\mathcal{N}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right){\int }_0^1{\int }_0^1{\upsilon }^{\alpha -1}{\kappa }^{\beta -1}[{\hslash }_1\left(1-\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(1-\kappa \right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\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)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\hslash }_1\left(\upsilon \right){\hslash }_2\left(\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(1-\kappa \right)]d\upsilon d\kappa \hfill \\ +Q\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right){\int }_0^1{\int }_0^1{\upsilon }^{\alpha -1}{\kappa }^{\beta -1}[{\hslash }_1\left(\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(1-\kappa \right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}+{\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)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}+{\hslash }_1\left(\upsilon \right){\hslash }_2\left(1-\upsilon \right){\hslash }_1\left(\kappa \right){\hslash }_2\left(1-\kappa \right)]d\upsilon d\kappa .\hfill \end{array}$$

Hence, the required result. □

Remark 4. 

If one assumes that $\mathsf{Ԓ}$ is coordinated left-$LR$-$\mathit{\hslash }$-convex with $\hslash \left(\upsilon \right)=\upsilon , \hslash \left(\kappa \right)=\kappa$ and $\alpha =1$ and $\beta =1$, then from (59), as a result, there will be inequality (see [42]): 

$$\begin{gathered} \frac{1}{\left(\mathfrak{i}-\mathcal{o}\right)\left(\mathfrak{v}-\epsilon \right)}{\int }_{\mathcal{o}}^{\mathfrak{i}}{\int }_{\epsilon }^{\mathfrak{v}}\mathsf{Ԓ}\left(x,y\right)\times \mathcal{J}\left(x,y\right)dydx \\ \supseteq \frac{1}{9}\mathcal{M}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)+\frac{1}{18}\left[P\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)+\mathcal{N}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)\right]+\frac{1}{36}Q\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right). \end{gathered}$$

(61)

If $\mathsf{Ԓ}$ is coordinated $LR$-$\mathit{\hslash }$-convex with $\hslash \left(\upsilon \right)=\upsilon , \hslash \left(\kappa \right)=\kappa$ and one assumes that $\alpha =1$ and $\beta =1$, then from (59), as a result, there will be inequality (see [43]): 

$$\begin{gathered} \frac{1}{\left(\mathfrak{i}-\mathcal{o}\right)\left(\mathfrak{v}-\epsilon \right)}{\int }_{\mathcal{o}}^{\mathfrak{i}}{\int }_{\epsilon }^{\mathfrak{v}}\mathsf{Ԓ}\left(x,y\right)\times \mathcal{J}\left(x,y\right)dydx \\ {\le }_p\frac{1}{9}\mathcal{M}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)+\frac{1}{18}\left[P\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)+\mathcal{N}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)\right]+\frac{1}{36}Q\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right). \end{gathered}$$

(62)

If ${\mathsf{Ԓ}}_{*}\left(x,y\right)\ne {\mathsf{Ԓ}}^{*}\left(x,y\right)$ and $\hslash \left(\upsilon \right)=\upsilon , \hslash \left(\kappa \right)=\kappa$ then, by (57), we succeed in bringing about the upcoming inequality (see [46]): 

$$\begin{array}{c}\frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{4\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }{\left(\mathfrak{v}-\epsilon \right)}^{\beta }}\left[\begin{array}{c}{\mathcal{I}}_{{\mathcal{o}}^{+},{\epsilon }^{+}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)\times J\left(\mathfrak{i},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathcal{o}}^{+},{\mathfrak{v}}^{-}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)\times J\left(\mathfrak{i},\epsilon \right)\end{array}\right]\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}+\frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{4\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }{\left(\mathfrak{v}-\epsilon \right)}^{\beta }}\left[\begin{array}{c}{\mathcal{I}}_{{\mathfrak{i}}^{-},{\epsilon }^{+}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)\times J\left(\mathcal{o},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{i}}^{-},{\mathfrak{v}}^{-}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)\times J\left(\mathcal{o},\epsilon \right)\end{array}\right]\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}{\le }_p\left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\mathcal{M}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{\alpha }{(\alpha +1)(\alpha +2)}\left(\frac{1}{2}-\frac{\beta }{(\beta +1)(\beta +2)}\right)P\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)\hfill \\ +\left(\frac{1}{2}-\frac{\alpha }{(\alpha +1)(\alpha +2)}\right)\frac{\beta }{(\beta +1)(\beta +2)}\mathcal{N}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)+\frac{\beta }{(\beta +1)(\beta +2)}\frac{\alpha }{(\alpha +1)(\alpha +2)}Q\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right).\hfill \end{array}$$

(63)

If $\hslash \left(\upsilon \right)=\upsilon , \hslash \left(\kappa \right)=\kappa$ and ${\mathsf{Ԓ}}_{*}\left(x,y\right)\ne {\mathsf{Ԓ}}^{*}\left(x,y\right)$, then by (57), we succeed in bringing about the upcoming inequality (see [43]): 

$$\begin{gathered} \frac{1}{\left(\mathfrak{i}-\mathcal{o}\right)\left(\mathfrak{v}-\epsilon \right)}{\int }_{\mathcal{o}}^{\mathfrak{i}}{\int }_{\epsilon }^{\mathfrak{v}}\mathsf{Ԓ}\left(x,y\right)\times \mathcal{J}\left(x,y\right)dydx \\ {\le }_p\frac{1}{9}\mathcal{M}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)+\frac{1}{18}\left[P\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)+\mathcal{N}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)\right]+\frac{1}{36}Q\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right). \end{gathered}$$

(64)

If ${\mathsf{Ԓ}}_{*}\left(x,y\right)={\mathsf{Ԓ}}^{*}\left(x,y\right)$ and ${\mathcal{J}}_{*}\left(x,y\right)={\mathcal{J}}^{*}\left(x,y\right)$ and $\hslash \left(\upsilon \right)=\upsilon , \hslash \left(\kappa \right)=\kappa$, then from (57), we succeed in bringing about the upcoming classical inequality: 

$$\begin{array}{c}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{4\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }{\left(\mathfrak{v}-\epsilon \right)}^{\beta }}\left[\begin{array}{c}{\mathcal{I}}_{{\mathcal{o}}^{+},{\epsilon }^{+}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)\times J\left(\mathfrak{i},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathcal{o}}^{+},{\mathfrak{v}}^{-}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)\times J\left(\mathfrak{i},\epsilon \right)\end{array}\right]\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}+\frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{4\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }{\left(\mathfrak{v}-\epsilon \right)}^{\beta }} \left[\begin{array}{c}+{\mathcal{I}}_{{\mathfrak{i}}^{-},{\epsilon }^{+}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)\times J\left(\mathcal{o},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{i}}^{-},{\mathfrak{v}}^{-}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)\times J\left(\mathcal{o},\epsilon \right)\end{array}\right]\hfill \\ \hspace{1em}\hspace{1em}\le \left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\mathcal{M}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)+\frac{\alpha }{(\alpha +1)(\alpha +2)}(\frac{1}{2}-\hfill \\ \hspace{1em}\frac{\beta }{(\beta +1)(\beta +2)})P\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)\hfill \\ +\left(\frac{1}{2}-\frac{\alpha }{(\alpha +1)(\alpha +2)}\right)\frac{\beta }{(\beta +1)(\beta +2)}\mathcal{N}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)+\frac{\beta }{(\beta +1)(\beta +2)}\frac{\alpha }{(\alpha +1)(\alpha +2)}Q\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right).\hfill \end{array}$$

(65)

Theorem 8. 

Let $\mathsf{Ԓ}, \mathcal{J}:\Omega \to {\mathbb{R}}_I^{+}$ be two coordinated $LR$-$\hslash$-convex ${\rm I}VM$s on $\Omega$, given by $\mathsf{Ԓ}\left(x,y\right)=\left[{\mathsf{Ԓ}}_{*}\left(x,y\right), {\mathsf{Ԓ}}^{*}\left(x,y\right)\right]$ and $\mathcal{J}\left(x,y\right)=\left[{\mathcal{J}}_{*}\left(x,y\right), {\mathcal{J}}^{*}\left(x,y\right)\right]$ for all $\left(x,y\right)\in \Omega$ and let $\hslash :\left[0, 1\right]\to {\mathbb{R}}^{+}$. If ${\mathsf{Ԓ}\times \mathcal{J}\in \mathfrak{T}\mathfrak{O}}_{\Omega }$, then the following inequalities hold: 

$$\begin{array}{c}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\frac{1}{2\alpha \beta {{\hslash }_1}^2\left(\frac{1}{2}\right){{\hslash }_2}^2\left(\frac{1}{2}\right)}\mathsf{Ԓ}\left(\frac{\mathcal{o}+\mathfrak{i}}{2}, \frac{\epsilon +\mathfrak{v}}{2}\right)\times \mathcal{J}\left(\frac{\mathcal{o}+\mathfrak{i}}{2}, \frac{\epsilon +\mathfrak{v}}{2}\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\le }_p\frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{2\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }{\left(\mathfrak{v}-\epsilon \right)}^{\beta }}\left[{\mathcal{I}}_{{\mathcal{o}}^{+},{\epsilon }^{+}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)\times \mathcal{J}\left(\mathfrak{i},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathcal{o}}^{+},{\mathfrak{v}}^{-}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)\times \mathcal{J}\left(\mathfrak{i},\epsilon \right)\right]\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{2{\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }{\left(\mathfrak{v}-\epsilon \right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{i}}^{-},{\epsilon }^{+}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)\times \mathcal{J}\left(\mathcal{o},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{i}}^{-},{\mathfrak{v}}^{-}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)\times \mathcal{J}\left(\mathcal{o},\epsilon \right)\right]\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\mathcal{M}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right){\int }_0^1{\upsilon }^{\alpha -1}{\kappa }^{\beta -1}\left[{\hslash }_1\left(\upsilon \right){\hslash }_1\left(\kappa \right)\right[{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)]\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\hslash }_1\left(\upsilon \right){\hslash }_1\left(1-\kappa \right)[{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)\left]\right]d\upsilon d\kappa \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+P\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right){\int }_0^1{\upsilon }^{\alpha -1}{\kappa }^{\beta -1}\left[{\hslash }_1\left(\upsilon \right){\hslash }_1\left(\kappa \right)\right[{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)+\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)]+{\hslash }_1\left(\upsilon \right){\hslash }_1\left(1-\kappa \right)[{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)+\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)\left]\right]d\upsilon d\kappa \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\mathcal{N}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right){\int }_0^1{\upsilon }^{\alpha -1}{\kappa }^{\beta -1}\left[{\hslash }_1\left(\upsilon \right){\hslash }_1\left(\kappa \right)\right[{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)]\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\hslash }_1\left(\upsilon \right){\hslash }_1\left(1-\kappa \right)[{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)\left]\right]d\upsilon d\kappa \hfill \\ +Q\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right){\int }_0^1{\upsilon }^{\alpha -1}{\kappa }^{\beta -1}\left[{\hslash }_1\left(\upsilon \right){\hslash }_1\left(\kappa \right)\right[{\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)]+\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}{\hslash }_1\left(\upsilon \right){\hslash }_1(1-\kappa )[{\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)]]d\upsilon d\kappa .\hfill \end{array}$$

(66)

If $\mathsf{Ԓ}$ and $\mathcal{J}$ both are coordinate $\mathit{\hslash }$-concave ${\rm I}VM$s on $\Omega$, then the inequality above can be expressed as follows: 

$$\begin{array}{c}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\frac{1}{2\alpha \beta {{\hslash }_1}^2\left(\frac{1}{2}\right){{\hslash }_2}^2\left(\frac{1}{2}\right)}\mathsf{Ԓ}\left(\frac{\mathcal{o}+\mathfrak{i}}{2}, \frac{\epsilon +\mathfrak{v}}{2}\right)\times \mathcal{J}\left(\frac{\mathcal{o}+\mathfrak{i}}{2}, \frac{\epsilon +\mathfrak{v}}{2}\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\ge }_p\frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{2\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }{\left(\mathfrak{v}-\epsilon \right)}^{\beta }}\left[{\mathcal{I}}_{{\mathcal{o}}^{+},{\epsilon }^{+}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)\times \mathcal{J}\left(\mathfrak{i},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathcal{o}}^{+},{\mathfrak{v}}^{-}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)\times \mathcal{J}\left(\mathfrak{i},\epsilon \right)\right]\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{2{\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }{\left(\mathfrak{v}-\epsilon \right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{i}}^{-},{\epsilon }^{+}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)\times \mathcal{J}\left(\mathcal{o},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{i}}^{-},{\mathfrak{v}}^{-}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)\times \mathcal{J}\left(\mathcal{o},\epsilon \right)\right]\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\mathcal{M}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right){\int }_0^1{\upsilon }^{\alpha -1}{\kappa }^{\beta -1}\left[{\hslash }_1\left(\upsilon \right){\hslash }_1\left(\kappa \right)\right[{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)]\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\hslash }_1\left(\upsilon \right){\hslash }_1\left(1-\kappa \right)[{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)\left]\right]d\upsilon d\kappa \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+P\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right){\int }_0^1{\upsilon }^{\alpha -1}{\kappa }^{\beta -1}\left[{\hslash }_1\left(\upsilon \right){\hslash }_1\left(\kappa \right)\right[{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)+\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)]+{\hslash }_1\left(\upsilon \right){\hslash }_1\left(1-\kappa \right)[{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)+\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)\left]\right]d\upsilon d\kappa \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\mathcal{N}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right){\int }_0^1{\upsilon }^{\alpha -1}{\kappa }^{\beta -1}\left[{\hslash }_1\left(\upsilon \right){\hslash }_1\left(\kappa \right)\right[{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)]\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\hslash }_1\left(\upsilon \right){\hslash }_1\left(1-\kappa \right)[{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)\left]\right]d\upsilon d\kappa \hfill \\ +Q\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right){\int }_0^1{\upsilon }^{\alpha -1}{\kappa }^{\beta -1}\left[{\hslash }_1\left(\upsilon \right){\hslash }_1\left(\kappa \right)\right[{\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)]+\hfill \\ \hspace{1em}\hspace{1em}{\hslash }_1\left(\upsilon \right){\hslash }_1(1-\kappa )[{\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)]]d\upsilon d\kappa .\hfill \end{array}$$

(67)

where $\mathcal{M}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)$, $P\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)$, $\mathcal{N}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)$, and $Q\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)$ are given in Theorem 7.

Proof. 

Since $\mathsf{Ԓ},\mathcal{J}:\Omega \to {\mathbb{R}}_I^{+}$ is two $LR$-$\hslash$-convex ${\rm I}VM$s, then from inequality (17), we have

$$\mathsf{Ԓ}\left(\frac{\mathcal{o}+\mathfrak{i}}{2},\frac{\epsilon +\mathfrak{v}}{2}\right)\times \mathcal{J}\left(\frac{\mathcal{o}+\mathfrak{i}}{2},\frac{\epsilon +\mathfrak{v}}{2}\right)$$

$$\begin{array}{c}=\mathsf{Ԓ}\left(\frac{\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i}}{2}+\frac{\left(1-\upsilon \right)\mathcal{o}+\upsilon \mathfrak{i}}{2},\frac{\kappa \epsilon +\left(1-\kappa \right)\mathfrak{v}}{2}+\frac{\epsilon +\mathfrak{v}}{2}\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times \mathcal{J}\left(\frac{\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i}}{2}+\frac{\left(1-\upsilon \right)\mathcal{o}+\upsilon \mathfrak{i}}{2},\frac{\kappa \epsilon +\left(1-\kappa \right)\mathfrak{v}}{2}+\frac{\left(1-\kappa \right)\epsilon +\kappa \mathfrak{v}}{2}\right)\hfill \\ \hspace{1em}{\le }_p{{\hslash }_1}^2\left(\frac{1}{2}\right){{\hslash }_2}^2\left(\frac{1}{2}\right)\times \left[\begin{array}{c}\mathsf{Ԓ}\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},\kappa \epsilon +\left(1-\kappa \right)\mathfrak{v}\right)+\mathsf{Ԓ}\left(\left(1-\upsilon \right)\mathcal{o}+\upsilon \mathfrak{i},\kappa \epsilon +\left(1-\kappa \right)\mathfrak{v}\right)\\ +\mathsf{Ԓ}\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},\left(1-\kappa \right)\epsilon +\kappa \mathfrak{v}\right)+\mathsf{Ԓ}\left(\left(1-\upsilon \right)\mathcal{o}+\upsilon \mathfrak{i},\left(1-\kappa \right)\epsilon +\kappa \mathfrak{v}\right)\end{array}\right]\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times \left[\begin{array}{c}J\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},\kappa \epsilon +\left(1-\kappa \right)\mathfrak{v}\right)+J\left(\left(1-\upsilon \right)\mathcal{o}+\upsilon \mathfrak{i},\kappa \epsilon +\left(1-\kappa \right)\mathfrak{v}\right)\\ +J\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},\left(1-\kappa \right)\epsilon +\kappa \mathfrak{v}\right)+J\left(\left(1-\upsilon \right)\mathcal{o}+\upsilon \mathfrak{i},\left(1-\kappa \right)\epsilon +\kappa \mathfrak{v}\right)\end{array}\right]\hfill \\ \hspace{1em}\hspace{1em}{\le }_p{{\hslash }_1}^2\left(\frac{1}{2}\right){{\hslash }_2}^2\left(\frac{1}{2}\right)\times \left[\begin{array}{c}\mathsf{Ԓ}\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},\kappa \epsilon +\left(1-\kappa \right)\mathfrak{v}\right)\times J\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},\kappa \epsilon +\left(1-\kappa \right)\mathfrak{v}\right)\\ +\mathsf{Ԓ}\left(\left(1-\upsilon \right)\mathcal{o}+\upsilon \mathfrak{i},\kappa \epsilon +\left(1-\kappa \right)\mathfrak{v}\right)\times J\left(\left(1-\upsilon \right)\mathcal{o}+\upsilon \mathfrak{i},\kappa \epsilon +\left(1-\kappa \right)\mathfrak{v}\right)\\ +\mathsf{Ԓ}\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},\left(1-\kappa \right)\epsilon +\kappa \mathfrak{v}\right)\times J\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},\left(1-\kappa \right)\epsilon +\kappa \mathfrak{v}\right)\\ +\mathsf{Ԓ}\left(\left(1-\upsilon \right)\mathcal{o}+\upsilon \mathfrak{i},\left(1-\kappa \right)\epsilon +\kappa \mathfrak{v}\right)\times J\left(\left(1-\upsilon \right)\mathcal{o}+\upsilon \mathfrak{i},\left(1-\kappa \right)\epsilon +\kappa \mathfrak{v}\right)\end{array}\right]\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}+{{\hslash }_1}^2\left(\frac{1}{2}\right){{\hslash }_2}^2\left(\frac{1}{2}\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\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(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}+{{\hslash }_1}^2\left(\frac{1}{2}\right){{\hslash }_2}^2\left(\frac{1}{2}\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\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(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}+{{\hslash }_1}^2\left(\frac{1}{2}\right){{\hslash }_2}^2\left(\frac{1}{2}\right)\hfill \\ \hspace{1em}\hspace{1em}\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(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{{\hslash }_1}^2\left(\frac{1}{2}\right){{\hslash }_2}^2\left(\frac{1}{2}\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\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(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right).\hfill \end{array}$$

Taking the multiplication of the above fuzzy inclusion with ${\upsilon }^{\alpha -1}{\kappa }^{\beta -1}$ 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

$${\int }_0^1{\int }_0^1{\upsilon }^{\alpha -1}{\kappa }^{\beta -1}\mathsf{Ԓ}\left(\frac{\mathcal{o}+\mathfrak{i}}{2},\frac{\epsilon +\mathfrak{v}}{2}\right)\times \mathcal{J}\left(\frac{\mathcal{o}+\mathfrak{i}}{2},\frac{\epsilon +\mathfrak{v}}{2}\right)d\upsilon d\kappa$$

$$\begin{array}{c}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\le }_p{{\hslash }_1}^2\left(\frac{1}{2}\right){{\hslash }_2}^2\left(\frac{1}{2}\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times {\int }_0^1{\int }_0^1{\upsilon }^{\alpha -1}{\kappa }^{\beta -1}\left[\begin{array}{c}\mathsf{Ԓ}\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},\kappa \epsilon +\left(1-\kappa \right)\mathfrak{v}\right)\times J\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},\kappa \epsilon +\left(1-\kappa \right)\mathfrak{v}\right)\\ +\mathsf{Ԓ}\left(\left(1-\upsilon \right)\mathcal{o}+\upsilon \mathfrak{i},\kappa \epsilon +\left(1-\kappa \right)\mathfrak{v}\right)\times J\left(\left(1-\upsilon \right)\mathcal{o}+\upsilon \mathfrak{i},\kappa \epsilon +\left(1-\kappa \right)\mathfrak{v}\right)\\ +\mathsf{Ԓ}\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},\left(1-\kappa \right)\epsilon +\kappa \mathfrak{v}\right)\times J\left(\upsilon \mathcal{o}+\left(1-\upsilon \right)\mathfrak{i},\left(1-\kappa \right)\epsilon +\kappa \mathfrak{v}\right)\\ +\mathsf{Ԓ}\left(\left(1-\upsilon \right)\mathcal{o}+\upsilon \mathfrak{i},\left(1-\kappa \right)\epsilon +\kappa \mathfrak{v}\right)\times J\left(\left(1-\upsilon \right)\mathcal{o}+\upsilon \mathfrak{i},\left(1-\kappa \right)\epsilon +\kappa \mathfrak{v}\right)\end{array}\right]d\upsilon d\kappa \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{{\hslash }_1}^2\left(\frac{1}{2}\right){{\hslash }_2}^2\left(\frac{1}{2}\right)\mathcal{M}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times {\int }_0^1{\int }_0^1{\upsilon }^{\alpha -1}{\kappa }^{\beta -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 \hfill \\ +{{\hslash }_1}^2\left(\frac{1}{2}\right){{\hslash }_2}^2\left(\frac{1}{2}\right)P\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)\hfill \\ \hspace{1em}\times {\int }_0^1{\int }_0^1{\upsilon }^{\alpha -1}{\kappa }^{\beta -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 \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} +{{\hslash }_1}^2\left(\frac{1}{2}\right){{\hslash }_2}^2\left(\frac{1}{2}\right)\mathcal{N}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)\hfill \\ \hspace{1em}\times {\int }_0^1{\int }_0^1{\upsilon }^{\alpha -1}{\kappa }^{\beta -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 \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} +{{\hslash }_1}^2\left(\frac{1}{2}\right){{\hslash }_2}^2\left(\frac{1}{2}\right)Q\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)\hfill \\ \hspace{1em}\hspace{1em}\times {\int }_0^1{\int }_0^1{\upsilon }^{\alpha -1}{\kappa }^{\beta -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 ,\hfill \end{array}$$

which implies that

$$\begin{array}{c}\frac{1}{\alpha \beta }\mathsf{Ԓ}\left(\frac{\mathcal{o}+\mathfrak{i}}{2}, \frac{\epsilon +\mathfrak{v}}{2}\right)\times \mathcal{J}\left(\frac{\mathcal{o}+\mathfrak{i}}{2}, \frac{\epsilon +\mathfrak{v}}{2}\right)\hfill \\ {\le }_p\frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right){{\hslash }_1}^2\left(\frac{1}{2}\right){{\hslash }_2}^2\left(\frac{1}{2}\right)}{{\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }{\left(\mathfrak{v}-\epsilon \right)}^{\beta }}\left[{\mathcal{I}}_{{\mathcal{o}}^{+},{\epsilon }^{+}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)\times \mathcal{J}\left(\mathfrak{i},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathcal{o}}^{+},{\mathfrak{v}}^{-}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)\times \mathcal{J}\left(\mathfrak{i},\epsilon \right)\right]\hfill \\ \hspace{1em}+\frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right){{\hslash }_1}^2\left(\frac{1}{2}\right){{\hslash }_2}^2\left(\frac{1}{2}\right)}{{\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }{\left(\mathfrak{v}-\epsilon \right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{i}}^{-},{\epsilon }^{+}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)\times \mathcal{J}\left(\mathcal{o},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{i}}^{-},{\mathfrak{v}}^{-}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)\times \mathcal{J}\left(\mathcal{o},\epsilon \right)\right]\hfill \\ \hspace{1em}\hspace{1em}+2{{\hslash }_1}^2\left(\frac{1}{2}\right){{\hslash }_2}^2\left(\frac{1}{2}\right)\mathcal{M}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right){\int }_0^1{\upsilon }^{\alpha -1}{\kappa }^{\beta -1}\left[{\hslash }_1\left(\upsilon \right){\hslash }_1\left(\kappa \right)\right[{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)]\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\hslash }_1\left(\upsilon \right){\hslash }_1\left(1-\kappa \right)[{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)\left]\right]d\upsilon d\kappa \hfill \\ \hspace{1em}\hspace{1em}+2{{\hslash }_1}^2\left(\frac{1}{2}\right){{\hslash }_2}^2\left(\frac{1}{2}\right)P\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right){\int }_0^1{\upsilon }^{\alpha -1}{\kappa }^{\beta -1}\left[{\hslash }_1\left(\upsilon \right){\hslash }_1\left(\kappa \right)\right[{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)]\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\hslash }_1\left(\upsilon \right){\hslash }_1\left(1-\kappa \right)[{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)\left]\right]d\upsilon d\kappa \hfill \\ \hspace{1em}\hspace{1em} +2{{\hslash }_1}^2\left(\frac{1}{2}\right){{\hslash }_2}^2\left(\frac{1}{2}\right)\mathcal{N}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right){\int }_0^1{\upsilon }^{\alpha -1}{\kappa }^{\beta -1}\left[{\hslash }_1\left(\upsilon \right){\hslash }_1\left(\kappa \right)\right[{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)]\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\hslash }_1\left(\upsilon \right){\hslash }_1\left(1-\kappa \right)[{\hslash }_2\left(\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)\left]\right]d\upsilon d\kappa \hfill \\ +2{{\hslash }_1}^2\left(\frac{1}{2}\right){{\hslash }_2}^2\left(\frac{1}{2}\right)Q\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right){\int }_0^1{\upsilon }^{\alpha -1}{\kappa }^{\beta -1}\left[{\hslash }_1\left(\upsilon \right){\hslash }_1\left(\kappa \right)\right[{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2(1-\hfill \\ \kappa )+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)]+{\hslash }_1\left(\upsilon \right){\hslash }_1\left(1-\kappa \right)[{\hslash }_2\left(1-\upsilon \right){\hslash }_2\left(1-\kappa \right)+{\hslash }_2\left(\upsilon \right){\hslash }_2\left(\kappa \right)+{\hslash }_2(\upsilon \left){\hslash }_2\right(1-\hfill \\ \kappa \left)\right]]d\upsilon d\kappa ,\hfill \end{array}$$

hence, the required result. □

Remark 5. 

If one assumes that $\mathsf{Ԓ}$ is coordinated left-$LR$-$\mathit{\hslash }$-convex with $\hslash \left(\upsilon \right)=\upsilon , \hslash \left(\kappa \right)=\kappa$ and $\alpha =1$ and $\beta =1$, then from (66), as a result, there will be inequality (see [42]): 

$$\begin{gathered} 4\mathsf{Ԓ}\left(\frac{\mathcal{o}+\mathfrak{i}}{2},\frac{\epsilon +\mathfrak{v}}{2}\right)\times \mathcal{J}\left(\frac{\mathcal{o}+\mathfrak{i}}{2},\frac{\epsilon +\mathfrak{v}}{2}\right) \\ \supseteq \frac{1}{\left(\mathfrak{i}-\mathcal{o}\right)\left(\mathfrak{v}-\epsilon \right)}{\int }_{\mathcal{o}}^{\mathfrak{i}}{\int }_{\epsilon }^{\mathfrak{v}}\mathsf{Ԓ}\left(x,y\right)\times \mathcal{J}\left(x,y\right)dydx+\frac{5}{36}\mathcal{M}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right) \\ +\frac{7}{36}\left[P\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)+\mathcal{N}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)\right]+\frac{2}{9}Q\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right). \end{gathered}$$

(68)

If $\mathsf{Ԓ}$ is coordinated $LR$-$\mathit{\hslash }$-convex with $\hslash \left(\upsilon \right)=\upsilon , \hslash \left(\kappa \right)=\kappa$ and one assumes that $\alpha =1$ and $\beta =1$, then from (66), as a result, there will be inequality (see [43]): 

$$\begin{gathered} 4\mathsf{Ԓ}\left(\frac{\mathcal{o}+\mathfrak{i}}{2},\frac{\epsilon +\mathfrak{v}}{2}\right)\times \mathcal{J}\left(\frac{\mathcal{o}+\mathfrak{i}}{2},\frac{\epsilon +\mathfrak{v}}{2}\right) \\ {\le }_p\frac{1}{\left(\mathfrak{i}-\mathcal{o}\right)\left(\mathfrak{v}-\epsilon \right)}{\int }_{\mathcal{o}}^{\mathfrak{i}}{\int }_{\epsilon }^{\mathfrak{v}}\mathsf{Ԓ}\left(x,y\right)\times \mathcal{J}\left(x,y\right)dydx+\frac{5}{36}\mathcal{M}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right) \\ +\frac{7}{36}\left[P\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)+\mathcal{N}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right)\right]+\frac{2}{9}Q\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right). \end{gathered}$$

(69)

If $\mathsf{Ԓ}$ is coordinated left-$LR$-$\mathit{\hslash }$-convex and ${\mathsf{Ԓ}}_{*}\left(x,y\right)\ne {\mathsf{Ԓ}}^{*}\left(x,y\right)$ with $\hslash \left(\upsilon \right)=\upsilon , \hslash \left(\kappa \right)=\kappa$, then from (66), we succeed in bringing about the upcoming inequality (see [41]): 

$$\begin{gathered} 4\mathsf{Ԓ}\left(\frac{\mathcal{o}+\mathfrak{i}}{2}, \frac{\epsilon +\mathfrak{v}}{2}\right)\times \mathcal{J}\left(\frac{\mathcal{o}+\mathfrak{i}}{2}, \frac{\epsilon +\mathfrak{v}}{2}\right) \\ \supseteq \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{4\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }{\left(\mathfrak{v}-\epsilon \right)}^{\beta }}\left[\begin{array}{c}{\mathcal{I}}_{{\mathcal{o}}^{+},{\epsilon }^{+}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)\times J\left(\mathfrak{i},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathcal{o}}^{+},{\mathfrak{v}}^{-}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)\times J\left(\mathfrak{i},\epsilon \right)\\ +{\mathcal{I}}_{{\mathfrak{i}}^{-},{\epsilon }^{+}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)\times J\left(\mathcal{o},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{i}}^{-},{\mathfrak{v}}^{-}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)\times J\left(\mathcal{o},\epsilon \right)\end{array}\right] \\ +\left[\frac{\alpha }{2\left(\alpha +1\right)\left(\alpha +2\right)}+\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\right]\mathcal{M}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right) \\ +\left[\frac{1}{2}\left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)+\frac{\alpha }{(\alpha +1)(\alpha +2)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]P\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right) \\ +\left[\frac{1}{2}\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)+\frac{\alpha }{(\alpha +1)(\alpha +2)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]\mathcal{N}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right) \\ +\left[\frac{1}{4}-\frac{\alpha }{(\alpha +1)(\alpha +2)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]Q\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right). \end{gathered}$$

(70)

If ${\mathsf{Ԓ}}_{*}\left(x,y\right)\ne {\mathsf{Ԓ}}^{*}\left(x,y\right)$ and $\hslash \left(\upsilon \right)=\upsilon , \hslash \left(\kappa \right)=\kappa$, then from (66), we succeed in bringing about the upcoming inequality (see [46]): 

$$\begin{gathered} 4\mathsf{Ԓ}\left(\frac{\mathcal{o}+\mathfrak{i}}{2}, \frac{\epsilon +\mathfrak{v}}{2}\right)\times \mathcal{J}\left(\frac{\mathcal{o}+\mathfrak{i}}{2}, \frac{\epsilon +\mathfrak{v}}{2}\right) \\ {\le }_p\frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{4\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }{\left(\mathfrak{v}-\epsilon \right)}^{\beta }} \left[\begin{array}{c}{\mathcal{I}}_{{\mathcal{o}}^{+},{\epsilon }^{+} }^{\alpha , \beta }\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)\times J\left(\mathfrak{i},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathcal{o}}^{+},{\mathfrak{v}}^{-}}^{\alpha , \beta } \mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)\times J\left(\mathfrak{i},\epsilon \right)\\ +{\mathcal{I}}_{{\mathfrak{i}}^{-},{\epsilon }^{+} }^{\alpha , \beta }\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)\times J\left(\mathcal{o},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{i}}^{-},{\mathfrak{v}}^{-}}^{\alpha , \beta } \mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)\times J\left(\mathcal{o},\epsilon \right)\end{array}\right] \\ +\left[\frac{\alpha }{2\left(\alpha +1\right)\left(\alpha +2\right)}+\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\right]\mathcal{M}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right) \\ +\left[\frac{1}{2}\left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)+\frac{\alpha }{(\alpha +1)(\alpha +2)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]P\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right) \\ +\left[\frac{1}{2}\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)+\frac{\alpha }{(\alpha +1)(\alpha +2)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]\mathcal{N}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right) \\ +\left[\frac{1}{4}-\frac{\alpha }{(\alpha +1)(\alpha +2)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]Q\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right). \end{gathered}$$

(71)

If ${\mathsf{Ԓ}}_{*}\left(x,y\right)={\mathsf{Ԓ}}^{*}\left(x,y\right)$ and ${\mathcal{J}}_{*}\left(x,y\right)={\mathcal{J}}^{*}\left(x,y\right)$ and $\hslash \left(\upsilon \right)=\upsilon , \hslash \left(\kappa \right)=\kappa$, then from (66), we succeed in bringing about the upcoming classical inequality. 

$$\begin{gathered} 4\mathsf{Ԓ}\left(\frac{\mathcal{o}+\mathfrak{i}}{2}, \frac{\epsilon +\mathfrak{v}}{2}\right)\times \mathcal{J}\left(\frac{\mathcal{o}+\mathfrak{i}}{2}, \frac{\epsilon +\mathfrak{v}}{2}\right) \\ \le \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{4\left(\mathfrak{i}-\mathcal{o}\right)}^{\alpha }{\left(\mathfrak{v}-\epsilon \right)}^{\beta }}\left[\begin{array}{c}{\mathcal{I}}_{{\mathcal{o}}^{+},{\epsilon }^{+}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathfrak{i},\mathfrak{v}\right)\times J\left(\mathfrak{i},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathcal{o}}^{+},{\mathfrak{v}}^{-}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathfrak{i},\epsilon \right)\times J\left(\mathfrak{i},\epsilon \right)\\ +{\mathcal{I}}_{{\mathfrak{i}}^{-},{\epsilon }^{+}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathcal{o},\mathfrak{v}\right)\times J\left(\mathcal{o},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{i}}^{-},{\mathfrak{v}}^{-}}^{\alpha , \beta }\mathsf{Ԓ}\left(\mathcal{o},\epsilon \right)\times J\left(\mathcal{o},\epsilon \right)\end{array}\right] \\ +\left[\frac{\alpha }{2\left(\alpha +1\right)\left(\alpha +2\right)}+\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\right]\mathcal{M}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right) \\ +\left[\frac{1}{2}\left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)+\frac{\alpha }{(\alpha +1)(\alpha +2)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]P\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right) \\ +\left[\frac{1}{2}\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)+\frac{\alpha }{(\alpha +1)(\alpha +2)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]\mathcal{N}\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right) \\ +\left[\frac{1}{4}-\frac{\alpha }{(\alpha +1)(\alpha +2)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]Q\left(\mathcal{o},\mathfrak{i},\epsilon ,\mathfrak{v}\right). \end{gathered}$$

(72)

4. Conclusions and Future Plans

To sum up, this study offers a new extension of interval-valued convexity. Through the use of fractional integral operators, various inequalities for $LR$-$\hslash$-convexity are produced by applying interval-valued mapping. Many exceptional cases are discussed and new and classical versions of integral inequalities are also acquired that can be considered as applications of this article’s outcomes. Some very interesting examples are also given to discuss the validation of the main results. The results of this research paper could potentially have applications in various areas of mathematics, physics, and engineering. The extension of the proposed iterative method to systems of equations could be an interesting future research problem. In the future, we will try to explore these concepts in quantum calculus.