Hermite–Hadamard-Mercer Type Inequalities for Interval-Valued Coordinated Convex Functions

Abstract

Determining the Jensen–Mercer inequality for interval-valued coordinated convex functions has been a challenging task for researchers in the fields of inequalities and interval analysis. We use ${\ominus }_g$ to establish the Jensen–Mercer inequality for interval-valued coordinated convex functions. In this paper, we make significant strides in establishing new results by introducing a novel approach. We present a Hermite–Hadamard (H.H.) Mercer-type inequality for interval-valued coordinated convex functions and show how it generalizes the traditional H.H. inequality. Specifically, the H.H. inequality for interval-valued coordinated convex functions can be derived as a special case by considering the endpoints of the H.H. Mercer-type inequality. Furthermore, we provide computational results that verify the accuracy of recent findings in the literature. Our results indicate that the proposed new results impose highly effective constraints on integrals of the specified functions and are valid for a broader class of functions. These new findings have significant implications for applications in fields such as economics, engineering, and physics, where they can improve the precision of system modeling and optimization.

1. Introduction

Despite having a long history dating back to the calculation of Archimedes’ circumference of a circle, notable work in this field was not available until the 1950s. Before the advent of interval analysis, single-point arithmetic was used for the majority of numerical calculations, which could lead to significant errors. Interval analysis is not a completely new phenomenon in mathematics; it has appeared numerous times throughout history under various names. For instance, Achimedes calculated lower and upper bound ${\displaystyle \frac{223}{71}}<\pi <{\displaystyle \frac{22}{7}}$ in the third century BC. An extensive article on interval algebra in numerical analysis was published by Teruo Sunaga in 1958. The 1966 book “Interval Analysis” by American mathematician Ramon Edgar Moore offers scholars studying contemporary interval analysis a fresh perspective [1]. Moore understood that numerical computations required a more thorough approach that took into consideration the precision of the original data as well as the computational process’s inherent limits. A mathematical technique called interval analysis is used to solve issues where the input data is ambiguous or inaccurate. It helps to achieve reliable constraints on the output of mathematical functions, which is very helpful in the field of numerical analysis. Extensive information on interval analysis has been presented in the literature [2,3].

Interval arithmetic theory is one of the foundational theories of interval analysis. The ranges of potential values for variables or functions are represented by intervals in this theory. In addition, subtraction, multiplication, and division are among the fundamental arithmetic operations that are extended to intervals in a way that yields an interval that contains every conceivable value of the corresponding operation applied to the intervals. Range analysis is primarily used to support the claim that ranges may contain this margin of error and that an item equal to another will inevitably contain one. Allow us to illustrate this with a brief example: In a market, the weights of identical food items arranged on identical shelves are accompanied by an error margin indicator. As an additional example, the air temperature is actually more accurate to be in the range of 27–29 °C than 28 °C. There will undoubtedly be a margin of error between two identical items when we perform accurate calculations. The intervals found are, therefore, more functional than those found in the classical analysis.

It was first applied in numerical analysis to ascertain the error bounds of numerical solutions of a finite state machine after being inspired by all of this. Following numerical analysis, the topic of quantum analysis, where numerous kinds of research are still conducted today, will be searched for answers to numerous questions. In interval analysis, quantum analysis has begun to provide answers to a number of issues in the literature. When applied to interval analysis, the findings of quantum analysis (q-analysis), the subject of numerous studies in the domains of physics, philosophy, cryptology, computer science, and mechanics, has provided answers to a new inquiry every day. It has answered several mathematical scholars’ inquiries on the theory of inequalities using the answers it has discovered thus far. Kac introduced the h-calculus without bounds, which is similar to the q-calculus. Recently, significant disparities have emerged for this analysis [4,5]. Convexity was already a topic of discussion among Greek philosophers, and its origins are likely in ancient Egypt and Babylon. Simple geometric shapes, such as circles and triangles, have been drawn since the beginning of human civilization, though not as long as numerals. Although it is difficult to determine who initially considered the concept of convexity, [6] provided additional details on the history of convexity. Karl Hermann Amandus Schwarz (1843–1921), a German mathematician, is credited with introducing the convex function in the late 1800s. His work on convex functions greatly influenced mathematical theory and applications. In 1913, Herman Weyl introduced the concept of convex functions in linear space. A convex function is defined as having a convex epigraph, which is the set of points above the function’s graph. Karl Menger proposed the broader concept of convex functions in metric space in 1928. He defined a convex function as one that meets a specific inequality between any two points in the metric space. This generalization enabled convex functions to be applied in a broader range of domains, including mathematics. In the same year, Stefan Banach independently presented a convex function on a normed linear space. He defined convexity as a function that fulfills an inequality akin to Menger’s concept. Mathematicians have extensively studied and applied the theory of convex functions on coordinates in fields such as optimization, economics, operations research, and machine learning. Convex functions play a crucial role in optimization problems such as linear programming, quadratic programming, and convex operations. It assists in the determination of the ideal value of an unknown under certain conditions. Convex functions are commonly employed in portfolio optimization to reduce risk while optimizing expected return. The convexity of some financial instruments, such as bonds, assures that declining interest rates always raise the price of these securities. Convex functions are utilized in control theory to provide optimal and robust feedback controllers for a variety of physical systems. They are also employed in signal processing, image analysis, and data compression. See [7,8] for further information on related applications. The feature of a function that any line segment joining two points on its graph lies above the graph itself is known as convexity in the setting of inequalities. The H.H. inequality was initially presented by two French mathematicians, Jacques Salomon Hadamard (1865–1963) and Charles Hermite (1822–1901). Major contributions to number theory, complex analysis, and many other areas of mathematics were made by C. Hermite and J. S. Hadamard; for additional information, see [9,10,11]. According to the inequality, if a function $𝐹:[\varsigma ,\phi ]\subset \mathbb{R}\to \mathbb{R}$ is convex, then

$${\displaystyle \frac{𝐹\left(\varsigma \right)+𝐹\left(\phi \right)}{2}}\ge {\displaystyle \frac{1}{\phi -\varsigma }}{\int }_{\varsigma }^{\phi }𝐹\left({\chi }_1\right)d{\chi }_1\ge 𝐹\left({\displaystyle \frac{\varsigma +\phi }{2}}\right).$$

(1)

Stated differently, a function is considered convex if the weighted average of its functional values at its endpoints is larger than or equal to the function’s value at the middle of any interval of a set of real numbers. Many branches of mathematics and economics use H.H. type inequality, while convexity is also crucial. In economics, for instance, H.H. inequality is used to demonstrate the existence and uniqueness of the solution to certain models, such as models of company behavior or general equilibrium models. There are numerous applications of H.H. inequality in the field of information theory. The study of error-correcting codes is one use. One can ascertain a code’s error-correcting capability by bonding the number of faults that the code can identify or fix. The inequality is particularly beneficial to evaluating the efficiency of data compression schemes because it limits the average amount of bits needed to describe a random variable. For more applications of H.H. inequality, one can consult [12,13,14]. Sadowska introduced the H.H. inequality for interval valued functions in [15], as stated as follows.

Let $F:[\varsigma ,\phi ]\to {\mathbb{R}}_I^{+}$ (${\mathbb{R}}_I^{+}$ is positive real intervals) be an interval valued function such that $F\left(\tau \right)=[F^{-}\left(\tau \right),F^{+}\left(\tau \right)]$, and $F\in I{R}_{\left(\right[\varsigma ,\phi \left]\right)}$, then

$$F\left({\displaystyle \frac{\varsigma +\phi }{2}}\right)\supseteq {\displaystyle \frac{1}{\phi -\varsigma }}\left(IR\right){\int }_{\varsigma }^{\phi }F\left({\chi }_1\right)d{\chi }_1\supseteq {\displaystyle \frac{F\left(\varsigma \right)+F\left(\phi \right)}{2}}.$$

(2)

Generally speaking, interval analysis in terms of the H.H. inequality offers a strong tool for carrying out calculations using interval values and meticulously evaluating a function’s range over the interval. In cases where input value variability or uncertainty must be considered, this can be especially helpful. Coordinated convex functions are motivated by the fact that every convex mapping $𝐹:[\varsigma ,\phi ]\to \mathbb{R}$ is convex on the coordinates. In addition, there are coordinated convex functions that are not convex (see, for instance [16,17]). For additional findings in the field of coordinated convex, interested readers are directed to [16,17,18,19,20,21,22]. The following H.H. type inequality for coordinated convex functions on the rectangle from the plane ${\mathbb{R}}^2$ was proved in [16], that is:

Let a function $𝐹:▵\subset {\mathbb{R}}^2\to \mathbb{R}$ be convex on coordinates. Then, one has the inequalities:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{𝐹\left(\varsigma ,\varrho \right)+𝐹\left(\varsigma ,\vartheta \right)+𝐹\left(\phi ,\varrho \right)+𝐹\left(\phi ,\vartheta \right)}{4}}& \ge & {\displaystyle \frac{1}{4}}\left[{\displaystyle \frac{1}{\phi -\varsigma }}{\int }_{\varsigma }^{\phi }𝐹\left({\chi }_1,\varrho \right)d{\chi }_1+{\displaystyle \frac{1}{\phi -\varsigma }}{\int }_{\varsigma }^{\phi }𝐹\left({\chi }_1,\vartheta \right)d{\chi }_1\right.\hfill \\ & & \left.+{\displaystyle \frac{1}{\vartheta -\varrho }}{\int }_{\varrho }^{\vartheta }𝐹\left(\delta ,{\chi }_2\right)d{\chi }_2+{\displaystyle \frac{1}{\vartheta -\varrho }}{\int }_{\varrho }^{\vartheta }𝐹\left(\phi ,{\chi }_2\right)d{\chi }_2\right]\hfill \\ & \ge & {\displaystyle \frac{1}{\left(\phi -\varsigma \right)\left(\vartheta -\varrho \right)}}{\int }_{\varsigma }^{\phi }{\int }_{\varrho }^{\vartheta }𝐹\left({\chi }_1,{\chi }_2\right)d{\chi }_{2}d{\chi }_1\hfill \\ & \ge & {\displaystyle \frac{1}{2}}\left[{\displaystyle \frac{1}{\phi -\varsigma }}{\int }_{\varsigma }^{\phi }𝐹\left({\chi }_1,{\displaystyle \frac{\varrho +\vartheta }{2}}\right)d{\chi }_1+{\displaystyle \frac{1}{\vartheta -\varrho }}{\int }_{\varrho }^{\vartheta }𝐹\left({\displaystyle \frac{\varsigma +\phi }{2}},{\chi }_2\right)d{\chi }_2\right]\hfill \\ & \ge & 𝐹\left({\displaystyle \frac{\varsigma +\phi }{2}},{\displaystyle \frac{\varrho +\vartheta }{2}}\right).\hfill \end{array}$$

(3)

Zhao et al. introduced the H.H. inequality for interval-valued coordinated convex functions, as stated in [23].

Let $𝐹:▵\subset {\mathbb{R}}^2\to {\mathbb{R}}_I^{+}$ be an interval valued coordinated convex function such that $𝐹\left(\tau \right)=\left[{𝐹}^{-}\left(\tau \right),{𝐹}^{+}\left(\tau \right)\right],$ and $𝐹\in I{R}_{\left(▵\right)}$ then

$$\begin{array}{ccc}& & 𝐹\left({\displaystyle \frac{\varsigma +\phi }{2}},{\displaystyle \frac{\varrho +\vartheta }{2}}\right)\hfill \\ & \supseteq & {\displaystyle \frac{1}{2}}\left[{\displaystyle \frac{1}{\phi -\varsigma }}{\int }_{\delta }^{\phi }𝐹\left({\chi }_1,{\displaystyle \frac{\varrho +\vartheta }{2}}\right)d{\chi }_1+{\displaystyle \frac{1}{\vartheta -\varrho }}{\int }_{\varrho }^{\vartheta }𝐹\left({\displaystyle \frac{\varsigma +\phi }{2}},{\chi }_2\right)d{\chi }_2\right]\hfill \\ & \supseteq & {\displaystyle \frac{1}{\left(\phi -\varsigma \right)\left(\vartheta -\varrho \right)}}{\int }_{\varsigma }^{\phi }{\int }_{\varrho }^{\vartheta }𝐹\left({\chi }_1,{\chi }_2\right)d{\chi }_{2}d{\chi }_1\hfill \\ & \supseteq & {\displaystyle \frac{1}{4}}\left[{\displaystyle \frac{1}{\phi -\varsigma }}{\int }_{\delta }^{\phi }𝐹\left({\chi }_1,\varrho \right)d{\chi }_1+{\displaystyle \frac{1}{\phi -\varsigma }}{\int }_{\varsigma }^{\phi }𝐹\left({\chi }_1,\vartheta \right)d{\chi }_1\right.\hfill \\ & & \left.+{\displaystyle \frac{1}{\vartheta -\varrho }}{\int }_{\varrho }^{\vartheta }𝐹\left(\varsigma ,{\chi }_2\right)d{\chi }_2+{\displaystyle \frac{1}{\vartheta -\varrho }}{\int }_{\varrho }^{\vartheta }𝐹\left(\phi ,{\chi }_2\right)d{\chi }_2\right]\hfill \\ & \supseteq & {\displaystyle \frac{𝐹\left(\varsigma ,\varrho \right)+𝐹\left(\varsigma ,\vartheta \right)+𝐹\left(\phi ,\varrho \right)+𝐹\left(\phi ,\vartheta \right)}{4}}.\hfill \end{array}$$

(4)

There is a connection in mathematics between the integral of a convex function and the value of a convex function of an integral. The Danish mathematician Johan Jensen created this relationship in 1906, and it is now known as Jensen’s inequality. The most well-known inequality in statistics and mathematics is the Jensen inequality for the convex function. It can be used to produce many other well-known inequalities, including Hölder’s inequality and Minkowski’s inequality, which are generic inequalities between means of order p and q. Jensen’s inequality for convex functions is the source of Minkowski’s inequality. Numerous variations, improvements, and generalizations of Jensen’s inequalities are provided; one can explore these inequalities by consulting [24,25,26,27,28]. A foundational finding in mathematics, Jensen’s inequality finds use in a variety of domains. It is connected to the idea of convex functions. For a convex function, the inequality asserts that the functional value at the average of a set of points is larger than or equal to the functional value at the average of those points. Numerous disciplines, including probability theory, statistics, economics, and optimization, use this inequality. In a variety of mathematical applications, it is employed to establish inequalities and obtain optimal solutions. Mercer wrote a paper on a variation of Jensen’s inequality in 2003 [29]. For operator convex with applications, Peĉarić et al. (2006) introduced the idea of Mercer-type Jensen inequality [30]. In 2009, Niezgoda worked on extending Mercer’s result on convex functions [31]. In 2013 [32], Kian et al. introduced the Mercer-type inequality.

If $𝐹$ is convex on $\left[\varsigma ,\phi \right],$ and for all ${\chi }_1,{\chi }_2\in \left[\varsigma ,\phi \right],$ then

$$𝐹\left(\varsigma \right)+𝐹\left(\phi \right)-{\displaystyle \frac{𝐹\left({\chi }_1\right)+𝐹\left({\chi }_2\right)}{2}}\ge {\displaystyle \frac{1}{{\chi }_2-{\chi }_1}}{\int }_{\varsigma +\phi -{\chi }_2}^{\delta +\phi -{\chi }_1}𝐹\left(\mu \right)d\mu \ge 𝐹\left(\delta +\phi -{\displaystyle \frac{{\chi }_1+{\chi }_2}{2}}\right),$$

(5)

or

$$\begin{array}{ccc}\hfill 𝐹\left(\varsigma \right)+𝐹\left(\phi \right)-𝐹\left({\displaystyle \frac{{\chi }_1+{\chi }_2}{2}}\right)& \ge & 𝐹\left(\varsigma \right)+𝐹\left(\phi \right)-{\displaystyle \frac{1}{\phi -\varsigma }}{\int }_{\varsigma }^{\phi }𝐹\left(\mu \right)d\mu \hfill \\ & \ge & 𝐹\left(\varsigma +\phi -{\displaystyle \frac{{\chi }_1+{\chi }_2}{2}}\right).\hfill \end{array}$$

(6)

Toseef et al. [33] established q-H.H. Mercer-type inequalities for generalized convex functions and analyzed them computationally. The H.H. Mercer-type inequality extends the standard H.H. inequality. The H.H. Mercer-type inequality is a valuable tool for mathematicians and scientists working on convex functions and related topics.

H.H. inequality applies to a large range of operator convex functions, allowing for intriguing inequalities in matrix analysis. A natural extension of the classical H.H. inequality to Hermitian matrices could be the double inequality.

$${\displaystyle \frac{𝐹\left(\varsigma \right)+𝐹\left(\phi \right)}{2}}\ge {\int }_0^1𝐹\left(\tau \varsigma +\left(1-\tau \right)\phi \right)d\tau \ge 𝐹\left({\displaystyle \frac{\varsigma +\phi }{2}}\right),$$

(7)

which is not true in general [34]. If $𝐹$ is convex on $\left[m,M\right],$ then

$$\begin{array}{cc}\hfill & 𝐹\left(m\right)+𝐹\left(M\right)-{\displaystyle \frac{\Lambda \left(𝐹\left(A\right)\right)+\Lambda \left(𝐹\left(B\right)\right)}{2}}\hfill \\ \hfill & \ge {\int }_0^1𝐹\left(m+M-\left(\tau \Lambda \left(A\right)+\left(1-\tau \right)\Lambda \left(B\right)\right)\right)d\tau \hfill \\ \hfill & \ge 𝐹\left(M+m-{\displaystyle \frac{\Lambda \left(A\right)+\Lambda \left(B\right)}{2}}\right),\hfill \end{array}$$

(8)

for all self-adjoint operators $A,B$ with spectra in $\left[m,M\right]$ and $\Lambda$ is a positive linear map. The H.H. Jensen–Mercer inequality is true in the operator version mentioned above. Matrix algebra or linear operators are crucial when discussing linear programming. As self-adjoint operators, A and B do not hold the classical H.H. inequality (7). Kian et al. [32] demonstrated the Operator Jensen–Mercer inequality (8) for self-adjoint operators to solve this problem in the field of inequalities, which is highly beneficial in the context of linear programming. Deriving H.H. Jensen’s Mercer inequality is the primary goal. It is the generalization of H.H. inequality and applies to matrices; however, it does not apply to self-adjoint matrices. Sitthiwirattham et al. presented the H.H. Mercer inequality for interval-valued convex functions in [35], as follows.

Let $𝐹:\left[\varsigma ,\phi \right]\to {\mathbb{R}}_I^{+}$ be an interval valued function such that $𝐹\left(\tau \right)=\left[{𝐹}^{-}\left(\tau \right),{𝐹}^{+}\left(\tau \right)\right],$ and $𝐹\in I{R}_{\left(\left[\varsigma ,\phi \right]\right)},{\chi }_1,{\chi }_2\in \left[\varsigma ,\phi \right]$ then

$$𝐹\left(\varsigma +\phi -{\displaystyle \frac{{\chi }_1+{\chi }_2}{2}}\right)\supseteq {\displaystyle \frac{1}{{\chi }_2-{\chi }_1}}\left(IR\right){\int }_{\varsigma +\phi -{\chi }_2}^{\varsigma +\phi -{\chi }_1}𝐹\left(\tau \right)d\tau \supseteq 𝐹\left(\varsigma \right)+𝐹\left(\phi \right){\ominus }_g{\displaystyle \frac{𝐹\left({\chi }_1\right)+𝐹\left({\chi }_2\right)}{2}}.$$

(9)

H.H. Mercer-type inequalities for coordinated convex functions was recently established by Toseef et al. in [36], which is stated as follows.

Let $𝐹:▵=\left[a,b\right]\times \left[c,d\right]\subseteq {\mathbb{R}}^2\to \mathbb{R}$ be coordinated convex function on $▵.$ Then, one has the inequalities:

$$\begin{array}{ccc}& & 𝐹\left(\varsigma +\phi -{\displaystyle \frac{\kappa +\rho }{2}},\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)\hfill \\ & \le & {\displaystyle \frac{1}{2}}\left[{\displaystyle \frac{1}{\left(\rho -\kappa \right)}}{\int }_{\delta +\phi -\rho }^{\varsigma +\phi -\kappa }𝐹\left({\chi }_1,\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)d{\chi }_1+{\displaystyle \frac{1}{\left(\mu -\nu \right)}}{\int }_{\varrho +\vartheta -\mu }^{\varrho +\vartheta -\nu }𝐹\left(\varsigma +\phi -{\displaystyle \frac{\kappa +\rho }{2}},{\chi }_2\right)d{\chi }_2\right]\hfill \\ & & \le {\displaystyle \frac{1}{\left(\rho -\kappa \right)\left(\mu -\nu \right)}}{\int }_{\varsigma +\phi -\rho }^{\varsigma +\phi -\kappa }{\int }_{\varrho +\vartheta -\mu }^{\varrho +\vartheta -\nu }𝐹\left({\chi }_1,{\chi }_2\right)d{\chi }_{2}d{\chi }_1\hfill \\ & \le & {\displaystyle \frac{1}{4}}\left[𝐹\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\mu \right)+𝐹\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\nu \right)\right.\hfill \\ & & \left.+𝐹\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu \right)+𝐹\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\nu \right)\right]\hfill \\ & \le & 𝐹\left(\varsigma ,\varrho \right)+𝐹\left(\varsigma ,\vartheta \right)+𝐹\left(\phi ,\varrho \right)+𝐹\left(\phi ,\vartheta \right)\hfill \\ & & -\left[{\displaystyle \frac{𝐹\left(\rho ,\mu \right)+𝐹\left(\rho ,\nu \right)+𝐹\left(\kappa ,\mu \right)+𝐹\left(\kappa ,\nu \right)}{4}}\right],\hfill \end{array}$$

(10)

where $\kappa ,\rho \in \left[\varsigma ,\phi \right]$ and $\mu ,\nu \in \left[\varrho ,\vartheta \right].$

Our definition of Jensen–Mercer inequality for interval-valued coordinated convex functions, which was a very challenging task in this field, is motivated by recent research in the theory of inequalities and interval-valued functions. To get around the problem, we employ the Cauchy–Schwarz inequality and ${\ominus }_g$ difference rather than the simple difference. The results are only verified if we use ${\ominus }_g$. Additionally, for interval-valued coordinated convex functions, we prove a few H.H. Mercer-type inequalities. We provide a few numerical examples to demonstrate our latest findings.

Section 2 presents preliminary information and fundamental definitions related to interval analysis and convexity. In Section 3, we prove the Jensen–Mercer inequality and the H.H. Mercer-type inequality for interval-valued coordinated convex functions. Additionally, Section 4 provides numerical examples to support the newly established results. Finally, Section 5 offers concluding remarks and suggests potential directions for future research.

2. Preliminaries

This section covers the essential principles and properties of convexity and interval calculus. To be brief, we use $𝐹=\left[{𝐹}^{-},{𝐹}^{+}\right]$, $G=\left[G^{-},G^{+}\right]$ and $H_1=\left[H_1^{-},H_1^{+}\right].$ We provide information on interval analysis. ${\mathcal{K}}_c$ represents the space of all closed intervals in $\mathbb{R}$. Suppose that ${\wp }_a$ is element of ${\mathcal{K}}_c$ and bounded, then we have the notation

$${\wp }_a=\left[{\wp }_a^{-},{\wp }_a^{+}\right]=\left\{\tau \in \mathbb{R}:{\wp }_a^{-}\le \tau \le {\wp }_a^{+}\right\},$$

where ${\wp }_a^{-}$, ${\wp }_a^{+}\in \mathbb{R}$ and ${\wp }_a^{-}\le {\wp }_a^{+}.$ The values ${\wp }_a^{-}$ and ${\wp }_a^{+}$ represent the left and right endpoints of the interval ${\wp }_a,$, respectively. ${\wp }_a$ is degenerate if ${\wp }_a^{-}=$${\wp }_a^{+},$ indicated by ${\wp }_a^{-}={\wp }_a^{+}=[{\wp }_a,{\wp }_a]={\wp }_a.$ Additionally, ${\wp }_a$ is positive when ${\wp }_a^{-}>0$, and negative when ${\wp }_a^{+}<0.$ ${\mathcal{K}}_c^{+}$ and ${\mathcal{K}}_c^{-}$ represent the sets of all closed positive and negative intervals in $\mathbb{R}$. The Hausdorff–Pompeiu distance between ${\wp }_a$ and ${\wp }_b$ is defined as

$$d\left({\wp }_a,{\wp }_b\right)=d\left(\left[{\wp }_a^{-},{\wp }_a^{+}\right],\left[{\wp }_b^{-},{\wp }_b^{+}\right]\right)=max\left\{\left|{\wp }_a^{-}-{\wp }_a^{-}\right|,\left|{\wp }_b^{+}-{\wp }_b^{+}\right|\right\}.$$

Furthermore, $\left(\Omega ,d\right)$ is a full metric space [37]. The features of fundamental interval analysis procedures for ${𝐹}_1$ and ${\wp }_b$ are as follows:

$$\begin{array}{ccc}\hfill {\wp }_a+{\wp }_b& =& \left[{\wp }_a^{-}+{\wp }_b^{-},{\wp }_a^{+}+{\wp }_b^{+}\right],\hfill \\ & & \\ \hfill {\wp }_a-{\wp }_b& =& \left[{\wp }_a^{-}-{\wp }_b^{-},{\wp }_a^{+}-{\wp }_b^{+}\right],\hfill \\ & & \\ \hfill {\wp }_a.{\wp }_b& =& \left[min\Lambda ,max\Lambda \right]\mathrm{where}\Lambda =\left\{{\wp }_a^{-}{\wp }_b^{-},{\wp }_a^{-}{\wp }_b^{+},{\wp }_a^{+}{\wp }_b^{-},{\wp }_a^{+}{\wp }_b^{+}\right\},\hfill \\ & & \\ \hfill {\wp }_a/{\wp }_b& =& \left[min\varsigma ,max\varsigma \right]\mathrm{where}\varsigma =\left\{{\wp }_a^{-}/{\wp }_b^{-},{\wp }_a^{-}/{\wp }_b^{+},{\wp }_a^{+}/{\wp }_b^{-},{\wp }_a^{+}/{\wp }_b^{+}\right\}\mathrm{and}0\notin {\wp }_b.\hfill \end{array}$$

Scalar multiplication of the interval ${\wp }_a$ is indicated by

$$\theta {\wp }_a=\theta \left[{\wp }_a^{-},{\wp }_a^{+}\right]=\left\{\begin{array}{cc}\left[\theta {\wp }_a^{-},\theta {\wp }_a^{+}\right],\hfill & \theta >0\hfill \\ \hfill & \\ \left\{0\right\},\hfill & \theta =0\hfill \\ \hfill & \\ \left[\theta {\wp }_a^{+},\theta {\wp }_a^{-}\right],\hfill & \theta <0,\hfill \end{array}\right.$$

where $\theta \in \mathbb{R}.$

The opposite of the interval ${\wp }_a$ is

$$-{\wp }_a:=(-1){\wp }_a=[-{\wp }_a^{+},-{\wp }_a^{-}],$$

for $\varsigma =-1$.

The subtraction is given by

$${\wp }_a-{\wp }_b={\wp }_a+(-{\wp }_b)=[{\wp }_a^{-}-{\wp }_b^{+},{\wp }_a^{+}-{\wp }_b^{-}].$$

In general, $-{\wp }_a$ is not additive inverse for ${\wp }_a$, i.e., ${\wp }_a-{\wp }_a\ne 0.$

$\Omega$ is a quasilinear space due to its algebraic features and operation definitions [38]. Moore, Markov, and Lupulescu describe the following qualities:

(1) Associative Property of Addition:

$$({\wp }_a+{\wp }_b)+{\wp }_c={\wp }_a+({\wp }_b+{\wp }_c),\forall {\wp }_a,{\wp }_b,{\wp }_c\in \Omega .$$

(2) Additive Element:

$${\wp }_a+0=0+{\wp }_a={\wp }_a,\forall {\wp }_a\in \Omega .$$

(3) Commutative Property of Addition:

$${\wp }_a+{\wp }_b={\wp }_b+{\wp }_a,\forall {\wp }_a,{\wp }_b\in \Omega .$$

(4) Cancellation Law:

$${\wp }_a+{\wp }_c={\wp }_b+{\wp }_c⟹{\wp }_a={\wp }_b,\forall {\wp }_a,{\wp }_b,{\wp }_c\in \Omega .$$

(5) Associative Property for Multiplication:

$$({\wp }_a.{\wp }_b).{\wp }_c={\wp }_a.({\wp }_b.{\wp }_c),\forall {\wp }_a,{\wp }_b,{\wp }_c\in \Omega .$$

(6) Commutative Property for Multiplication:

$${\wp }_a.{\wp }_b={\wp }_b.{\wp }_a,\forall {\wp }_a,{\wp }_b\in \Omega .$$

(7) Identity Property for Multiplication:

$${\wp }_a.1=1.{\wp }_a,\forall {\wp }_a\in \Omega .$$

(8) Associative Law:

$$\theta \left(\varrho {\wp }_a\right)=\left(\theta \varrho \right){\wp }_a,\forall {\wp }_a\in \Omega ,\forall \theta ,\varrho \in \mathbb{R}.$$

(9) First Distributive Law:

$$\theta ({\wp }_a+{\wp }_b)=\theta {\wp }_a+\theta {\wp }_b,\forall {\wp }_a,{\wp }_b\in \Omega \mathrm{and}\theta \in \mathbb{R}.$$

(10) Second Distributive Law:

$$(\theta +\varrho ){\wp }_a=\theta {\wp }_a+\varrho {\wp }_a,\forall {\wp }_a\in \Omega \mathrm{and}\theta ,\varrho \in \mathbb{R}.$$

There is another property, “⊆” that can be defined as:

$${\wp }_a\subseteq {\wp }_b⟺{\wp }_b\le {\wp }_a\mathrm{and}{\wp }_a^{+}\le {\wp }_b^{+}.$$

A function $𝐹$ is said to be an interval-valued function of $\tau$ on $[\varsigma ,\phi ]$ if it assigns a nonempty interval to each $\tau \in [\varsigma ,\phi ]$

$$𝐹\left(\tau \right)=\left[{𝐹}^{-}\left(\tau \right),{𝐹}^{+}\left(\tau \right)\right].$$

A partition of $\left[\varsigma ,\phi \right]$ is any finite ordered subset D having the form

$$D:\varsigma ={\tau }_0<{\tau }_1<\dots <{\tau }_n=\phi .$$

The mesh of a partition D is indicated by

$$mesh\left(D\right)=max\left\{{\tau }_i-{\tau }_{i-1}:i=1,2,\dots ,n\right\}.$$

We denote by $D\left(\left[\varsigma ,\phi \right]\right)$ the set of all partition of $\left[\varsigma ,\phi \right].$ Suppose that $D\left(\varsigma ,\left[\varsigma ,\phi \right]\right)$ is the set of all $D\in D\left(\left[\varsigma ,\phi \right]\right)$ such that $mesh\left(D\right)<\varsigma .$ We chose an arbitrary point ${\chi }_i$ in the interval $\left[{\tau }_{i-1},{\tau }_i\right],$ $i=1,2,\dots ,n,$ and we define the Sum

$$S(𝐹,D,\varsigma )=\sum _{i=1}^n𝐹\left({\chi }_i\right)\left[{\tau }_i-{\tau }_{i-1}\right],$$

where $𝐹:\left[\varsigma ,\phi \right]\to {\mathcal{K}}_c.$ The Sum $S(𝐹,D,\varsigma )$ is said to be a Riemann Sum of $𝐹$ corresponding to $D\in D\left(\varsigma ,\left[\varsigma ,\phi \right]\right).$

Definition 1

([39,40,41]). A function $𝐹:\left[\varsigma ,\phi \right]\to {\mathcal{K}}_c$ is said to be an interval Riemann-integrable function ($IR$-integrable) on $\left[\varsigma ,\phi \right]$ if there exist $\varsigma \in {\mathcal{K}}_c$ and $\varsigma >0,$ for each $\epsilon >0$

$$d\left(S(𝐹,D,\varsigma ),\varsigma \right)<\epsilon ,$$

for every Riemann sum S of ϝ corresponding to each $D\in D\left(\varsigma ,\left[\varsigma ,\phi \right]\right)$ and independent of choice of ${\chi }_{1i}\in \left[{\tau }_{i-1},{\tau }_i\right]$, $1\le i\le n.$ In this case, ς is called as the $IR$-integral of ϝ on $\left[\varsigma ,\phi \right]$ and is denoted by

$$\varsigma =\left(IR\right)\underset{\varsigma }{\overset{\phi }{\int }}𝐹\left(\tau \right)d\tau .$$

The collection of all functions that are $IR$-integrable on $\left[\delta ,\phi \right]$ will be denoted by ${\mathcal{IR}}_{\left(\left[\varsigma ,\phi \right]\right)}.$

Definition 2

([42]). For the intervals ${\wp }_a$ and ${\wp }_b,$ we state the g-difference of ${\wp }_a$ and ${\wp }_b$ as the interval ${\wp }_c$ such that

$${\wp }_a{\ominus }_g{\wp }_b={\wp }_c\iff \left\{\begin{array}{c}{\wp }_a={\wp }_b+{\wp }_c,\\ or\\ {\wp }_c={\wp }_a+\left(-{\wp }_b\right).\end{array}\right.$$

It looks beyond dispute that

$${\wp }_a{\ominus }_g{\wp }_b=\left\{\begin{array}{c}\left[{\wp }_a^{-}-{\wp }_b^{-},{\wp }_a^{+}-{\wp }_b^{+}\right],if\ell \left({\wp }_a\right)\ge \ell \left({\wp }_b\right),\\ \left[{\wp }_a^{+}-{\wp }_b^{+},{\wp }_a^{-}-{\wp }_b^{-}\right],if\ell \left({\wp }_a\right)<\ell \left({\wp }_b\right),\end{array}\right.$$

where $\ell \left({\wp }_a\right)={\wp }_a^{+}-{\wp }_a^{-}.$

• Moreover, another set feature is the inclusion ⊆ that is defined by

$${\wp }_a\subseteq {\wp }_b⟺{\wp }_a^{-}\ge {\wp }_b^{-}\mathrm{and}{\wp }_a^{+}\le {\wp }_b^{+}.$$

The next theorem explains the connection between $IR$-integrable and Riemann-integrable (R-integrable).

Theorem 1.

Assume that $𝐹:\left[\varsigma ,\phi \right]\to {\mathcal{K}}_c$ is an interval-valued function. A function $𝐹\in {\mathcal{IR}}_{\left(\left[\varsigma ,\phi \right]\right)}$ if and only if ${𝐹}^{-}\left(\tau \right)$, ${𝐹}^{+}\left(\tau \right)\in {\mathcal{R}}_{\left(\left[\varsigma ,\phi \right]\right)}$ and

$$\left(IR\right)\underset{\varsigma }{\overset{\phi }{\int }}𝐹\left(\tau \right)d\tau =\left[\left(R\right)\underset{\varsigma }{\overset{\phi }{\int }}{𝐹}^{-}\left(\tau \right)d\tau ,\left(R\right)\underset{\delta }{\overset{\phi }{\int }}{𝐹}^{+}\left(\tau \right)d\tau \right],$$

where ${\mathcal{R}}_{\left(\left[\varsigma ,\phi \right]\right)}$ denotes the R-integrable function.

It is easy to see that if $𝐹\left(\tau \right)\subseteq \psi \left(\tau \right)$ for all $\tau \in [\varsigma ,\phi ],$ then $\left(IR\right)\underset{\varsigma }{\overset{\phi }{\int }}𝐹\left(\tau \right)d\tau \subseteq \left(IR\right)\underset{\varsigma }{\overset{\phi }{\int }}\psi \left(\tau \right)d\tau .$

Definition 3

(Convex Set [6]). A set $I\subset {\mathbb{R}}^n$ is convex, for any two points $\varsigma ,\phi \in I,$ the entire segment joining ς and φ lies in $I.$ The points in the segments are of the form

$$\tau \varsigma +(1-\tau )\phi \in I,\forall \tau \in [0,1].$$

(11)

Definition 4

(Convex Function [6]). Let I be a convex subset of a real vector space and $𝐹:I\subset \mathbb{R}\to \mathbb{R}$ is said to be convex if

$$\tau 𝐹\left({\chi }_1\right)+(1-\tau )𝐹\left({\chi }_2\right)\ge 𝐹(\tau {\chi }_1+(1-\tau ){\chi }_2)$$

(12)

for all $\tau \in [0,1],$ and ${\chi }_1,{\chi }_2\in I.$

Definition 5

(Interval-Valued Convex Function (IV-CF) [15]). A set-valued function ϝ is said to be convex if it satisfies:

$$\tau 𝐹\left({\chi }_1\right)+(1-\tau )𝐹\left({\chi }_2\right)\subseteq 𝐹(\tau {\chi }_1+(1-\tau ){\chi }_2)$$

for all $\tau \in \left[0,1\right]$ and ${\chi }_1,{\chi }_2$ from its domain.

Definition 6

(coordinated Convex Function (C-CF) [16]). A mapping $𝐹:▵\subset {\mathbb{R}}^2\to \mathbb{R}$ is convex on coordinates, if the following inequality holds:

$$\tau 𝐹\left({\chi }_1,{\chi }_2\right)+\left(1-\tau \right)𝐹\left(\rho ,\kappa \right)\ge 𝐹\left(\tau {\chi }_1+\left(1-\tau \right)\rho ,\tau {\chi }_2+\left(1-\tau \right)\kappa \right)$$

(13)

for all $\left({\chi }_1,{\chi }_2\right),\left(\rho ,\kappa \right)\in ▵$ and $\tau \in \left[0,1\right].$

A modification for convex functions on coordinates, which are also known as coordinated convex functions, was introduced by Dragomir [16,17] as follows: a function $𝐹:▵\subset {\mathbb{R}}^2\to \mathbb{R}$ is convex on the coordinates on $▵$ if the partial differentiable mappings ${𝐹}_{{\chi }_2}:\left[\varsigma ,\phi \right]\to \mathbb{R},{𝐹}_{{\chi }_2}\left(\mu \right)=𝐹\left(\mu ,{\chi }_2\right)$ and ${𝐹}_{{\chi }_1}:\left[\varrho ,\vartheta \right]\to \mathbb{R},{𝐹}_{{\chi }_1}\left(\nu \right)=𝐹\left({\chi }_1,\nu \right)$ are convex for all ${\chi }_1\in \left[\varsigma ,\phi \right],{\chi }_2\in \left[\varrho ,\vartheta \right].$

A formal definition for the coordinated convex functions can be stated as:

Definition 7

(Modified definition of coordinated convex functions [22]). A mapping $𝐹:▵\subset {\mathbb{R}}^2\to \mathbb{R}$ is convex on coordinates, if the following inequality holds:

$$\theta \tau 𝐹\left({\chi }_1,{\chi }_2\right)+\theta \left(1-\tau \right)𝐹\left({\chi }_1,\kappa \right)+\left(1-\theta \right)\tau 𝐹\left(\rho ,{\chi }_2\right)+\left(1-\tau \right)\left(1-\theta \right)𝐹\left(\rho ,\kappa \right)\ge 𝐹\left(\theta {\chi }_1+\left(1-\theta \right)\rho ,\tau {\chi }_2+\left(1-\tau \right)\kappa \right)$$

(14)

for all $\left({\chi }_1,{\chi }_2\right),\left(\rho ,\kappa \right)\in ▵$ and $\theta ,\tau \in \left[0,1\right].$

Definition 8

(Interval-Valued coordinated Convex Function). A function $𝐹:▵\subset {\mathbb{R}}^2\to {\mathbb{R}}_I^{+}$ is said to be an interval-valued coordinated convex function if the following inequality holds:

$$\theta \tau 𝐹\left({\chi }_1,{\chi }_2\right)+\theta \left(1-\tau \right)𝐹\left({\chi }_1,\kappa \right)+\left(1-\theta \right)\tau 𝐹\left(\rho ,{\chi }_2\right)+\left(1-\tau \right)\left(1-\theta \right)𝐹\left(\rho ,\kappa \right)\subseteq 𝐹\left(\theta {\chi }_1+\left(1-\theta \right)\rho ,\tau {\chi }_2+\left(1-\tau \right)\kappa \right)$$

(15)

for all $\left({\chi }_1,{\chi }_2\right),\left(\rho ,\kappa \right)\in ▵$ and $\theta ,\tau \in \left[0,1\right].$

Definition 9

(Jensen’s Inequality [43]). Let ϝ be a convex function defined on the real interval $\chi \subset \mathbb{R}.$ If ${\chi }_1,{\chi }_2,{\chi }_3,\dots ,{\chi }_n\in \chi$ and ${\mu }_1,{\mu }_2,{\mu }_3,\dots ,{\mu }_n\ge 0,$ then

$$\sum _{i=1}^n{\mu }_i𝐹\left({\chi }_i\right)\ge 𝐹\left(\sum _{i=1}^n{\mu }_i{\chi }_i\right),$$

(16)

where ${\sum }_{i=1}^n{\mu }_i=1.$

Definition 10

([35]). Let ϝ be a convex interval-valued function on $\left[\varsigma ,\phi \right]$ then the following inclusion is true:

$$\sum _{i=1}^n{\mu }_i𝐹\left({\chi }_i\right)\subseteq 𝐹\left(\sum _{i=1}^n{\mu }_i{\chi }_i\right),$$

where ${\sum }_{i=1}^n{\mu }_i=1.$

Definition 11

(Jensen’s Mercer’s Inequality [29]). Let $𝐹:[\varsigma ,\phi ]\subset \mathbb{R}\to \mathbb{R}$ be a convex function. Then,

$$𝐹\left(\varsigma \right)+𝐹\left(\phi \right)-\sum _{i=1}^n{\mu }_i𝐹\left({\chi }_i\right)\ge 𝐹\left(\varsigma +\phi -\sum _{i=1}^n{\mu }_i{\chi }_i\right)$$

(17)

for every ${\chi }_i\in [\varsigma ,\phi ]$ and ${\mu }_i\in [0,1],(i=1,2,3,\dots ,n)$ with ${\sum }_{i=1}^n{\mu }_i=1.$ The inequality was established by Mercer in [29], and considered as a variant of Jensen’s inequality.

Theorem 2

([35]). Let ϝ be a convex interval-valued function on $\left[\delta ,\phi \right]$ such that $L\left(𝐹\left(\phi \right)\right)\ge$$L\left(𝐹\left({\varsigma }_0\right)\right)$ for all ${\varsigma }_0\in \left[\varsigma ,\phi \right]$, then the following identity is true:

$$𝐹\left(\varsigma \right)+𝐹\left(\phi \right){\ominus }_g\sum _{i=1}^n{\mu }_i𝐹\left({\chi }_i\right)\subseteq 𝐹\left(\varsigma +\phi -\sum _{i=1}^n{\mu }_i{\chi }_i\right),$$

where ${\sum }_{i=1}^n{\mu }_i=1.$

Augustin-Louis Cauchy and Hermann Amandus Schwarz separately defined the Cauchy-Schwarz inequality (also known as the Cauchy–Bunyakovsky–Schwarz inequality).

Definition 12

(Cauchy–Schwarz Inequality [44]). It states that for any two vectors, denoted as μ and ν, in the inner product space, the following inequality holds:

$$∥\mu ∥∥\nu ∥\ge \left|〈\mu ,\nu 〉\right|,$$

(18)

where $〈\mu ,\nu 〉$ represent the inner product of μ and $\nu ,$ and $∥\mu ∥,∥\nu ∥$ represent the norms of μ and ν, respectively.

Theorem 3

([45]). Let $𝐹,G:\left[\varsigma ,\phi \right]\to {\mathbb{R}}_I^{+}$ be two interval-valued coordinated convex functions such that $𝐹\left(\tau \right)=\left[{𝐹}^{-}\left(\tau \right),{𝐹}^{+}\left(\tau \right)\right]$ and $G\left(\tau \right)=\left[G^{-}\left(\tau \right),G^{+}\left(\tau \right)\right],$ also $h_1,h_2:\left[0,1\right]\to \mathbb{R}$ are two non-negative functions and $h_1\left({\displaystyle \frac{1}{2}}\right)h_2\left({\displaystyle \frac{1}{2}}\right)\ne 0.$ If $𝐹,G\in SX\left(h,\left[\delta ,\phi \right],{\mathbb{R}}_I^{+}\right),$ then

$$\begin{array}{ccc}& & {\displaystyle \frac{1}{2h_1\left(\frac{1}{2}\right)h_2\left(\frac{1}{2}\right)}}𝐹\left({\displaystyle \frac{\varsigma +\phi }{2}}\right)G\left({\displaystyle \frac{\varsigma +\phi }{2}}\right)\hfill \\ & \supseteq & {\displaystyle \frac{1}{\phi -\varsigma }}\left(IR\right){\int }_{\varsigma }^{\phi }𝐹\left({\chi }_1\right)G\left({\chi }_1\right)d{\chi }_1+M\left(\varsigma ,\phi \right){\int }_0^1{h}_1\left(\tau \right)h_2\left(1-\tau \right)d\tau \hfill \\ & & +N\left(\varsigma ,\phi \right){\int }_0^1{h}_1\left(\tau \right)h_2\left(1-\tau \right)d\tau ,\hfill \end{array}$$

(19)

and

$$\begin{array}{ccc}\hfill {\displaystyle \frac{1}{\phi -\varsigma }}\left(IR\right){\int }_{\varsigma }^{\phi }𝐹\left({\chi }_1\right)G\left({\chi }_1\right)d{\chi }_1& \supseteq & M\left(\varsigma ,\phi \right){\int }_0^1{h}_1\left(\tau \right)h_2\left(\tau \right)d\tau \hfill \\ & & +N\left(\varsigma ,\phi \right){\int }_0^1{h}_1\left(\tau \right)h_2\left(1-\tau \right)d\tau ,\hfill \end{array}$$

(20)

where

$$\begin{array}{ccc}\hfill M\left(\varsigma ,\phi \right)& =& 𝐹\left(\varsigma \right)G\left(\delta \right)+𝐹\left(\phi \right)G\left(\phi \right),\hfill \\ \hfill N\left(\varsigma ,\phi \right)& =& 𝐹\left(\varsigma \right)G\left(\phi \right)+𝐹\left(\phi \right)G\left(\varsigma \right).\hfill \end{array}$$

Remark 1.

The identities (19) and (20) simplify to the following if $h\left(\tau \right)=\tau$:

$$\begin{array}{ccc}& & 2𝐹\left({\displaystyle \frac{\varsigma +\phi }{2}}\right)G\left({\displaystyle \frac{\varsigma +\phi }{2}}\right)\hfill \\ & \supseteq & {\displaystyle \frac{1}{\phi -\varsigma }}\left(IR\right){\int }_{\varsigma }^{\phi }𝐹\left({\chi }_1\right)G\left({\chi }_1\right)d{\chi }_1+{\displaystyle \frac{1}{6}}M\left(\varsigma ,\phi \right)+{\displaystyle \frac{1}{3}}N\left(\varsigma ,\phi \right),\hfill \end{array}$$

(21)

and

$${\displaystyle \frac{1}{\phi -\varsigma }}\left(IR\right){\int }_{\varsigma }^{\phi }𝐹\left({\chi }_1\right)G\left({\chi }_1\right)d{\chi }_1\supseteq {\displaystyle \frac{1}{3}}M\left(\varsigma ,\phi \right)+{\displaystyle \frac{1}{6}}N\left(\varsigma ,\phi \right).$$

(22)

The definition of the Jensen–Mercer inequality for interval-valued coordinated convex functions and the demonstration of new H.H. Jensen–Mercer-type inequalities for interval-valued coordinated convex functions are the primary goals of the discussion mentioned above.

3. Main Results

We provide Jensen–Mercer and a few H.H. Mercer-type identities for interval-valued coordinated convex functions.

Lemma 1.

A function $𝐹:▵=\left[{\chi }_1,{\chi }_n\right]\times \left[w_1,w_n\right]\to {\mathbb{R}}_{\varrho }^{+}$ is $IV-CF$ on $▵,$ if ${𝐹}^{-}$ is coordinated convex on $▵ and {𝐹}^{+}$ is a coordinated concave function on $▵ such that L\left(𝐹\left({\chi }_n,w_n\right)\right)\ge L\left(𝐹\left({\chi }_i,w_i\right)\right)$ for all ${\chi }_i\in \left[{\chi }_1,{\chi }_n\right]$ and $w_i\in \left[w_1,w_n\right]$, we have:

$$𝐹\left({\chi }_1+{\chi }_n-{\chi }_k,w_1+w_n-w_k\right)\supseteq 𝐹\left({\chi }_1,w_1\right)+𝐹\left({\chi }_1,w_n\right)+𝐹\left({\chi }_n,w_1\right)+𝐹\left({\chi }_n,w_n\right){\ominus }_g𝐹\left({\chi }_k,w_k\right),$$

(23)

where $1\le k\le n.$

Proof. 

Write $y_k={\chi }_1+{\chi }_n-{\chi }_k.$ Then, ${\chi }_1+{\chi }_n={\chi }_k+y_k$ so that the pairs ${\chi }_1,{\chi }_n$ and ${\chi }_k,y_k$ possess the same midpoint. Since that is the case, there exists $\theta$ such that

$$\begin{array}{ccc}\hfill {\chi }_k& =& \left(1-\theta \right){\chi }_1+\theta {\chi }_n,\hfill \\ \hfill y_k& =& \theta {\chi }_1+\left(1-\theta \right){\chi }_n.\hfill \end{array}$$

Similarly, write ${\rho }_k=w_1+w_n-w_k.$ Then $w_1+w_n=w_k+{\rho }_k$ so that the pairs $w_1,w_n$ and $w_k,{\rho }_k$ possess the same midpoint. Since that is the case, there exists $\tau$ such that

$$\begin{array}{ccc}\hfill w_k& =& \left(1-\tau \right)w_1+\tau w_n,\hfill \\ \hfill {\rho }_k& =& \tau w_1+\left(1-\tau \right)w_n,\hfill \end{array}$$

where $\theta ,\tau \in \left[0,1\right]$ and $1\le k\le n.$

$$\begin{array}{cc}\hfill 𝐹\left(y_k,{\rho }_k\right)& =𝐹\left(\theta {\chi }_1+\left(1-\theta \right){\chi }_n,\tau w_1+\left(1-\tau \right)w_n\right)\hfill \\ \hfill & \supseteq \theta \tau 𝐹\left({\chi }_1,w_1\right)+\theta \left(1-\tau \right)𝐹\left({\chi }_1,w_n\right)+\left(1-\theta \right)\tau 𝐹\left({\chi }_n,w_1\right)+\left(1-\theta \right)\left(1-\tau \right)𝐹\left({\chi }_n,w_n\right),\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill 𝐹\left({\chi }_k,w_k\right)& =𝐹\left(\left(1-\theta \right){\chi }_1+\theta {\chi }_n,\left(1-\tau \right)w_1+\tau w_n\right)\hfill \\ \hfill & \supseteq \left(1-\theta \right)\left(1-\tau \right)𝐹\left({\chi }_1,w_1\right)+\left(1-\theta \right)\tau 𝐹\left({\chi }_1,w_n\right)+\theta \left(1-\tau \right)𝐹\left({\chi }_n,w_1\right)+\tau \theta 𝐹\left({\chi }_n,w_n\right),\hfill \end{array}$$

(25)

Now, by Jensen inequality for interval-valued coordinated convex functions and using (25)

$$\begin{array}{ccc}\hfill 𝐹\left(y_k,{\rho }_k\right)& \supseteq & \theta \tau 𝐹\left({\chi }_1,w_1\right)+\theta \left(1-\tau \right)𝐹\left({\chi }_1,w_n\right)+\left(1-\theta \right)\tau 𝐹\left({\chi }_n,w_1\right)+\left(1-\theta \right)\left(1-\tau \right)𝐹\left({\chi }_n,w_n\right)\hfill \\ & =& \theta \tau 𝐹\left({\chi }_1,w_1\right)+𝐹\left({\chi }_1,w_1\right){\ominus }_g𝐹\left({\chi }_1,w_1\right)+\theta \left(1-\tau \right)𝐹\left({\chi }_1,w_n\right)+𝐹\left({\chi }_1,w_n\right){\ominus }_g𝐹\left({\chi }_1,w_n\right)\hfill \\ & & +\left(1-\theta \right)\tau 𝐹\left({\chi }_n,w_1\right)+𝐹\left({\chi }_n,w_1\right){\ominus }_g𝐹\left({\chi }_n,w_1\right)+\left(1-\tau -\theta +\theta \tau \right)𝐹\left({\chi }_n,w_n\right)\hfill \\ & =& 𝐹\left({\chi }_1,w_1\right)+𝐹\left({\chi }_1,w_n\right)+𝐹\left({\chi }_n,w_1\right)+𝐹\left({\chi }_n,w_n\right)\hfill \\ & & {\ominus }_g\left[\left(1-\theta \tau \right)𝐹\left({\chi }_1,w_1\right)+\left(1-\theta +\theta \tau \right)𝐹\left({\chi }_1,w_n\right)+\left(1-\tau +\theta \tau \right)𝐹\left({\chi }_n,w_1\right)+\left(\theta +\tau -\theta \tau \right)𝐹\left({\chi }_n,w_n\right)\right]\hfill \\ & \supseteq & 𝐹\left({\chi }_1,w_1\right)+𝐹\left({\chi }_1,w_n\right)+𝐹\left({\chi }_n,w_1\right)+𝐹\left({\chi }_n,w_n\right)\hfill \\ & & {\ominus }_g\left[\left(1-\theta \right)\left(1-\tau \right)𝐹\left({\chi }_1,w_1\right)+\left(1-\theta \right)\tau 𝐹\left({\chi }_1,w_n\right)+\theta \left(1-\tau \right)𝐹\left({\chi }_n,w_1\right)+\tau \theta 𝐹\left({\chi }_n,w_n\right)\right]\hfill \\ & \supseteq & 𝐹\left({\chi }_1,w_1\right)+𝐹\left({\chi }_1,w_n\right)+𝐹\left({\chi }_n,w_1\right)+𝐹\left({\chi }_n,w_n\right){\ominus }_g𝐹\left({\chi }_k,w_k\right),\hfill \end{array}$$

where $1-\theta \tau \ge 1-\tau -\theta +\theta \tau ,1-\tau +\theta \tau \ge \theta -\tau \theta ,1-\theta +\theta \tau \ge \tau -\theta \tau ,$ and $\tau +\theta -\theta \tau \ge \theta \tau ,$ by using (18).

Hence, we have

$$𝐹\left({\chi }_1+{\chi }_n-{\chi }_k,w_1+w_n-w_k\right)\supseteq 𝐹\left({\chi }_1,w_1\right)+𝐹\left({\chi }_1,w_n\right)+𝐹\left({\chi }_n,w_1\right)+𝐹\left({\chi }_n,w_n\right){\ominus }_g𝐹\left({\chi }_k,w_k\right).$$

The proof of Lemma 1 is completed. □

Theorem 4.

Let $𝐹:▵=\left[{\chi }_1,{\chi }_n\right]\times \left[w_1,w_n\right]\to {\mathbb{R}}_{\varrho }^{+}$ be interval-valued coordinated convex functions on $▵ with {\chi }_k\in \left[{\chi }_1,{\chi }_n\right],w_k\in \left[w_1,w_n\right].$ If ${𝐹}^{-}$ is coordinated convex on $▵ and {𝐹}^{+}$ is coordinated concave function on $▵ such that L\left(𝐹\left({\chi }_n,w_n\right)\right)\ge L\left(𝐹\left({\chi }_i,w_i\right)\right)$ for all ${\chi }_i\in \left[{\chi }_1,{\chi }_n\right]$ and $w_i\in \left[w_1,w_n\right],$ then

$$𝐹\left({\chi }_1+{\chi }_n\sum {\mu }_k{\chi }_k,w_1+w_n-\sum {\nu }_k{w}_k\right)\supseteq 𝐹\left({\chi }_1,w_1\right)+𝐹\left({\chi }_1,w_n\right)+𝐹\left({\chi }_n,w_1\right)+𝐹\left({\chi }_n,w_n\right){\ominus }_g\sum {\mu }_k{\nu }_k𝐹\left({\chi }_k,w_k\right).$$

(26)

Proof. 

By Jensen Inequality for interval-valued functions and Lemma 1, we have:

$$\begin{array}{ccc}\hfill 𝐹\left({\chi }_1+{\chi }_n-\sum {\mu }_k{\chi }_k,w_1+w_n-\sum {\nu }_k{w}_k\right)& =& 𝐹\left(\sum {\mu }_k\left({\chi }_1+{\chi }_n-{\chi }_k\right),\sum {\nu }_k\left(w_1+w_n-w_k\right)\right)\hfill \\ & \supseteq & \sum {\mu }_k{\nu }_k𝐹\left({\chi }_1+{\chi }_n-{\chi }_k,w_1+w_n-w_k\right)\hfill \\ & \supseteq & \sum {\mu }_k{\nu }_k\left[𝐹\left({\chi }_1,w_1\right)+𝐹\left({\chi }_1,w_n\right)+𝐹\left({\chi }_n,w_1\right)+𝐹\left({\chi }_n,w_n\right){\ominus }_g𝐹\left({\chi }_k,w_k\right)\right]\hfill \\ & =& 𝐹\left({\chi }_1,w_1\right)+𝐹\left({\chi }_1,w_n\right)+𝐹\left({\chi }_n,w_1\right)+𝐹\left({\chi }_n,w_n\right){\ominus }_g\sum {\mu }_k{\nu }_k𝐹\left({\chi }_k,w_k\right).\hfill \end{array}$$

Hence, the proof of Theorem 4 is completed. □

Theorem 5.

Let $𝐹:▵=\left[\varsigma ,\phi \right]\times \left[\varrho ,\vartheta \right]\to {\mathbb{R}}_I^{+}$ be interval-valued coordinated convex functions on Δ such that $𝐹\left(\tau \right)=\left[{𝐹}^{-}\left(\tau \right),{𝐹}^{+}\left(\tau \right)\right].$ Then, one has the inequalities:

$$\begin{array}{ccc}& & 𝐹\left(\varsigma +\phi -{\displaystyle \frac{\kappa +\rho }{2}},\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)\hfill \\ & \supseteq & {\displaystyle \frac{1}{2}}\left[{\displaystyle \frac{1}{\left(\rho -\kappa \right)}}{\int }_{\varsigma +\phi -\rho }^{\varsigma +\phi -\kappa }𝐹\left({\chi }_1,\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)d{\chi }_1+{\displaystyle \frac{1}{\left(\mu -\nu \right)}}{\int }_{\varrho +\vartheta -\mu }^{\varrho +\vartheta -\nu }𝐹\left(\delta +\phi -{\displaystyle \frac{\kappa +\rho }{2}},{\chi }_2\right)d{\chi }_2\right]\hfill \\ & \supseteq & {\displaystyle \frac{1}{\left(\rho -\kappa \right)\left(\mu -\nu \right)}}{\int }_{\varsigma +\phi -\rho }^{\varsigma +\phi -\kappa }{\int }_{\varrho +\vartheta -\mu }^{\varrho +\vartheta -\nu }𝐹\left({\chi }_1,{\chi }_2\right)d{\chi }_{2}d{\chi }_1\hfill \\ & \supseteq & {\displaystyle \frac{1}{4}}\left[𝐹\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\mu \right)+𝐹\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\nu \right)\right.\left.+𝐹\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu \right)+𝐹\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\nu \right)\right]\hfill \\ & \supseteq & 𝐹\left(\varsigma ,\varrho \right)+𝐹\left(\varsigma ,\vartheta \right)+𝐹\left(\phi ,\varrho \right)+𝐹\left(\phi ,\vartheta \right){\ominus }_g\left[{\displaystyle \frac{𝐹\left(\rho ,\mu \right)+𝐹\left(\rho ,\nu \right)+𝐹\left(\kappa ,\mu \right)+𝐹\left(\kappa ,\nu \right)}{4}}\right],\hfill \end{array}$$

(27)

where $\kappa ,\rho \in \left[\varsigma ,\phi \right]$ and $\mu ,\nu \in \left[\varrho ,\vartheta \right].$

Proof. 

Since $𝐹:▵\to \mathbb{R}$ is interval-valued convex on coordinates and $F_{{\chi }_1}:\left[\varrho ,\vartheta \right]\to \mathbb{R},F_{{\chi }_1}\left({\chi }_2\right)=𝐹\left({\chi }_1,{\chi }_2\right)$ is interval-valued convex on $\left[\varrho ,\vartheta \right],$ for all ${\chi }_1\in \left[\varsigma ,\phi \right].$ Then, by H.H. Jensen–Mercer Inequality

$$F_{{\chi }_1}\left(\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)\supseteq {\displaystyle \frac{1}{\left(\mu -\nu \right)}}{\int }_{\varrho +\vartheta -\mu }^{\varrho +\vartheta -\nu }{F}_{{\chi }_1}\left({\chi }_2\right)d{\chi }_2\supseteq {\displaystyle \frac{F_{{\chi }_1}\left(\varrho +\vartheta -\mu \right)+F_{{\chi }_1}\left(\varrho +\vartheta -\nu \right)}{2}},$$

so,

$$\begin{array}{cc}\hfill 𝐹\left({\chi }_1,\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)& \supseteq {\displaystyle \frac{1}{\left(\mu -\nu \right)}}{\int }_{\varrho +\vartheta -\mu }^{\varrho +\vartheta -\nu }𝐹\left({\chi }_1,{\chi }_2\right)d{\chi }_2\hfill \\ \hfill & \supseteq {\displaystyle \frac{𝐹\left({\chi }_1,\varrho +\vartheta -\mu \right)+𝐹\left({\chi }_1,\varrho +\vartheta -\nu \right)}{2}},\hfill \end{array}$$

(28)

integrating (28) with respect to ${\chi }_1$, we get

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{\left(\rho -\kappa \right)}}{\int }_{\varsigma +\phi -\rho }^{\delta +\phi -\kappa }𝐹\left({\chi }_1,\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)d{\chi }_1\hfill \\ \hfill & \supseteq {\displaystyle \frac{1}{\left(\rho -\kappa \right)\left(\mu -\nu \right)}}{\int }_{\varsigma +\phi -\rho }^{\varsigma +\phi -\kappa }{\int }_{\varrho +\vartheta -\mu }^{\varrho +\vartheta -\nu }𝐹\left({\chi }_1,{\chi }_2\right)d{\chi }_{2}d{\chi }_1\hfill \\ \hfill & \supseteq {\displaystyle \frac{1}{2\left(\rho -\kappa \right)}}{\int }_{\varsigma +\phi -\rho }^{\varsigma +\phi -\kappa }𝐹\left({\chi }_1,\varrho +\vartheta -\mu \right)d{\chi }_1+{\displaystyle \frac{1}{2\left(\rho -\kappa \right)}}{\int }_{\delta +\phi -\rho }^{\varsigma +\phi -\kappa }𝐹\left({\chi }_1,\varrho +\vartheta -\nu \right)d{\chi }_1.\hfill \end{array}$$

(29)

Similarly, for the mapping $F_{{\chi }_1}:\left[\varsigma ,\phi \right]\to \mathbb{R},F_{{\chi }_1}\left({\chi }_2\right)=𝐹\left({\chi }_1,{\chi }_2\right),$ we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{\left(\mu -\nu \right)}}{\int }_{\varrho +\vartheta -\mu }^{\varrho +\vartheta -\nu }𝐹\left(\varsigma +\phi -{\displaystyle \frac{\kappa +\rho }{2}},{\chi }_2\right)d{\chi }_2\hfill \\ \hfill & \supseteq {\displaystyle \frac{1}{\left(\rho -\kappa \right)\left(\mu -\nu \right)}}{\int }_{\varsigma +\phi -\rho }^{\varsigma +\phi -\kappa }{\int }_{\varrho +\vartheta -\mu }^{\varrho +\vartheta -\nu }𝐹\left({\chi }_1,{\chi }_2\right)d{\chi }_{2}d{\chi }_1\hfill \\ \hfill & \supseteq {\displaystyle \frac{1}{2\left(\mu -\nu \right)}}{\int }_{\varrho +\vartheta -\mu }^{\varrho +\vartheta -\nu }𝐹\left(\varsigma +\phi -\rho ,{\chi }_2\right)d{\chi }_2+{\displaystyle \frac{1}{2\left(\mu -\nu \right)}}{\int }_{\varrho +\vartheta -\mu }^{\varrho +\vartheta -\nu }𝐹\left(\varsigma +\phi -\kappa ,{\chi }_2\right)d{\chi }_2.\hfill \end{array}$$

(30)

The second and third inequalities of (27) can be obtained by adding the inequalities (29) and (30). Next, using the H. H. Mercer inequality, we have:

$$𝐹\left(\varsigma +\phi -{\displaystyle \frac{\kappa +\rho }{2}},\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)\supseteq {\displaystyle \frac{1}{\left(\rho -\kappa \right)}}{\int }_{\delta +\phi -\rho }^{\varsigma +\phi -\kappa }𝐹\left({\chi }_1,\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)d{\chi }_1,$$

(31)

$$𝐹\left(\varsigma +\phi -{\displaystyle \frac{\kappa +\rho }{2}},\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)\supseteq {\displaystyle \frac{1}{\left(\mu -\nu \right)}}{\int }_{\varrho +\vartheta -\mu }^{\varrho +\vartheta -\nu }𝐹\left(\varsigma +\phi -{\displaystyle \frac{\kappa +\rho }{2}},{\chi }_2\right)d{\chi }_2,$$

(32)

from (31) and (32), we get the first inequality of (27). From (29) and (30)

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{\left(\rho -\kappa \right)}}{\int }_{\varsigma +\phi -\rho }^{\delta +\phi -\kappa }𝐹\left({\chi }_1,\varrho +\vartheta -\mu \right)d{\chi }_1& \supseteq {\displaystyle \frac{𝐹\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\mu \right)+𝐹\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu \right)}{2}},\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{\left(\rho -\kappa \right)}}{\int }_{\varsigma +\phi -\rho }^{\delta +\phi -\kappa }𝐹\left({\chi }_1,\varrho +\vartheta -\nu \right)d{\chi }_1& \supseteq {\displaystyle \frac{𝐹\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\nu \right)+𝐹\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\nu \right)}{2}},\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{\left(\mu -\nu \right)}}{\int }_{\varrho +\vartheta -\mu }^{\varrho +\vartheta -\nu }𝐹\left(\varsigma +\phi -\rho ,{\chi }_2\right)d{\chi }_2& \supseteq {\displaystyle \frac{𝐹\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\mu \right)+𝐹\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\nu \right)}{2}},\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{\left(\mu -\nu \right)}}{\int }_{\varrho +\vartheta -\mu }^{\varrho +\vartheta -\nu }𝐹\left(\varsigma +\phi -\kappa ,{\chi }_2\right)d{\chi }_2& \supseteq {\displaystyle \frac{𝐹\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu \right)+𝐹\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\nu \right)}{2}}.\hfill \end{array}$$

(36)

The second last inequality of (27) is obtained by applying inequalities (33)–(36). Additionally, we obtain the final inequality of (27) by applying the Jensen–Mercer inequality for interval-valued coordinated convex functions. This completes the proof of Theorem 5. □

Remark 2.

Let $\kappa =\varsigma ,\rho =\phi ,\mu =\varrho ,$ and $\nu =\vartheta$ in Theorem 5, we obtain result given by Zhao in [23].

Remark 3.

We obtain the result proved by Toseef et al. in [36] upon substituting ${𝐹}^{-}={𝐹}^{+}$ in Theorem 5.

Remark 4.

We obtain the result proved by Dragomir in [16] by substituting ${𝐹}^{-}={𝐹}^{+}$ and $\kappa =\varsigma ,\rho =\phi ,\mu =\varrho ,$ and $\nu =\vartheta$ in Theorem 5.

Theorem 6.

Let $𝐹,G:▵=\left[\varsigma ,\phi \right]\times \left[\varrho ,\vartheta \right]\to {\mathbb{R}}_I^{+}$ be two interval-valued coordinated convex functions with $\kappa ,\rho \in \left[\varsigma ,\phi \right]$ and $\mu ,\nu \in \left[\varrho ,\vartheta \right],$ such that $𝐹\left(\tau \right)=\left[{𝐹}^{-}\left(\tau \right),{𝐹}^{+}\left(\tau \right)\right]$ and $G\left(\tau \right)=\left[G^{-}\left(\tau \right),G^{+}\left(\tau \right)\right],$ then the Subsequent inequality holds:

$$\begin{array}{ccc}& & {\displaystyle \frac{1}{\left(\rho -\kappa \right)\left(\mu -\nu \right)}}{\int }_{\delta +\phi -\rho }^{\varsigma +\phi -\kappa }{\int }_{\varrho +\vartheta -\mu }^{\varrho +\vartheta -\nu }𝐹\left({\chi }_1,{\chi }_2\right)G\left({\chi }_1,{\chi }_2\right)d{\chi }_{2}d{\chi }_1\hfill \\ & \supseteq & {\displaystyle \frac{1}{9}}P\left(\varsigma +\phi -\rho ,\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu ,\varrho +\vartheta -\nu \right)+{\displaystyle \frac{1}{18}}M\left(\varsigma +\phi -\rho ,\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu ,\varrho +\vartheta -\nu \right)\hfill \\ & & +{\displaystyle \frac{1}{36}}N\left(\varsigma +\phi -\rho ,\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu ,\varrho +\vartheta -\nu \right),\hfill \end{array}$$

(37)

where

$$\begin{array}{ccc}& & P\left(\varsigma +\phi -\rho ,\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu ,\varrho +\vartheta -\nu \right)\hfill \\ & =& 𝐹\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\mu \right)G\left(\delta +\phi -\rho ,\varrho +\vartheta -\mu \right)+𝐹\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\nu \right)G\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\nu \right)\hfill \\ & & +𝐹\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu \right)G\left(\delta +\phi -\kappa ,\varrho +\vartheta -\mu \right)+𝐹\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\nu \right)G\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\nu \right),\hfill \end{array}$$

$$\begin{array}{ccc}& & M\left(\varsigma +\phi -\rho ,\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu ,\varrho +\vartheta -\nu \right)\hfill \\ & =& 𝐹\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\mu \right)G\left(\delta +\phi -\rho ,\varrho +\vartheta -\nu \right)+𝐹\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\nu \right)G\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\mu \right)\hfill \\ & & +𝐹\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu \right)G\left(\delta +\phi -\kappa ,\varrho +\vartheta -\nu \right)+𝐹\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\nu \right)G\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu \right)\hfill \\ & & +𝐹\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu \right)G\left(\delta +\phi -\rho ,\varrho +\vartheta -\mu \right)+𝐹\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\mu \right)G\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu \right)\hfill \\ & & +𝐹\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\nu \right)G\left(\delta +\phi -\rho ,\varrho +\vartheta -\nu \right)+𝐹\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\nu \right)G\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\nu \right),\hfill \end{array}$$

$$\begin{array}{ccc}& & N\left(\varsigma +\phi -\rho ,\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu ,\varrho +\vartheta -\nu \right)\hfill \\ & =& 𝐹\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu \right)G\left(\delta +\phi -\kappa ,\varrho +\vartheta -\nu \right)+𝐹\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\nu \right)G\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu \right)\hfill \\ & & +𝐹\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\nu \right)G\left(\delta +\phi -\rho ,\varrho +\vartheta -\mu \right)+𝐹\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\mu \right)G\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\nu \right).\hfill \end{array}$$

Proof. 

Since $𝐹$ and G are interval-valued coordinated convex functions on $▵,$ hence

$$\begin{array}{ccc}\hfill F_{{\chi }_1}\left({\chi }_2\right)& :& \left[\varrho ,\vartheta \right]\to {\mathbb{R}}_I^{+},F_{{\chi }_1}\left({\chi }_2\right)=𝐹\left({\chi }_1,{\chi }_2\right)\hfill \\ \hfill G_{{\chi }_1}\left({\chi }_2\right)& :& \left[\varrho ,\vartheta \right]\to {\mathbb{R}}_I^{+},G_{{\chi }_1}\left({\chi }_2\right)=G\left({\chi }_1,{\chi }_2\right)\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill F_{{\chi }_2}\left({\chi }_1\right)& :& \left[\varsigma ,\phi \right]\to {\mathbb{R}}_I^{+},F_{{\chi }_2}\left({\chi }_1\right)=𝐹\left({\chi }_1,{\chi }_2\right)\hfill \\ \hfill G_{{\chi }_2}\left({\chi }_1\right)& :& \left[\varsigma ,\phi \right]\to {\mathbb{R}}_I^{+},G_{{\chi }_2}\left({\chi }_1\right)=G\left({\chi }_1,{\chi }_2\right)\hfill \end{array}$$

are interval-valued functions on $\left[\varsigma ,\phi \right]$ and $\left[\varrho ,\vartheta \right]$, respectively, for all ${\chi }_1\in \left[\delta ,\phi \right]$ and ${\chi }_2\in \left[\varrho ,\vartheta \right].$

Now, from (21), we have:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{1}{\mu -\nu }}{\int }_{\varrho +\vartheta -\mu }^{\varrho +\vartheta -\nu }{F}_{{\chi }_1}\left({\chi }_2\right)G_{{\chi }_1}\left({\chi }_2\right)d{\chi }_2& \supseteq & {\displaystyle \frac{1}{3}}\left[F_{{\chi }_1}\left(\varrho +\vartheta -\mu \right)G_{{\chi }_1}\left(\varrho +\vartheta -\mu \right)+F_{{\chi }_1}\left(\varrho +\vartheta -\nu \right)G_{{\chi }_1}\left(\varrho +\vartheta -\nu \right)\right]\hfill \\ & & +{\displaystyle \frac{1}{6}}\left[F_{{\chi }_1}\left(\varrho +\vartheta -\mu \right)G_{{\chi }_1}\left(\varrho +\vartheta -\nu \right)+F_{{\chi }_1}\left(\varrho +\vartheta -\nu \right)G_{{\chi }_1}\left(\varrho +\vartheta -\mu \right)\right],\hfill \end{array}$$

that can be expressed as:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{1}{\mu -\nu }}{\int }_{\varrho +\vartheta -\mu }^{\varrho +\vartheta -\nu }𝐹\left({\chi }_1,{\chi }_2\right)G\left({\chi }_1,{\chi }_2\right)d{\chi }_2& \supseteq & {\displaystyle \frac{1}{3}}\left[𝐹\left({\chi }_1,\varrho +\vartheta -\mu \right)G\left({\chi }_1,\varrho +\vartheta -\mu \right)+𝐹\left({\chi }_1,\varrho +\vartheta -\nu \right)G\left({\chi }_1,\varrho +\vartheta -\nu \right)\right]\hfill \\ & & +{\displaystyle \frac{1}{6}}\left[𝐹\left({\chi }_1,\varrho +\vartheta -\mu \right)G\left({\chi }_1,\varrho +\vartheta -\nu \right)+𝐹\left({\chi }_1,\varrho +\vartheta -\nu \right)G\left({\chi }_1,\varrho +\vartheta -\mu \right)\right].\hfill \end{array}$$

When the aforementioned inequality is integrated with regard to ${\chi }_1$ and both sides are divided by $\left(\rho -\kappa \right),$ we obtain

$$\begin{array}{ccc}& & {\displaystyle \frac{1}{\left(\rho -\kappa \right)\left(\mu -\nu \right)}}{\int }_{\delta +\phi -\rho }^{\varsigma +\phi -\kappa }{\int }_{\varrho +\vartheta -\mu }^{\varrho +\vartheta -\nu }𝐹\left({\chi }_1,{\chi }_2\right)G\left({\chi }_1,{\chi }_2\right)d{\chi }_{2}d{\chi }_1\hfill \\ & \supseteq & {\displaystyle \frac{1}{3\left(\rho -\kappa \right)}}{\int }_{\varsigma +\phi -\rho }^{\varsigma +\phi -\kappa }\left[F_{{\chi }_1}\left(\varrho +\vartheta -\mu \right)G_{{\chi }_1}\left(\varrho +\vartheta -\mu \right)+F_{{\chi }_1}\left(\varrho +\vartheta -\nu \right)G_{{\chi }_1}\left(\varrho +\vartheta -\nu \right)\right]d{\chi }_1\hfill \\ & & +{\displaystyle \frac{1}{6\left(\rho -\kappa \right)}}{\int }_{\varsigma +\phi -\rho }^{\varsigma +\phi -\kappa }\left[𝐹\left({\chi }_1,\varrho +\vartheta -\mu \right)G\left({\chi }_1,\varrho +\vartheta -\nu \right)+𝐹\left({\chi }_1,\varrho +\vartheta -\nu \right)G\left({\chi }_1,\varrho +\vartheta -\mu \right)\right]d{\chi }_1.\hfill \end{array}$$

(38)

We apply (21) to each integral on the right side of (38), yielding

$$\begin{array}{ccc}& & {\displaystyle \frac{1}{\left(\rho -\kappa \right)}}{\int }_{\varsigma +\phi -\rho }^{\delta +\phi -\kappa }𝐹\left({\chi }_1,\varrho +\vartheta -\mu \right)G\left({\chi }_1,\varrho +\vartheta -\mu \right)d{\chi }_1\hfill \\ & \supseteq & {\displaystyle \frac{1}{3}}\left[𝐹\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\mu \right)G\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\mu \right)+𝐹\left(\delta +\phi -\kappa ,\varrho +\vartheta -\nu \right)G\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\nu \right)\right]\hfill \\ & & +{\displaystyle \frac{1}{6}}\left[𝐹\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\mu \right)G\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\nu \right)+𝐹\left(\delta +\phi -\kappa ,\varrho +\vartheta -\nu \right)G\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\mu \right)\right],\hfill \end{array}$$

(39)

$$\begin{array}{ccc}& & {\displaystyle \frac{1}{\left(\rho -\kappa \right)}}{\int }_{\varsigma +\phi -\rho }^{\delta +\phi -\kappa }𝐹\left({\chi }_1,\varrho +\vartheta -\nu \right)G\left({\chi }_1,\varrho +\vartheta -\nu \right)d{\chi }_1\hfill \\ & \supseteq & {\displaystyle \frac{1}{3}}\left[𝐹\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\nu \right)G\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\nu \right)+𝐹\left(\delta +\phi -\kappa ,\varrho +\vartheta -\nu \right)G\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\nu \right)\right]\hfill \\ & & +{\displaystyle \frac{1}{6}}\left[𝐹\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\nu \right)G\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\nu \right)+𝐹\left(\delta +\phi -\kappa ,\varrho +\vartheta -\nu \right)G\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\nu \right)\right],\hfill \end{array}$$

(40)

$$\begin{array}{ccc}& & {\displaystyle \frac{1}{\left(\rho -\kappa \right)}}{\int }_{\varsigma +\phi -\rho }^{\delta +\phi -\kappa }𝐹\left({\chi }_1,\varrho +\vartheta -\mu \right)G\left({\chi }_1,\varrho +\vartheta -\nu \right)d{\chi }_1\hfill \\ & \supseteq & {\displaystyle \frac{1}{3}}\left[𝐹\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\mu \right)G\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\nu \right)+𝐹\left(\delta +\phi -\kappa ,\varrho +\vartheta -\mu \right)G\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\nu \right)\right]\hfill \\ & & +{\displaystyle \frac{1}{6}}\left[𝐹\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\mu \right)G\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\nu \right)+𝐹\left(\delta +\phi -\kappa ,\varrho +\vartheta -\mu \right)G\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\nu \right)\right],\hfill \end{array}$$

(41)

$$\begin{array}{ccc}& & {\displaystyle \frac{1}{\left(\rho -\kappa \right)}}{\int }_{\varsigma +\phi -\rho }^{\delta +\phi -\kappa }𝐹\left({\chi }_1,\varrho +\vartheta -\nu \right)G\left({\chi }_1,\varrho +\vartheta -\mu \right)d{\chi }_1\hfill \\ & \supseteq & {\displaystyle \frac{1}{3}}\left[𝐹\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\nu \right)G\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\mu \right)+𝐹\left(\delta +\phi -\kappa ,\varrho +\vartheta -\nu \right)G\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu \right)\right]\hfill \\ & & +{\displaystyle \frac{1}{6}}\left[𝐹\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\nu \right)G\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu \right)+𝐹\left(\delta +\phi -\kappa ,\varrho +\vartheta -\nu \right)G\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\mu \right)\right].\hfill \end{array}$$

(42)

Substituting from (39)–(42) in (38), we get the desired result (37). Likewise, we can establish a similar inequality by utilizing $F_{{\chi }_2}\left({\chi }_1\right)G_{{\chi }_2}\left({\chi }_1\right)$ on $\left[\varsigma ,\phi \right].$ □

Remark 5.

Consider $\kappa =\varsigma ,\rho =\phi ,\mu =\varrho ,$ and $\nu =\vartheta$ in Theorem 6, we obtain the result which Zhao proved in [23].

Remark 6.

Upon substituting ${𝐹}^{-}={𝐹}^{+}$ in Theorem 6, we obtain the result proved by Toseef et al. in [36].

Remark 7.

We obtain the result demonstrated by Latif et al. in [22] by Substituting ${𝐹}^{-}={𝐹}^{+}$ and $\kappa =\varsigma ,\rho =\phi ,\mu =\varrho ,$ and $\nu =\vartheta$ in Theorem 6.

Theorem 7.

Let ϝ and G be two coordinated convex functions with interval values defined in the domain $▵=\left[\varsigma ,\phi \right]\times \left[\varrho ,\vartheta \right]$ with values in ${\mathbb{R}}_I^{+}$. Suppose that $\kappa ,\rho \in \left[\varsigma ,\phi \right]$ and $\mu ,\nu \in \left[\varrho ,\vartheta \right]$, and that for each τ, we have $𝐹\left(\tau \right)=\left[{𝐹}^{-}\left(\tau \right),{𝐹}^{+}\left(\tau \right)\right]$ and $G\left(\tau \right)=\left[G^{-}\left(\tau \right),G^{+}\left(\tau \right)\right]$. Then, the subsequent inequality holds:

$$\begin{array}{ccc}& & 4𝐹\left(\varsigma +\phi -{\displaystyle \frac{\kappa +\rho }{2}},\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)G\left(\varsigma +\phi -{\displaystyle \frac{\kappa +\rho }{2}},\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)\hfill \\ & \supseteq & {\displaystyle \frac{1}{\left(\rho -\kappa \right)\left(\mu -\nu \right)}}{\int }_{\varsigma +\phi -\rho }^{\varsigma +\phi -\kappa }{\int }_{\varrho +\vartheta -\mu }^{\varrho +\vartheta -\nu }𝐹\left({\chi }_1,{\chi }_2\right)G\left({\chi }_1,{\chi }_2\right)d{\chi }_{2}d{\chi }_1\hfill \\ & & +{\displaystyle \frac{5}{36}}P\left(\varsigma +\phi -\rho ,\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu ,\varrho +\vartheta -\nu \right)+{\displaystyle \frac{7}{36}}M\left(\varsigma +\phi -\rho ,\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu ,\varrho +\vartheta -\nu \right)\hfill \\ & & +{\displaystyle \frac{2}{9}}N\left(\varsigma +\phi -\rho ,\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu ,\varrho +\vartheta -\nu \right),\hfill \end{array}$$

(43)

where $P\left(\varsigma +\phi -\rho ,\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu ,\varrho +\vartheta -\nu \right),$ $M(\varsigma +\phi -\rho ,\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu ,$ $\varrho +\vartheta -\nu ),$ and $N\left(\varsigma +\phi -\rho ,\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu ,\varrho +\vartheta -\nu \right)$ are outlined in Theorem 6.

Proof. 

As $𝐹$ and G are interval-valued coordinated convex functions on $▵,$ hence

$$\begin{array}{ccc}& & 2𝐹\left(\varsigma +\phi -{\displaystyle \frac{\kappa +\rho }{2}},\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)G\left(\varsigma +\phi -{\displaystyle \frac{\kappa +\rho }{2}},\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)\hfill \\ & \supseteq & {\displaystyle \frac{1}{\left(\rho -\kappa \right)}}{\int }_{\varsigma +\phi -\rho }^{\varsigma +\phi -\kappa }𝐹\left({\chi }_1,\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)G\left({\chi }_1,\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)d{\chi }_1\hfill \\ & & +{\displaystyle \frac{1}{6}}\left[𝐹\left(\varsigma +\phi -\rho ,\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)G\left(\varsigma +\phi -\rho ,\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)\right.\hfill \\ & & \left.+𝐹\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)G\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)\right]\hfill \\ & & +{\displaystyle \frac{1}{3}}\left[𝐹\left(\varsigma +\phi -\rho ,\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)G\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)\right.\hfill \\ & & \left.+𝐹\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)G\left(\varsigma +\phi -\rho ,\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)\right]\hfill \end{array}$$

(44)

and

$$\begin{array}{ccc}& & 2𝐹\left(\varsigma +\phi -{\displaystyle \frac{\kappa +\rho }{2}},\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)G\left(\varsigma +\phi -{\displaystyle \frac{\kappa +\rho }{2}},\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)\hfill \\ & \supseteq & {\displaystyle \frac{1}{\left(\mu -\nu \right)}}{\int }_{\varrho +\vartheta -\mu }^{\varrho +\vartheta -\nu }𝐹\left(\varsigma +\phi -{\displaystyle \frac{\kappa +\rho }{2}},{\chi }_2\right)G\left(\varsigma +\phi -{\displaystyle \frac{\kappa +\rho }{2}},{\chi }_2\right)d{\chi }_2\hfill \\ & & +{\displaystyle \frac{1}{6}}\left[𝐹\left(\varsigma +\phi -{\displaystyle \frac{\kappa +\rho }{2}},\varrho +\vartheta -\mu \right)G\left(\varsigma +\phi -{\displaystyle \frac{\kappa +\rho }{2}},\varrho +\vartheta -\mu \right)\right.\hfill \\ & & \left.+𝐹\left(\varsigma +\phi -{\displaystyle \frac{\kappa +\rho }{2}},\varrho +\vartheta -\nu \right)G\left(\varsigma +\phi -{\displaystyle \frac{\kappa +\rho }{2}},\varrho +\vartheta -\nu \right)\right]\hfill \\ & & +{\displaystyle \frac{1}{3}}\left[𝐹\left(\varsigma +\phi -{\displaystyle \frac{\kappa +\rho }{2}},\varrho +\vartheta -\mu \right)G\left(\varsigma +\phi -{\displaystyle \frac{\kappa +\rho }{2}},\varrho +\vartheta -\nu \right)\right.\hfill \\ & & \left.+𝐹\left(\varsigma +\phi -{\displaystyle \frac{\kappa +\rho }{2}},\varrho +\vartheta -\nu \right)G\left(\varsigma +\phi -{\displaystyle \frac{\kappa +\rho }{2}},\varrho +\vartheta -\mu \right)\right].\hfill \end{array}$$

(45)

Combining (44) and (45), then scaling both sides of the outcome by 2, we obtain

$$\begin{array}{ccc}& & 8𝐹\left(\varsigma +\phi -{\displaystyle \frac{\kappa +\rho }{2}},\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)G\left(\varsigma +\phi -{\displaystyle \frac{\kappa +\rho }{2}},\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)\hfill \\ & \supseteq & {\displaystyle \frac{2}{\left(\rho -\kappa \right)}}{\int }_{\varsigma +\phi -\rho }^{\varsigma +\phi -\kappa }𝐹\left({\chi }_1,\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)G\left({\chi }_1,\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)d{\chi }_1\hfill \\ & & +{\displaystyle \frac{2}{\left(\mu -\nu \right)}}{\int }_{\varrho +\vartheta -\mu }^{\varrho +\vartheta -\nu }𝐹\left(\varsigma +\phi -{\displaystyle \frac{\kappa +\rho }{2}},{\chi }_2\right)G\left(\varsigma +\phi -{\displaystyle \frac{\kappa +\rho }{2}},{\chi }_2\right)d{\chi }_2\hfill \\ & & +{\displaystyle \frac{1}{6}}\left[2𝐹\left(\varsigma +\phi -\rho ,\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)G\left(\varsigma +\phi -\rho ,\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)\right.\hfill \\ & & \left.+2𝐹\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)G\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)\right]\hfill \\ & & +{\displaystyle \frac{1}{6}}\left[2𝐹\left(\varsigma +\phi -\rho ,\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)G\left(\varsigma +\phi -\rho ,\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)\right.\hfill \\ & & \left.+2𝐹\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)G\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)\right]\hfill \\ & & +{\displaystyle \frac{1}{3}}\left[2𝐹\left(\varsigma +\phi -\rho ,\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)G\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)\right.\hfill \\ & & \left.+𝐹\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)G\left(\varsigma +\phi -\rho ,\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)\right]\hfill \\ & & +{\displaystyle \frac{1}{3}}\left[𝐹\left(\varsigma +\phi -\rho ,\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)G\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)\right.\hfill \\ & & \left.+𝐹\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)G\left(\varsigma +\phi -\rho ,\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)\right].\hfill \end{array}$$

(46)

Now, from (22) we have:

$$\begin{array}{ccc}& & 2𝐹\left(\varsigma +\phi -\rho ,\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)G\left(\varsigma +\phi -\rho ,\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)\hfill \\ & \supseteq & {\displaystyle \frac{1}{\left(\mu -\nu \right)}}{\int }_{\varrho +\vartheta -\mu }^{\varrho +\vartheta -\nu }𝐹\left(\varsigma +\phi -\rho ,{\chi }_2\right)G\left(\varsigma +\phi -\rho ,{\chi }_2\right)d{\chi }_2\hfill \\ & & +{\displaystyle \frac{1}{6}}\left[𝐹\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\mu \right)G\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\mu \right)+𝐹\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\nu \right)G\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\nu \right)\right]\hfill \\ & & +{\displaystyle \frac{1}{3}}\left[𝐹\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\mu \right)G\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\nu \right)+𝐹\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\nu \right)G\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\mu \right)\right],\hfill \end{array}$$

(47)

$$\begin{array}{ccc}& & 2𝐹\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)G\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)\hfill \\ & \supseteq & {\displaystyle \frac{1}{\left(\mu -\nu \right)}}{\int }_{\varrho +\vartheta -\mu }^{\varrho +\vartheta -\nu }𝐹\left(\varsigma +\phi -\kappa ,{\chi }_2\right)G\left(\varsigma +\phi -\kappa ,{\chi }_2\right)d{\chi }_2\hfill \\ & & +{\displaystyle \frac{1}{6}}\left[𝐹\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu \right)G\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu \right)+𝐹\left(\delta +\phi -\kappa ,\varrho +\vartheta -\nu \right)G\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\nu \right)\right]\hfill \\ & & +{\displaystyle \frac{1}{3}}\left[𝐹\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu \right)G\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\nu \right)+𝐹\left(\delta +\phi -\kappa ,\varrho +\vartheta -\nu \right)G\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu \right)\right],\hfill \end{array}$$

(48)

$$\begin{array}{ccc}& & 2𝐹\left(\varsigma +\phi -{\displaystyle \frac{\rho +\kappa }{2}},\varrho +\vartheta -\mu \right)G\left(\varsigma +\phi -{\displaystyle \frac{\rho +\kappa }{2}},\varrho +\vartheta -\mu \right)\hfill \\ & \supseteq & {\displaystyle \frac{1}{\left(\rho -\kappa \right)}}{\int }_{\varsigma +\phi -\rho }^{\varsigma +\phi -\kappa }𝐹\left({\chi }_1,\varrho +\vartheta -\mu \right)G\left({\chi }_1,\varrho +\vartheta -\mu \right)d{\chi }_1\hfill \\ & & +{\displaystyle \frac{1}{6}}\left[𝐹\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\mu \right)G\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\mu \right)+𝐹\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu \right)G\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu \right)\right]\hfill \\ & & +{\displaystyle \frac{1}{3}}\left[𝐹\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu \right)G\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\mu \right)+𝐹\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu \right)G\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\mu \right)\right],\hfill \end{array}$$

(49)

$$\begin{array}{ccc}& & 2𝐹\left(\varsigma +\phi -{\displaystyle \frac{\rho +\kappa }{2}},\varrho +\vartheta -\nu \right)G\left(\varsigma +\phi -{\displaystyle \frac{\rho +\kappa }{2}},\varrho +\vartheta -\nu \right)\hfill \\ & \supseteq & {\displaystyle \frac{1}{\left(\rho -\kappa \right)}}{\int }_{\varsigma +\phi -\rho }^{\varsigma +\phi -\kappa }𝐹\left({\chi }_1,\varrho +\vartheta -\nu \right)G\left({\chi }_1,\varrho +\vartheta -\nu \right)d{\chi }_1\hfill \\ & & +{\displaystyle \frac{1}{6}}\left[𝐹\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\nu \right)G\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\nu \right)+𝐹\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\nu \right)G\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\nu \right)\right]\hfill \\ & & +{\displaystyle \frac{1}{3}}\left[𝐹\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\nu \right)G\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\nu \right)+𝐹\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\nu \right)G\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\nu \right)\right],\hfill \end{array}$$

(50)

$$\begin{array}{ccc}& & 2𝐹\left(\varsigma +\phi -\rho ,\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)G\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)\hfill \\ & \supseteq & {\displaystyle \frac{1}{\left(\mu -\nu \right)}}{\int }_{\varrho +\vartheta -\mu }^{\varrho +\vartheta -\nu }𝐹\left(\varsigma +\phi -\rho ,{\chi }_2\right)G\left(\varsigma +\phi -\kappa ,{\chi }_2\right)d{\chi }_2\hfill \\ & & +{\displaystyle \frac{1}{6}}\left[𝐹\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\mu \right)G\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu \right)+𝐹\left(\delta +\phi -\rho ,\varrho +\vartheta -\nu \right)G\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\nu \right)\right]\hfill \\ & & +{\displaystyle \frac{1}{3}}\left[𝐹\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\mu \right)G\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\nu \right)+𝐹\left(\delta +\phi -\rho ,\varrho +\vartheta -\nu \right)G\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu \right)\right],\hfill \end{array}$$

(51)

$$\begin{array}{ccc}& & 2𝐹\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)G\left(\varsigma +\phi -\rho ,\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)\hfill \\ & \supseteq & {\displaystyle \frac{1}{\left(\mu -\nu \right)}}{\int }_{\varrho +\vartheta -\mu }^{\varrho +\vartheta -\nu }𝐹\left(\varsigma +\phi -\kappa ,{\chi }_2\right)G\left(\varsigma +\phi -\rho ,{\chi }_2\right)d{\chi }_2\hfill \\ & & +{\displaystyle \frac{1}{6}}\left[𝐹\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu \right)G\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\mu \right)+𝐹\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\nu \right)G\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\nu \right)\right]\hfill \\ & & +{\displaystyle \frac{1}{3}}\left[𝐹\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu \right)G\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\nu \right)+𝐹\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\nu \right)G\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\mu \right)\right],\hfill \end{array}$$

(52)

$$\begin{array}{ccc}& & 2𝐹\left(\varsigma +\phi -{\displaystyle \frac{\rho +\kappa }{2}},\varrho +\vartheta -\mu \right)G\left(\varsigma +\phi -{\displaystyle \frac{\rho +\kappa }{2}},\varrho +\vartheta -\nu \right)\hfill \\ & \supseteq & {\displaystyle \frac{1}{\left(\rho -\kappa \right)}}{\int }_{\varsigma +\phi -\rho }^{\varsigma +\phi -\kappa }𝐹\left({\chi }_1,\varrho +\vartheta -\mu \right)G\left({\chi }_1,\varrho +\vartheta -\nu \right)d{\chi }_1\hfill \\ & & +{\displaystyle \frac{1}{6}}\left[𝐹\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\mu \right)G\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\nu \right)+𝐹\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu \right)G\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\nu \right)\right]\hfill \\ & & +{\displaystyle \frac{1}{3}}\left[𝐹\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\mu \right)G\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\nu \right)+𝐹\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu \right)G\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\nu \right)\right],\hfill \end{array}$$

(53)

$$\begin{array}{ccc}& & 2𝐹\left(\varsigma +\phi -{\displaystyle \frac{\rho +\kappa }{2}},\varrho +\vartheta -\nu \right)G\left(\varsigma +\phi -{\displaystyle \frac{\rho +\kappa }{2}},\varrho +\vartheta -\mu \right)\hfill \\ & \supseteq & {\displaystyle \frac{1}{\left(\rho -\kappa \right)}}{\int }_{\varsigma +\phi -\rho }^{\varsigma +\phi -\kappa }𝐹\left({\chi }_1,\varrho +\vartheta -\nu \right)G\left({\chi }_1,\varrho +\vartheta -\mu \right)d{\chi }_1\hfill \\ & & +{\displaystyle \frac{1}{6}}\left[𝐹\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\nu \right)G\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\mu \right)+𝐹\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\nu \right)G\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu \right)\right]\hfill \\ & & +{\displaystyle \frac{1}{3}}\left[𝐹\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\nu \right)G\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu \right)+𝐹\left(\delta +\phi -\kappa ,\varrho +\vartheta -\nu \right)G\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\mu \right)\right].\hfill \end{array}$$

(54)

By using (47)–(54), put in (46), we have:

$$\begin{array}{ccc}& & 8𝐹\left(\varsigma +\phi -{\displaystyle \frac{\kappa +\rho }{2}},\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)G\left(\varsigma +\phi -{\displaystyle \frac{\kappa +\rho }{2}},\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)\hfill \\ & \supseteq & {\displaystyle \frac{1}{6\left(\mu -\nu \right)}}{\int }_{\varrho +\vartheta -\mu }^{\varrho +\vartheta -\nu }𝐹\left(\varsigma +\phi -\rho ,{\chi }_2\right)G\left(\varsigma +\phi -\rho ,{\chi }_2\right)d{\chi }_2+{\displaystyle \frac{1}{6\left(\rho -\kappa \right)}}{\int }_{\varsigma +\phi -\rho }^{\varsigma +\phi -\kappa }𝐹\left({\chi }_1,\varrho +\vartheta -\mu \right)G\left({\chi }_1,\varrho +\vartheta -\mu \right)d{\chi }_1\hfill \\ & & +{\displaystyle \frac{1}{6\left(\mu -\nu \right)}}{\int }_{\varrho +\vartheta -\mu }^{\varrho +\vartheta -\nu }𝐹\left(\varsigma +\phi -\kappa ,{\chi }_2\right)G\left(\delta +\phi -\kappa ,{\chi }_2\right)d{\chi }_2+{\displaystyle \frac{1}{6\left(\rho -\kappa \right)}}{\int }_{\varsigma +\phi -\rho }^{\varsigma +\phi -\kappa }𝐹\left({\chi }_1,\varrho +\vartheta -\nu \right)G\left({\chi }_1,\varrho +\vartheta -\nu \right)d{\chi }_1\hfill \\ & & +{\displaystyle \frac{1}{3\left(\mu -\nu \right)}}{\int }_{\varrho +\vartheta -\mu }^{\varrho +\vartheta -\nu }𝐹\left(\varsigma +\phi -\rho ,{\chi }_2\right)G\left(\varsigma +\phi -\kappa ,{\chi }_2\right)d{\chi }_2+{\displaystyle \frac{1}{3\left(\mu -\nu \right)}}{\int }_{\varrho +\vartheta -\mu }^{\varrho +\vartheta -\nu }𝐹\left(\varsigma +\phi -\kappa ,{\chi }_2\right)G\left(\varsigma +\phi -\rho ,{\chi }_2\right)d{\chi }_2\hfill \\ & & +{\displaystyle \frac{1}{3\left(\rho -\kappa \right)}}{\int }_{\varsigma +\phi -\rho }^{\varsigma +\phi -\kappa }𝐹\left({\chi }_1,\varrho +\vartheta -\mu \right)G\left({\chi }_1,\varrho +\vartheta -\nu \right)d{\chi }_1+{\displaystyle \frac{1}{3\left(\rho -\kappa \right)}}{\int }_{\varsigma +\phi -\rho }^{\varsigma +\phi -\kappa }𝐹\left({\chi }_1,\varrho +\vartheta -\nu \right)G\left({\chi }_1,\varrho +\vartheta -\mu \right)d{\chi }_1\hfill \\ & & +{\displaystyle \frac{1}{18}}P\left(\varsigma +\phi -\rho ,\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu ,\varrho +\vartheta -\nu \right)+{\displaystyle \frac{1}{9}}M\left(\varsigma +\phi -\rho ,\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu ,\varrho +\vartheta -\nu \right)\hfill \\ & & +{\displaystyle \frac{2}{9}}N\left(\varsigma +\phi -\rho ,\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu ,\varrho +\vartheta -\nu \right).\hfill \end{array}$$

(55)

Utilizing (22) another time, we obtain the following relationship:

$$\begin{array}{ccc}& & {\displaystyle \frac{2}{\left(\mu -\nu \right)}}{\int }_{\varrho +\vartheta -\mu }^{\varrho +\vartheta -\nu }𝐹\left(\varsigma +\phi -{\displaystyle \frac{\kappa +\rho }{2}},{\chi }_2\right)G\left(\varsigma +\phi -{\displaystyle \frac{\kappa +\rho }{2}},{\chi }_2\right)d{\chi }_2\hfill \\ & \supseteq & {\displaystyle \frac{1}{\left(\rho -\kappa \right)\left(\mu -\nu \right)}}{\int }_{\varsigma +\phi -\rho }^{\varsigma +\phi -\kappa }{\int }_{\varrho +\vartheta -\mu }^{\varrho +\vartheta -\nu }𝐹\left({\chi }_1,{\chi }_2\right)G\left({\chi }_1,{\chi }_2\right)d{\chi }_{2}d{\chi }_1+{\displaystyle \frac{1}{6\left(\mu -\nu \right)}}{\int }_{\varrho +\vartheta -\mu }^{\varrho +\vartheta -\nu }𝐹\left(\varsigma +\phi -\rho ,{\chi }_2\right)G\left(\varsigma +\phi -\rho ,{\chi }_2\right)d{\chi }_2\hfill \\ & & +{\displaystyle \frac{1}{6\left(\mu -\nu \right)}}{\int }_{\varrho +\vartheta -\mu }^{\varrho +\vartheta -\nu }𝐹\left(\varsigma +\phi -\kappa ,{\chi }_2\right)G\left(\delta +\phi -\kappa ,{\chi }_2\right)d{\chi }_2+{\displaystyle \frac{1}{3\left(\mu -\nu \right)}}{\int }_{\varrho +\vartheta -\mu }^{\varrho +\vartheta -\nu }𝐹\left(\varsigma +\phi -\rho ,{\chi }_2\right)G\left(\varsigma +\phi -\kappa ,{\chi }_2\right)d{\chi }_2\hfill \\ & & +{\displaystyle \frac{1}{3\left(\mu -\nu \right)}}{\int }_{\varrho +\vartheta -\mu }^{\varrho +\vartheta -\nu }𝐹\left(\varsigma +\phi -\kappa ,{\chi }_2\right)G\left(\delta +\phi -\rho ,{\chi }_2\right)d{\chi }_2\hfill \end{array}$$

(56)

$$\begin{array}{ccc}& & {\displaystyle \frac{2}{\left(\rho -\kappa \right)}}{\int }_{\varsigma +\phi -\rho }^{\delta +\phi -\kappa }𝐹\left({\chi }_1,\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)G\left({\chi }_1,\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)d{\chi }_1\hfill \\ & \supseteq & {\displaystyle \frac{1}{\left(\rho -\kappa \right)\left(\mu -\nu \right)}}{\int }_{\varsigma +\phi -\rho }^{\varsigma +\phi -\kappa }{\int }_{\varrho +\vartheta -\mu }^{\varrho +\vartheta -\nu }𝐹\left({\chi }_1,{\chi }_2\right)G\left({\chi }_1,{\chi }_2\right)d{\chi }_{2}d{\chi }_1+{\displaystyle \frac{1}{6\left(\rho -\kappa \right)}}{\int }_{\varsigma +\phi -\rho }^{\varsigma +\phi -\kappa }𝐹\left({\chi }_1,\varrho +\vartheta -\mu \right)G\left({\chi }_1,\varrho +\vartheta -\mu \right)d{\chi }_1\hfill \\ & & +{\displaystyle \frac{1}{6\left(\rho -\kappa \right)}}{\int }_{\varsigma +\phi -\rho }^{\varsigma +\phi -\kappa }𝐹\left({\chi }_1,\varrho +\vartheta -\nu \right)G\left({\chi }_1,\varrho +\vartheta -\nu \right)d{\chi }_1++{\displaystyle \frac{1}{3\left(\rho -\kappa \right)}}{\int }_{\varsigma +\phi -\rho }^{\varsigma +\phi -\kappa }𝐹\left({\chi }_1,\varrho +\vartheta -\mu \right)G\left({\chi }_1,\varrho +\vartheta -\nu \right)d{\chi }_1\hfill \\ & & +{\displaystyle \frac{1}{6\left(\rho -\kappa \right)}}{\int }_{\varsigma +\phi -\rho }^{\varsigma +\phi -\kappa }𝐹\left({\chi }_1,\varrho +\vartheta -\nu \right)G\left({\chi }_1,\varrho +\vartheta -\mu \right)d{\chi }_1.\hfill \end{array}$$

(57)

Using (56) and (57) in (55), we have

$$\begin{array}{ccc}& & 8𝐹\left(\varsigma +\phi -{\displaystyle \frac{\kappa +\rho }{2}},\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)G\left(\varsigma +\phi -{\displaystyle \frac{\kappa +\rho }{2}},\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)\hfill \\ & \supseteq & {\displaystyle \frac{2}{\left(\rho -\kappa \right)\left(\mu -\nu \right)}}{\int }_{\varsigma +\phi -\rho }^{\varsigma +\phi -\kappa }{\int }_{\varrho +\vartheta -\mu }^{\varrho +\vartheta -\nu }𝐹\left({\chi }_1,{\chi }_2\right)G\left({\chi }_1,{\chi }_2\right)d{\chi }_{2}d{\chi }_1\hfill \\ & & +{\displaystyle \frac{1}{3\left(\mu -\nu \right)}}{\int }_{\varrho +\vartheta -\mu }^{\varrho +\vartheta -\nu }𝐹\left(\varsigma +\phi -\rho ,{\chi }_2\right)G\left(\varsigma +\phi -\rho ,{\chi }_2\right)d{\chi }_2+{\displaystyle \frac{1}{3\left(\mu -\nu \right)}}{\int }_{\varrho +\vartheta -\mu }^{\varrho +\vartheta -\nu }𝐹\left(\varsigma +\phi -\kappa ,{\chi }_2\right)G\left(\varsigma +\phi -\kappa ,{\chi }_2\right)d{\chi }_2\hfill \\ & & +{\displaystyle \frac{1}{3\left(\rho -\kappa \right)}}{\int }_{\varsigma +\phi -\rho }^{\varsigma +\phi -\kappa }𝐹\left({\chi }_1,\varrho +\vartheta -\mu \right)G\left({\chi }_1,\varrho +\vartheta -\mu \right)d{\chi }_1+{\displaystyle \frac{1}{3\left(\rho -\kappa \right)}}{\int }_{\varsigma +\phi -\rho }^{\varsigma +\phi -\kappa }𝐹\left({\chi }_1,\varrho +\vartheta -\nu \right)G\left({\chi }_1,\varrho +\vartheta -\nu \right)d{\chi }_1\hfill \\ & & +{\displaystyle \frac{2}{3\left(\mu -\nu \right)}}{\int }_{\varrho +\vartheta -\mu }^{\varrho +\vartheta -\nu }𝐹\left(\varsigma +\phi -\rho ,{\chi }_2\right)G\left(\varsigma +\phi -\kappa ,{\chi }_2\right)d{\chi }_2+{\displaystyle \frac{2}{3\left(\mu -\nu \right)}}{\int }_{\varrho +\vartheta -\mu }^{\varrho +\vartheta -\nu }𝐹\left(\varsigma +\phi -\kappa ,{\chi }_2\right)G\left(\varsigma +\phi -\rho ,{\chi }_2\right)d{\chi }_2\hfill \\ & & +{\displaystyle \frac{2}{3\left(\rho -\kappa \right)}}{\int }_{\varsigma +\phi -\rho }^{\varsigma +\phi -\kappa }𝐹\left({\chi }_1,\varrho +\vartheta -\mu \right)G\left({\chi }_1,\varrho +\vartheta -\nu \right)d{\chi }_1+{\displaystyle \frac{2}{3\left(\rho -\kappa \right)}}{\int }_{\varsigma +\phi -\rho }^{\varsigma +\phi -\kappa }𝐹\left({\chi }_1,\varrho +\vartheta -\nu \right)G\left({\chi }_1,\varrho +\vartheta -\mu \right)d{\chi }_1\hfill \\ & & +{\displaystyle \frac{1}{18}}P\left(\varsigma +\phi -\rho ,\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu ,\varrho +\vartheta -\nu \right)+{\displaystyle \frac{1}{9}}M\left(\varsigma +\phi -\rho ,\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu ,\varrho +\vartheta -\nu \right)\hfill \\ & & +{\displaystyle \frac{2}{9}}N\left(\varsigma +\phi -\rho ,\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu ,\varrho +\vartheta -\nu \right).\hfill \end{array}$$

(58)

Our intended outcome is obtained by applying (21) to each integral in (58). □

Remark 8.

Upon setting $\kappa =\varsigma ,\rho =\phi ,\mu =\varrho ,$ and $\nu =\vartheta$ in Theorem 7, we obtain result that was demonstrated by Zhao in [23].

Remark 9.

Substituting ${𝐹}^{-}={𝐹}^{+}$ in Theorem 7, we obtain the result given by Toseef et al. in [36].

Remark 10.

By equating ${𝐹}^{-}$ with ${𝐹}^{+}$ and assigning the values $\kappa =\varsigma ,\rho =\phi ,\mu =\varrho ,$ and $\nu =\vartheta$ in Theorem 7, we obtain the result proved by Latif et al. in [22].

4. Numerical Examples

In this section, we provide numerical examples to illustrate and verify the validity of the newly established results. These examples demonstrate the practical applicability of the theoretical findings and offer insight into the behavior of interval-valued coordinated convex functions under the proposed inequalities.

Example 1.

Consider $𝐹\left({\chi }_1,{\chi }_2\right)=\left[{\chi }_1{\chi }_2,\left(6-e^{{\chi }_1}\right)\left(6-e^{{\chi }_2}\right)\right]$ is an interval-valued coordinated convex functions on $△=\left[\varsigma ,\phi \right]\times \left[\varrho ,\vartheta \right]=\left[-1,2\right]\times \left[-1,2\right],$ and $\kappa =0,\rho =1,\nu =0,$ and $\mu =1.$ Then,

$$\begin{array}{cc}\hfill 𝐹\left(\varsigma +\phi -{\displaystyle \frac{\kappa +\rho }{2}},\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)& =𝐹\left({\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}\right)\hfill \\ \hfill & =\left[{\displaystyle \frac{1}{4}},{\left(6-e^{{\displaystyle \frac{1}{2}}}\right)}^2\right],\hfill \end{array}$$

(59)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2}}\left[{\displaystyle \frac{1}{\rho -\kappa }}{\int }_{\varsigma +\phi -\rho }^{\varsigma +\phi -\kappa }𝐹\left({\chi }_1,\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)d{\chi }_1+{\displaystyle \frac{1}{\mu -\nu }}{\int }_{\varrho +\vartheta -\mu }^{\varrho +\vartheta -\nu }𝐹\left(\varsigma +\phi -{\displaystyle \frac{\kappa +\rho }{2}},{\chi }_2\right)d{\chi }_2\right]\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}\left[{\int }_0^1𝐹\left({\chi }_1,{\displaystyle \frac{1}{2}}\right)d{\chi }_1+{\int }_0^1𝐹\left({\displaystyle \frac{1}{2}},{\chi }_2\right)d{\chi }_2\right]\hfill \\ \hfill & =\left[{\displaystyle \frac{1}{4}},\left(7-e\right)\left(6-e^{{\displaystyle \frac{1}{2}}}\right)\right],\hfill \end{array}$$

(60)

$$\begin{array}{ccc}& & {\displaystyle \frac{1}{\left(\rho -\kappa \right)\left(\mu -\nu \right)}}{\int }_{\delta +\phi -\rho }^{\varsigma +\phi -\kappa }{\int }_{\varrho +\vartheta -\mu }^{\varrho +\vartheta -\nu }𝐹\left({\chi }_1,{\chi }_2\right)d{\chi }_{2}d{\chi }_1={\int }_0^1{\int }_0^1𝐹\left({\chi }_1,{\chi }_2\right)d{\chi }_{2}d{\chi }_1\hfill \\ & =& \left[{\displaystyle \frac{1}{4}},{\left(7-e\right)}^2\right],\hfill \end{array}$$

(61)

and

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{4}}\left[𝐹\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\mu \right)+𝐹\left(\varsigma +\phi -\rho ,\varrho +\vartheta -\nu \right)\right.\hfill \\ \hfill & \left.+𝐹\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu \right)+𝐹\left(\varsigma +\phi -\kappa ,\varrho +\vartheta -\nu \right)\right]\hfill \\ \hfill & ={\displaystyle \frac{𝐹\left(0,0\right)+𝐹\left(0,1\right)+𝐹\left(1,0\right)+𝐹\left(1,1\right)}{4}}\hfill \\ \hfill & =\left[{\displaystyle \frac{1}{4}},{\displaystyle \frac{\left(6-e\right)\left(16-e\right)+25}{4}}\right],\hfill \end{array}$$

(62)

Theorem 5 is thus validated by Example 1.

Example 2.

Assume $𝐹\left({\chi }_1,{\chi }_2\right)=\left[{\chi }_1,\left(6-e^{{\chi }_1}\right)\right]$ and $G\left({\chi }_1,{\chi }_2\right)=\left[{\chi }_2,\left(6-e^{{\chi }_2}\right)\right]$ are two interval-valued coordinated convex functions on $△=\left[\varsigma ,\phi \right]\times \left[\varrho ,\vartheta \right]=\left[-1,2\right]\times \left[-1,2\right],$ and $\kappa =0,\rho =1,\nu =0,$ and $\mu =1.$ Then,

$$\begin{array}{ccc}& & {\displaystyle \frac{1}{\left(\rho -\kappa \right)\left(\mu -\nu \right)}}{\int }_{\delta +\phi -\rho }^{\varsigma +\phi -\kappa }{\int }_{\varrho +\vartheta -\mu }^{\varrho +\vartheta -\nu }𝐹\left({\chi }_1,{\chi }_2\right)G\left({\chi }_1,{\chi }_2\right)d{\chi }_{2}d{\chi }_1\hfill \\ & =& {\int }_0^1{\int }_0^1\left[{\chi }_1,\left(6-e^{{\chi }_1}\right)\right]\left[{\chi }_2,\left(6-e^{{\chi }_2}\right)\right]d{\chi }_{2}d{\chi }_1\hfill \\ & =& {\int }_0^1{\int }_0^1\left[{\chi }_1{\chi }_2,\left(6-e^{{\chi }_1}\right)\left(6-e^{{\chi }_2}\right)\right]d{\chi }_{2}d{\chi }_1\hfill \\ & =& \left[{\displaystyle \frac{1}{4}},{\left(7-e\right)}^2\right]=\left[0.25,18.33\right],\hfill \end{array}$$

(63)

$$\begin{array}{ccc}& & {\displaystyle \frac{1}{9}}P\left(\varsigma +\phi -\rho ,\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu ,\varrho +\vartheta -\nu \right)+{\displaystyle \frac{1}{18}}M\left(\varsigma +\phi -\rho ,\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu ,\varrho +\vartheta -\nu \right)\hfill \\ & & +{\displaystyle \frac{1}{36}}N\left(\varsigma +\phi -\rho ,\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu ,\varrho +\vartheta -\nu \right)\hfill \\ & =& {\displaystyle \frac{1}{9}}\left[1,68.56\right]+{\displaystyle \frac{1}{18}}\left[4,137.12\right]+{\displaystyle \frac{1}{36}}\left[2,59.96\right]\hfill \\ & =& \left[0.39,16.90\right].\hfill \end{array}$$

(64)

Theorem 6 has been demonstrated by (63) and (64).

Example 3.

Assume $𝐹\left({\chi }_1,{\chi }_2\right)=\left[{\chi }_1,\left(6-e^{{\chi }_1}\right)\right]$ and $G\left({\chi }_1,{\chi }_2\right)=\left[{\chi }_2,\left(6-e^{{\chi }_2}\right)\right]$ are two interval-valued coordinated convex functions on $△=\left[\varsigma ,\phi \right]\times \left[\varrho ,\vartheta \right]=\left[-1,2\right]\times \left[-1,2\right],$ and $\kappa =0,\rho =1,\nu =0,$ and $\mu =1.$ Then,

$$\begin{array}{ccc}& & 4𝐹\left(\varsigma +\phi -{\displaystyle \frac{\kappa +\rho }{2}},\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)G\left(\varsigma +\phi -{\displaystyle \frac{\kappa +\rho }{2}},\varrho +\vartheta -{\displaystyle \frac{\mu +\nu }{2}}\right)\hfill \\ & =& 4𝐹\left({\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}\right)G\left({\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}\right)=4\left[{\displaystyle \frac{1}{2}},4.35\right]\left[{\displaystyle \frac{1}{2}},4.35\right]\hfill \\ & =& \left[1,75.69\right],\hfill \end{array}$$

(65)

$$\begin{array}{ccc}& & {\displaystyle \frac{1}{\left(\rho -\kappa \right)\left(\mu -\nu \right)}}{\int }_{\delta +\phi -\rho }^{\varsigma +\phi -\kappa }{\int }_{\varrho +\vartheta -\mu }^{\varrho +\vartheta -\nu }𝐹\left({\chi }_1,{\chi }_2\right)G\left({\chi }_1,{\chi }_2\right)d{\chi }_{2}d{\chi }_1\hfill \\ & & +{\displaystyle \frac{5}{36}}P\left(\varsigma +\phi -\rho ,\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu ,\varrho +\vartheta -\nu \right)+{\displaystyle \frac{7}{36}}M\left(\varsigma +\phi -\rho ,\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu ,\varrho +\vartheta -\nu \right)\hfill \\ & & +{\displaystyle \frac{2}{9}}N\left(\varsigma +\phi -\rho ,\varsigma +\phi -\kappa ,\varrho +\vartheta -\mu ,\varrho +\vartheta -\nu \right)\hfill \\ & =& {\int }_0^1{\int }_0^1\left[{\chi }_1{\chi }_2,\left(6-e^{{\chi }_1}\right)\left(6-e^{{\chi }_2}\right)\right]d{\chi }_{2}d{\chi }_1+{\displaystyle \frac{5}{36}}\left[1,68.56\right]+{\displaystyle \frac{7}{36}}\left[4,137.12\right]+{\displaystyle \frac{2}{9}}\left[2,59.96\right]\hfill \\ & =& \left[0.25,18.33\right]+\left[1.36,49.50\right]\hfill \\ & =& \left[1.61,67.83\right].\hfill \end{array}$$

(66)

By (65) and (66), Theorem 7 is verified.

Remark 11.

Upon setting $\kappa =\varsigma ,\rho =\phi ,\mu =\varrho ,$ and $\nu =\vartheta$ in the above Examples 1–3, we obtain the Examples that were given by Zhao in [23]. Mercer gives better results than those in the simple version.

5. Concluding Remarks and Future Direction

In conclusion, this research successfully addresses the significant challenge of establishing the Jensen–Mercer inequality for interval-valued coordinated convex functions by using the generalized difference operation (${\ominus }_g$). We introduced a novel approach leading to the development of H.H. Mercer-type inequalities specifically for this class of functions. This new inequality fundamentally generalizes the classical H.H. inequality for interval-valued coordinated convex functions, as the latter emerges directly as a special case when considering the endpoints of our broader Mercer-type result. Computational validation rigorously confirms the accuracy and robustness of new findings against recent literature. Crucially, the proposed inequalities provide highly effective constraints for integrals involving these functions, significantly extending their applicability beyond previously established classes. These advancements hold considerable promise for enhancing precision in modeling, optimization, and analysis across diverse applied fields such as economics, engineering, and physics, where interval-valued functions naturally arise to capture uncertainty.

The results presented in this work open several promising avenues for further research in the field of interval-valued analysis and convex inequalities. Future studies could explore the following directions:

• Investigate Jensen–Mercer and H.H. Mercer inequalities for other generalized convex functions, such as quasi-convex, h-convex, or Godunova-Levin interval-valued functions.
• Study coordinated versions of other convexity classes (e.g., s-convex, strongly convex, or logarithmically convex interval-valued functions).
• Incorporate fractional calculus operators (e.g., Riemann–Liouville or Caputo fractional integrals) into interval-valued Mercer-type inequalities.
• Explore the use of alternative generalized difference operations (beyond ${\ominus }_g$) to derive sharper bounds.