Some (p, q)-Estimates of Hermite-Hadamard-Type Inequalities for Coordinated Convex and Quasi- Convex Functions

Abstract

In this paper, we present the preliminaries of $(p,q)$-calculus for functions of two variables. Furthermore, we prove some new Hermite-Hadamard integral-type inequalities for convex functions on coordinates over $[a,b]\times [c,d]$ by using the $(p,q)$-calculus of the functions of two variables. Furthermore, we establish an identity for the right-hand side of the Hermite-Hadamard-type inequalities on coordinates that is proven by using the $(p,q)$-calculus of the functions of two variables. Finally, we use the new identity to prove some trapezoidal-type inequalities with the assumptions of convexity and quasi-convexity on coordinates of the absolute values of the partial derivatives defined in the $(p,q)$-calculus of the functions of two variables.

1. Introduction

In mathematics, q-calculus, also known as quantum calculus, is the study of calculus with no limits. In quantum calculus, we obtain q-analogues of mathematical objects that can be recaptured as $q\to 1^{-}$. The concept was presented by renowned mathematician Euler in the Eighteenth Century, introducing the number in q-infinite series defined by Newton. Later on, Jackson [1] introduced the concept of q-definite integrals extending the concept of q-calculus. Quantum calculus has numerous applications in various fields of mathematics and physics, especially in orthogonal polynomials, number theory, hypergeometric functions, mechanics, and relativity theory. For the background study of quantum calculus theory and the theory of inequalities in quantum calculus, interested, readers are referred to [2,3,4].

The following double inequality is well known in the literature as Hermit-Hadamard’s inequality. Let $h:J\subset \mathbb{R}\to \mathbb{R}$ be convex on $[a,b]$, where $a,b\in J$ with $a<b$:

$$h\left(\frac{a+b}{2}\right)\le \frac{1}{b-a}{\int }_a^{b}h\left(t\right)dt\le \frac{h\left(a\right)+h\left(b\right)}{2}.$$

(1)

Due to its role in various fields of modern mathematics, such as numerical analysis and functional and mathematical analyses, the validity of Hermit-Hadamard’s inequality is important; see [5,6,7].

The concept of q-derivatives over the definite interval $\left[a,b\right]\subset \mathbb{R}$ was attended by Tariboon et al. [8,9] and addressed numerous problems on quantum analogues such as the q-Hölder inequality, the q-Ostrowski inequality, the q-Cauchy-Schwarz inequality, the q-Grüss-Čebyšev integral inequality, the q-Grüss inequality, and other integral inequalities by classical convexity. Most recently, Alp et al. [10] proved the q-Hermite-Hadamard inequality and then acquired the generalized q-Hermite-Hadamard inequality; also, they studied some integral inequalities, which provide quantum estimates for the left part of the quantum analogue of the q-Hermite-Hadamard inequality through q-differentiable convex and quasi-convex functions, for more details and interesting applications see References [11,12,13,14,15,16]. A recent development in the context of the above concept was presented by Tunç and Göv [17,18,19], named as $(p,q)$-derivatives and $(p,q)$-integrals over $[a,b]\subset \mathbb{R}$. Some well-known results that depend on $(p,q)$-calculus are the $(p,q)$-Minkowski inequality, the $(p,q)$-Hölder inequality, the $(p,q)$-Ostrowski inequality, the $(p,q)$-Cauchy-Schwarz inequality, the $(p,q)$-Grüss-Čebyšev integral inequality, the $(p,q)$-Grüss inequality, and other integral inequalities by classical convexity. Dragomir et al. [20] studied some integral inequalities that provide $(p,q)$ estimates for the right part of the quantum analogue of the Hermit-Hadamard inequality through $(p,q)$- differentiable convex and quasi-convex functions.

Dragomir [20] proved the following inequalities of the Hermite-Hadamard type for coordinated convex functions on a rectangle from the plane ${\mathbb{R}}^2$.

Theorem 1.

Let $h:▵=[a,b]\times [c,d]\subset {\mathbb{R}}^2\to \mathbb{R}$ be convex on the coordinates on $▵$ with $a<b$ and $c<d$. Then, one has the inequalities:

$$\begin{array}{c}h\left(\frac{a+b}{2},\frac{c+d}{2}\right)\le \frac{1}{2}\left[\frac{1}{b-a}{\int }_a^{b}h\left(s,\frac{c+d}{2}\right)ds+\frac{1}{c-d}{\int }_c^{d}h\left(\frac{a+b}{2},t\right)dt\right]\hfill \\ \le \frac{1}{(b-a)(d-c)}{\int }_a^b{\int }_c^{d}h(s,t)dsdt\le \frac{1}{4}[\frac{1}{b-a}\left({\int }_a^{b}h(s,c)ds+{\int }_a^{b}h(s,d)ds\right)\hfill \\ +\frac{1}{d-c}\left({\int }_c^{d}h(a,t)dt+{\int }_c^{d}h(b,t)dt\right)]\le \frac{h(a,c)+h(b,c)+h(a,d)+h(b,d)}{4}.\hfill \end{array}$$

Kunt et al. [21] proved the following $(p,q)$-Hermite-Hadamard inequalities for convex functions by the quantum integral approach:

Theorem 2.

Let $0<q<p\le 1$ and h: $[a,b]\to \mathbb{R}$ be a convex differentiable function on $[a,b]$. Then, the following integral inequality holds:

$$h\left(\frac{qa+pb}{p+q}\right)\le \frac{1}{p(b-a)}{\int }_a^{bp+(1-p)a}h\left(s\right){}_a{d}_{p,q}s\le \frac{qh\left(a\right)+ph\left(b\right)}{p+q}.$$

(2)

Latif et al. [22] for the first time introduced the idea of the $q_1{q}_2$-Hermite-Hadamard inequality and obtained the $q_1{q}_2$-Hermite-Hadamard-type inequalities for coordinated convex functions on the quantum integral. Furthermore, Alp et al. [23], proved the corrected $q_1{q}_2$-Hermite-Hadamard-type inequalities for coordinated convex functions by using quantum calculus.

Theorem 3.

Suppose that $h:▵=[a,b]\times [c,d]\subset {\mathbb{R}}^2\to \mathbb{R}$ is coordinated convex on $▵$; the following inequalities hold for all $q_1,q_2\in (0,1)$:

$$\begin{array}{c}h\left(\frac{a{q}_1+b}{1+q_1},\frac{c{q}_2+d}{1+q_2}\right)\le \frac{1}{2}\left[\frac{1}{b-a}{\int }_a^{b}h\left(z,\frac{c{q}_2+d}{1+q_2}\right){}_a{d}_{q_1}z+\frac{1}{d-c}{\int }_c^{d}h\left(\frac{a{q}_1+b}{1+q_1},w\right){}_c{d}_{q_2}w\right]\hfill \\ \le \frac{1}{(b-a)(d-c)}{\int }_a^b{\int }_c^{d}h(z,w){}_c{d}_{q_2}w{}_a{d}_{q_1}z\le \frac{q_2}{2(1+q_2)(b-a)}{\int }_a^{b}h(z,c){}_a{d}_{q_1}z\hfill \end{array}$$

$$\begin{array}{c}+\frac{1}{2(1+q_2)(b-a)}{\int }_a^{b}h(z,d){}_a{d}_{q_1}z+\frac{q_1}{2(1+q_1)(d-c)}{\int }_c^{d}h(a,w){}_c{d}_{q_2}w\hfill \\ +\frac{1}{2(1+q_1)(d-c)}{\int }_c^{d}h(b,w){}_c{d}_{q_2}w\le \frac{q_1{q}_{2}h(a,c)+q_{1}h(a,d)+q_{2}h(b,c)+h(b,d)}{(1+q_1)(1+q_2)}.\hfill \end{array}$$

This paper is organized in the following way. After this Introduction, in Section 2, we define some new concepts regarding the $\left(p,q\right)$-calculus of the function of two variables and discuss some properties and examples. Moreover, our aim is to describe the Hermite-Hadamard-type inequalities for functions of two variables using $(p,q)$-calculus, which is a generalization of Theorem 3 presented by Alp et al. [23]. Next, we provide some new estimates for $\left(p,q\right)$-analogues of the Hermite-Hadamard-type inequalities of the functions of two variables using convexity and quasi-convexity on coordinates; these results are a generalization of all those results that were presented in [22].

2. Preliminaries and Main Results

Definition 1.

Let $h:▵=[a,b]\times [c,d]\subset {\mathbb{R}}^2\to \mathbb{R}$ be a continuous function in each variable; let $0<q_i<p_i\le 1$, $i=1,2$; the partial $(p_1,q_1)$-, $(p_2,q_2)$- and $(p_1{p}_2,q_1{q}_2)$-derivatives at $(s,t)\in [a,b]\times [c,d]$ are, respectively, defined as:

$$\begin{array}{ccc}\hfill \frac{{}_a{\partial }_{p_1,q_1}h(s,t)}{{}_a{\partial }_{p_1,q_1}s}& =& \frac{h(p_{1}s+(1-p_1)a,t)-h(q_{1}s+(1-q_1)a,t)}{(p_1-q_1)(s-a)},s\ne a\hfill \\ \hfill \frac{{}_c{\partial }_{p_2,q_2}h(s,t)}{{}_c{\partial }_{p_2,q_2}t}& =& \frac{h(s,p_{2}t+(1-p_2)c)-h(s,q_{2}t+(1-q_2)c)}{(p_2-q_2)(t-c)},t\ne c\hfill \\ \hfill \frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h(s,t)}{{}_a{\partial }_{p_1,q_1}s{}_c{\partial }_{p_2,q_2}t}& =& \frac{h\left(q_{1}s+(1-q_1)a,q_{2}t+(1-q_2)c\right)}{(p_1-q_1)(p_2-q_2)(s-a)(t-b)}\hfill \\ \hfill & -& \frac{h(q_{1}s+(1-q_1)a,p_{2}t+(1-p_2)c)}{(p_1-q_1)(p_2-q_2)(s-a)(t-b)}\hfill \\ \hfill & -& \frac{h(p_{1}s+(1-p_1)a,q_{2}t+(1-q_2)c)}{(p_1-q_1)(p_2-q_2)(s-a)(t-b)}\hfill \\ \hfill & +& \frac{h(p_{1}s+(1-p_1)a,p_{2}t+(1-p_2)c)}{(p_1-q_1)(p_2-q_2)(s-a)(t-b)},s\ne a,t\ne c.\hfill \end{array}$$

In a similar way, higher order partial derivatives can be defined.

Definition 2

([24]). For any real number $n$, the $(p,q)$-analogue of n is defined as:

$${\left[n\right]}_{p,q}=\frac{p^n-q^n}{p-q}.$$

Example 1.

Let $h(s,t)={(s-a)}^n{(t-c)}^m$ and $n,m\in {\mathbb{R}}_{+},$ then:

$$\begin{array}{cc}\hfill \frac{{}_a{\partial }_{p_1,q_1}}{{}_a{\partial }_{p_1,q_1}s}\left[{(s-a)}^n{(t-c)}^m\right]& =\frac{{(t-c)}^m\left[(p_1^n{(s-a)}^n-q_1^n{(s-a)}^n\right]}{(p_1-q_1)(s-a)}\hfill \\ \hfill & =\frac{{(s-a)}^n{(t-c)}^m(p_1^n-q_1^n)}{(p_1-q_1)(s-a)}\hfill \\ \hfill & ={\left[n\right]}_{p_1,q_1}{(s-a)}^{n-1}{(t-c)}^m,\hfill \\ \hfill \frac{{}_c{\partial }_{p_2,q_2}}{{}_c{\partial }_{p_2,q_2}t}\left[{(s-a)}^n{(t-c)}^m\right]& =\frac{{(s-a)}^n\left[(p_2^m{(t-c)}^m-q_2^m{(t-c)}^m\right]}{(p_2-q_2)(t-c)}\hfill \\ \hfill & =\frac{{(s-a)}^n{(t-c)}^m(p_2^m-q_2^m)}{(p_2-q_2)(t-c)}\hfill \\ \hfill & ={\left[m\right]}_{p_2,q_2}{(s-a)}^n{(t-c)}^{m-1}\hfill \end{array}$$

and:

$$\begin{array}{ccc}\hfill \frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^2}{{}_a{\partial }_{p_1,q_1}s{}_c{\partial }_{p_2,q_2}t}\left[{(s-a)}^n{(t-c)}^m\right]& =& \frac{{(s-a)}^n{(t-c)}^m(p_1^n-q_1^n)(p_2^m-q_2^m)}{(p_1-q_1)(p_2-q_2)(s-a)(t-a)}\hfill \\ \hfill & =& {\left[n\right]}_{p_1,q_1}{\left[m\right]}_{p_2,q_2}{(s-a)}^{n-1}{(t-c)}^{m-1},\hfill \end{array}$$

which are the $(p_1{p}_2,q_1{q}_2)$-analogues of the usual partial derivatives:

$$\frac{\partial }{\partial s}\left[{(s-a)}^n{(t-c)}^m\right]=n{(s-a)}^{n-1}{(t-c)}^m,$$

$$\frac{\partial }{\partial t}\left[{(s-a)}^n{(t-c)}^m\right]=m{(s-a)}^n{(t-c)}^{m-1}$$

and

$$\frac{{\partial }^2}{\partial s\partial t}\left[{(s-a)}^n{(t-c)}^m\right]=nm{(s-a)}^{n-1}{(t-c)}^{m-1}.$$

The $(p,q)$-exponential functions are defined as:

$$e_{p,q}\left(x\right)={\displaystyle \sum _{n=0}^{\infty }}{p}^{\left(\genfrac{}{}{0pt}{}{n}{2}\right)}\frac{x^n}{{\left[n\right]}_{p,q}!},$$

(3)

$$E_{p,q}\left(x\right)={\displaystyle \sum _{n=0}^{\infty }}{q}^{\left(\genfrac{}{}{0pt}{}{n}{2}\right)}\frac{x^n}{{\left[n\right]}_{p,q}!}.$$

(4)

In the limiting case, as $q\to 1^{-}$ and for $p=1$, the exponential function defined by (3) reduces to the exponential function:

$$e^x={\displaystyle \sum _{n=0}^{\infty }}\frac{x^n}{n!}.$$

Replacing x by $s-a$ in (3), we obtain:

$$e_{p,q}(s-a)={\displaystyle \sum _{n=0}^{\infty }}{p}^{\left(\genfrac{}{}{0pt}{}{n}{2}\right)}\frac{{\left(s-a\right)}^n}{{\left[n\right]}_{p,q}!}.$$

Example 2.

Now, we consider the following exponential function in the $\left(p,q\right)$-calculus of the functions of two variables defined by:

$$e_{p_1,q_1}\left(s-a\right)e_{p_2,q_2}\left(t-c\right)={\displaystyle \sum _{m=0}^{\infty }}{\displaystyle \sum _{n=0}^{\infty }}{p}_1^{\left(\genfrac{}{}{0pt}{}{n}{2}\right)}{p}_2^{\left(\genfrac{}{}{0pt}{}{m}{2}\right)}\frac{{\left(s-a\right)}^n{\left(t-c\right)}^m}{{\left[n\right]}_{p_1,q_1}!{\left[m\right]}_{p_2,q_2}!}.$$

We find the partial derivative of this function with respect to s using the $\left(p,q\right)$-calculus of the functions of two variables is:

$$\begin{array}{cc}\frac{{}_a{\partial }_{p_1,q_1}}{{}_a{\partial }_{p_1,q_1}s}\left[e_{p_1,q_1}(s-a)e_{p_2,q_2}(t-c)\right]\hfill & =e_{p_2,q_2}(t-c){\displaystyle {\sum }_{n=0}^{\infty }}{p}_1^{\left(\genfrac{}{}{0pt}{}{n}{2}\right)}\frac{\frac{{}_a{\partial }_{p_1,q_1}}{{}_a{\partial }_{p_1,q_1}s}\left[{\left(s-a\right)}^n\right]}{{\left[n\right]}_{p_1,q_1}!}\hfill \\ \hfill & =e_{p_2,q_2}(t-c){\displaystyle {\sum }_{n=0}^{\infty }}{p}_1^{\left(\genfrac{}{}{0pt}{}{n}{2}\right)}\frac{{\left[n\right]}_{p_1,q_1}{\left(s-a\right)}^{n-1}}{{\left[n\right]}_{p_1,q_1}!}\hfill \\ \hfill & =e_{p_2,q_2}(t-c){\displaystyle {\sum }_{n=1}^{\infty }}{p}_1^{\left(\genfrac{}{}{0pt}{}{n}{2}\right)}\frac{{\left(s-a\right)}^{n-1}}{{[n-1]}_{p_1,q_1}!}\hfill \\ \hfill & =e_{p_2,q_2}(t-c){\displaystyle {\sum }_{n=0}^{\infty }}{p}_1^{\left(\genfrac{}{}{0pt}{}{n+1}{2}\right)}\frac{{\left(s-a\right)}^n}{{\left[n\right]}_{p_1,q_1}!}\hfill \\ \hfill & =e_{p_2,q_2}(t-c){\displaystyle {\sum }_{n=0}^{\infty }}{p}_1^{\left(\genfrac{}{}{0pt}{}{n}{2}\right)+\left(\genfrac{}{}{0pt}{}{n}{1}\right)}\frac{{\left(s-a\right)}^n}{{\left[n\right]}_{p_1,q_1}!}\hfill \\ \hfill & =e_{p_2,q_2}(t-c){\displaystyle {\sum }_{n=0}^{\infty }}{p}_1^{\left(\genfrac{}{}{0pt}{}{n}{2}\right)}\frac{{\left(p\left(s-a\right)\right)}^n}{{\left[n\right]}_{p_1,q_1}!}\hfill \\ \hfill & =e_{p_1,q_1}\left(p_1(s-a)\right)e_{p_2,q_2}(t-c).\hfill \end{array}$$

(5)

Again, taking the partial derivative with respect to t on both sides of Equation (5) using the $\left(p,q\right)$-calculus of the functions of two variables, we have:

$$\begin{array}{c}\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^2}{{}_a{\partial }_{p_1,q_1}s{}_c{\partial }_{p_2,q_2}t}\left[e_{p_1,q_1}(s-a)e_{p_2,q_2}(t-c)\right]\hfill \\ =e_{p_1,q_1}\left(p_1(s-a)\right){\displaystyle \sum _{m=0}^{\infty }}{p}_2^{\left(\genfrac{}{}{0pt}{}{m}{2}\right)}\frac{\frac{{}_c{\partial }_{p_2,q_2}}{{}_c{\partial }_{p_2,q_2}t}\left[{\left(t-c\right)}^m\right]}{{\left[m\right]}_{p_2,q_2}!}\hfill \\ =e_{p_1,q_1}\left(p_1(s-a)\right){\displaystyle \sum _{m=0}^{\infty }}{p}_2^{\left(\genfrac{}{}{0pt}{}{m}{2}\right)}\frac{{\left[m\right]}_{p_2,q_2}{\left(t-c\right)}^{m-1}}{{\left[m\right]}_{p_1,q_2}!}\hfill \\ =e_{p_1,q_1}\left(p_1(s-a)\right){\displaystyle \sum _{m=1}^{\infty }}{p}_2^{\left(\genfrac{}{}{0pt}{}{m}{2}\right)}\frac{{\left(t-c\right)}^{m-1}}{{[m-1]}_{p_2,q_2}!}\hfill \\ =e_{p_1,q_1}\left(p_1(s-a)\right){\displaystyle \sum _{m=0}^{\infty }}{p}_2^{\left(\genfrac{}{}{0pt}{}{m+1}{2}\right)}\frac{{\left(t-c\right)}^m}{{\left[m\right]}_{p_2,q_2}!}\hfill \\ =e_{p_1,q_1}\left(p_1(s-a)\right){\displaystyle \sum _{m=0}^{\infty }}{p}_2^{\left(\genfrac{}{}{0pt}{}{m}{2}\right)+\left(\genfrac{}{}{0pt}{}{m}{1}\right)}\frac{{\left(t-c\right)}^m}{{\left[m\right]}_{p_2,q_2}!}\hfill \\ =e_{p_1,q_1}\left(p_1(s-a)\right){\displaystyle \sum _{m=0}^{\infty }}{p}_2^{\left(\genfrac{}{}{0pt}{}{m}{2}\right)}\frac{{\left(p_2\left(t-c\right)\right)}^m}{{\left[m\right]}_{p_2,q_2}!}\hfill \\ =e_{p_1,q_1}\left(p_1(s-a)\right)e_{p_2,q_2}\left(p_2(t-c)\right),\hfill \end{array}$$

which is the $(p_1{p}_2,q_1{q}_2)$-analogue of the usual partial derivative:

$$\frac{{\partial }^2}{\partial s\partial t}\left[e^{(s-a)}{e}^{(t-c)}\right]=e^{(s-a)}{e}^{(t-c)}.$$

The $(p,q)$-trigonometric functions are defined as:

$$\begin{array}{cc}\hfill {sin}_{p,q}\left(x\right)& =\frac{e_{p,q}\left(ix\right)-e_{p,q}(-ix)}{2i},\hfill \\ \hfill {cos}_{p,q}\left(x\right)& =\frac{e_{p,q}\left(ix\right)+e_{p,q}(-ix)}{2}.\hfill \end{array}$$

(6)

Replacing x by $s-a$ in (6), we obtain:

$$\begin{array}{cc}\hfill {sin}_{p,q}(s-a)& =\frac{e_{p_1,q_1}\left(i(s-a)\right)-e_{p_1,q_1}(-i(s-a))}{2i},\hfill \\ \hfill {cos}_{p,q}(s-a)& =\frac{e_{p,q}\left(i(s-a)\right)+e_{p,q}(-i(s-a))}{2}.\hfill \end{array}$$

Example 3.

Now, we consider the following trigonometric function of two variables in the $(p,q)$-calculus of the functions of two variables defined by:

$$\begin{array}{c}{sin}_{p_1,q_1}\left(s-a\right){cos}_{p_2,q_2}\left(t-c\right)\hfill \\ =\left(\frac{e_{p_1,q_1}\left(i(s-a)\right)-e_{p_1,q_1}(-i(s-a))}{2i}\right)\left(\frac{e_{p_2,q_2}\left(i(t-c)\right)+e_{p_2,q_2}(-i(t-c))}{2}\right).\hfill \end{array}$$

The partial derivative with respect to s using the $(p,q)$-calculus of functions of two variables is:

$$\begin{array}{cc}\hfill & \frac{{}_a{\partial }_{p_1,q_1}}{{}_a{\partial }_{p_1,q_1}s}\left[{sin}_{p_1,q_1}(s-a){cos}_{p_2,q_2}(t-c)\right]\hfill \\ \hfill & =\left(\frac{\frac{{}_a{\partial }_{p_1,q_1}}{{}_a{\partial }_{p_1,q_1}s}{e}_{p_1,q_1}\left(i(s-a)\right)-\frac{{}_a{\partial }_{p_1,q_1}}{{}_a{\partial }_{p_1,q_1}s}{e}_{p_1,q_1}(-i(s-a))}{2i}\right)\hfill \\ \hfill & \times \left(\frac{e_{p_2,q_2}\left(i(t-c)\right)+e_{p_2,q_2}(-i(t-c))}{2}\right)\hfill \\ \hfill & =\left(\frac{e_{p_1,q_1}\left(i{p}_1(s-a)\right)+e_{p_1,q_1}(-i{p}_1(s-a))}{2}\right)\hfill \\ \hfill & \times \left(\frac{e_{p_2,q_2}\left(i(t-c)\right)+e_{p_2,q_2}(-i(t-c))}{2}\right)\hfill \\ \hfill & =cos\left(p_1(s-a)\right)\left(\frac{e_{p_2,q_2}\left(i(t-c)\right)+e_{p_2,q_2}(-i(t-c))}{2}\right).\hfill \end{array}$$

(7)

Again, taking the partial derivative with respect to t on both sides of Equation (7) using the $(p,q)$-calculus of functions of two variables, we have:

$$\begin{array}{c}\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^2}{{}_a{\partial }_{p_1,q_1}s{}_c{\partial }_{p_2,q_2}t}\left[{cos}_{p_1,q_1}\left(p_1(s-a)\right)co{s}_{p_2,q_2}(t-c)\right]\hfill \\ =cos\left(p_1(s-a)\right)\left(\frac{\frac{{}_c{\partial }_{p_2,q_2}}{{}_c{\partial }_{p_2,q_2}t}{e}_{p_2,q_2}\left(i(t-c)\right)+\frac{{}_c{\partial }_{p_2,q_2}}{{}_c{\partial }_{p_2,q_2}t}{e}_{p_2,q_2}(-i(t-c))}{2}\right)\hfill \\ =-cos\left(p_1(s-a)\right)\left(\frac{e_{p_2,q_2}\left(i{p}_2(t-c)\right)-e_{p_2,q_2}(-i{p}_2(t-c))}{2i}\right)\hfill \\ =-cos\left(p_1(s-a)\right)sin\left(p_2(t-c)\right),\hfill \end{array}$$

which is the $(p_1{p}_2,q_1{q}_2)$-analogue of the usual partial derivative:

$$\frac{{\partial }^2}{\partial s\partial t}\left[sin(s-a)cos(t-c)\right]=-cos(s-a)sin(t-c).$$

Definition 3.

Let $h:▵=[a,b]\times [c,d]\subset {\mathbb{R}}^2\to \mathbb{R}$ be a continuous function in each variable; let $0<q_i<p_i\le 1$ where $i=1,2,$ then the definite $(p_1{p}_2,q_1{q}_2)$-integral on $[a,b]\times [c,d]$ is defined as:

$$\begin{array}{c}{\int }_c^t{\int }_a^{s}h(z,w){}_a{d}_{p_1,q_1}z{}_c{d}_{p_2,q_2}w=(p_1-q_1)(p_2-q_2)(s-a)(t-c)\hfill \\ \times {\displaystyle \sum _{m=0}^{\infty }}{\displaystyle \sum _{n=0}^{\infty }}\frac{q_1^n}{p_1^{n+1}}\frac{q_2^m}{p_2^{m+1}}h\left(\frac{q_1^n}{p_1^{n+1}}s+\left(1-\frac{q_1^n}{p_1^{n+1}}\right)a,\frac{q_2^m}{p_2^{m+1}}t+\left(1-\frac{q_2^m}{p_2^{m+1}}\right)c\right)\hfill \end{array}$$

(8)

for $(s,t)\in [a,b]\times [c,d].$

Example 4.

Let $h(z,w)={(z-a)}^n{(w-c)}^m$ for $(s,t)\in [a,b]\times [c,d]$, then we have:

$$\begin{array}{ccc}\hfill {\int }_c^t{\int }_a^{s}h(z,w){}_a{d}_{p_1,q_1}z{}_c{d}_{p_2,q_2}w& =& {\int }_c^t{\int }_a^s{(z-a)}^n{(w-c)}^m{}_a{d}_{p_1,q_1}z{}_c{d}_{p_2,q_2}w\hfill \\ \hfill & =& \left((p_1-q_1){(s-a)}^{n+1}{p}_1^{-n-1}{\displaystyle \sum _{i=0}^{\infty }}{\left(\frac{q_1^{n+1}}{p_1^{n+1}}\right)}^i\right)\hfill \\ \hfill & & \times \left((p_2-q_2){(t-c)}^{m+1}{p}_2^{-m-1}{\displaystyle \sum _{j=0}^{\infty }}{\left(\frac{q_2^{m+1}}{p_2^{m+1}}\right)}^j\right)\hfill \\ \hfill & =& \left(\frac{{(s-a)}^{n+1}}{[n+1]{{}_p}_1{}_{,}{{}_q}_1}\right)\left(\frac{{(t-c)}^{m+1}}{[m+1]{{}_p}_2{}_{,}{{}_q}_2}\right),\hfill \end{array}$$

which is the $(p_1{p}_2,q_1{q}_2)$-analogue of the usual integral:

$${\int }_c^t{\int }_a^s{(z-a)}^n{(w-c)}^{m}dzdw=\left(\frac{{(s-a)}^{n+1}}{n+1}\right)\left(\frac{{(t-c)}^{m+1}}{m+1}\right).$$

Remark 1.

By Definition 3, it directly follows:

• $${\int }_c^t{\int }_a^{s}h(z,w){}_a{d}_{p_1,q_1}z{}_c{d}_{p_2,q_2}w={\int }_a^s{\int }_c^{t}h(z,w){}_c{d}_{p_2,q_2}w{}_a{d}_{p_1,q_1}z.$$
• $$\begin{array}{ccc}\hfill {\int }_{t_1}^t{\int }_{s_1}^{s}h(z,w){}_a{d}_{p_1,q_1}z{}_c{d}_{p_2,q_2}w& =& {\int }_c^t{\int }_a^{s}h(z,w){}_a{d}_{p_1,q_1}z{}_c{d}_{p_2,q_2}w\hfill \\ \hfill & & -{\int }_c^{t_1}{\int }_a^{s}h(z,w){}_a{d}_{p_1,q_1}z{}_c{d}_{p_2,q_2}w\hfill \\ \hfill & & -{\int }_c^t{\int }_a^{s_1}h(z,w){}_a{d}_{p_1,q_1}z{}_c{d}_{p_2,q_2}w\hfill \\ \hfill & & +{\int }_c^{t_1}{\int }_a^{s_1}h(z,w){}_a{d}_{p_1,q_1}z{}_c{d}_{p_2,q_2}w,\hfill \end{array}$$

  for $(s_1,t_1)\in (a,s)\times (c,t).$
• Setting $q_i\to 1^{-}$ and $p_i=1$ for $i=1,2$, Definitions 1 and 3 reduce to the classical definitions of partial derivatives and the classical integration of double integrals, respectively.
• For $p_i=1$ where $i=1,2$, Definitions 1 and 3 reduce to Definitions 1 and 2 in [22], respectively.

Theorem 4.

Let $h_1,h_2:▵=[a,b]\times [c,d]\subset {\mathbb{R}}^2\to \mathbb{R}$ be continuous functions in each variable, then we can prove the following easily:

• $$\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^2}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}{\int }_c^t{\int }_a^s{h}_1(z,w){}_a{d}_{p_1,q_1}z{}_c{d}_{p_2,q_2}w=h_1(s,t).$$
• $${\int }_c^t{\int }_a^s\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^2{h}_1(z,w)}{{}_c{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}{}_a{d}_{p_1,q_1}z{}_c{d}_{p_2,q_2}w=h_1(s,t).$$
  For $(s_1,t_1)\in (a,s)\times (c,t)$:
• $${\int }_{t_1}^t{\int }_{s_1}^s\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^2{h}_1(z,w)}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}{}_a{d}_{p_1,q_1}z{}_c{d}_{p_2,q_2}w$$

  $$=h_1(s,t)-h_1(s,t_1)-h_1(s_1,t)+h_1(s_1,t_1).$$
• $$\begin{array}{c}{\int }_c^t{\int }_a^s\left[h_1(z,w)+h_2(z,w)\right]{}_a{d}_{p_1,q_1}z{}_c{d}_{p_2,q_2}w\hfill \\ ={\int }_c^t{\int }_a^s{h}_1(z,w){}_a{d}_{p_1,q_1}z{}_c{d}_{p_2,q_2}w+{\int }_c^t{\int }_a^s{h}_2(z,w){}_a{d}_{p_1,q_1}z{}_c{d}_{p_2,q_2}w\hfill \end{array}$$
• $${\int }_c^t{\int }_a^s\beta h_1(z,w){}_a{d}_{p_1,q_1}z{}_c{d}_{p_2,q_2}w=\beta {\int }_c^t{\int }_a^s{h}_1(z,w){}_a{d}_{p_1,q_1}z{}_c{d}_{p_2,q_2}w.$$
• $$\begin{array}{c}{\int }_{t_1}^t{\int }_{s_1}^{s}g(z,w)\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^2{h}_1(z,w)}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}{}_a{d}_{p_1,q_1}z{}_c{d}_{p_2,q_2}w\hfill \\ =g(s,t)h_1(s,t)-g(s,t_1)h_1(s,t_1)-g(s_1,t)h_1(s_1,t)+g(s_1,t_1)h_1(s_1,t_1)\hfill \\ -{\int }_{t_1}^{t}g(p_{1}s+(1-p_1)a,q_{2}w+(1-q_2)c)\frac{{}_c{\partial }_{p_2,q_2}g(s,w)}{{}_c{\partial }_{p_2,q_2}w}{}_c{d}_{p_2,q_2}w\hfill \\ +{\int }_{t_1}^t{h}_1(p_1{s}_1+(1-p_1)a,q_{2}w+(1-q_2)c)\frac{{}_c{\partial }_{p_2,q_2}g(s_1,w)}{{}_c{\partial }_{p_2,q_2}w}{}_c{d}_{p_2,q_2}w\hfill \\ -{\int }_{s_1}^s{h}_1(q_{1}z+(1-q_1)a,p_{2}t+(1-p_2)c)\frac{{}_a{\partial }_{p_1,q_1}g(z,t)}{{}_a{\partial }_{p_1,q_1}z}{}_a{d}_{p_1,q_1}z\hfill \\ +{\int }_{s_1}^s{h}_1(q_{1}z+(1-q_1)a,p_2{t}_1+(1-p_2)c)\frac{{}_a{\partial }_{p_1,q_1}g(z,t_1)}{{}_a{\partial }_{p_1,q_1}z}{}_a{d}_{p_1,q_1}z\hfill \\ +{\int }_{t_1}^t{\int }_{s_1}^s{h}_1(q_{1}z+(1-q_1)a,q_{2}w+(1-q_2)c)\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}g(z,w)}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}{}_a{d}_{p_1,q_1}z{}_c{d}_{p_2,q_2}w.\hfill \end{array}$$

Proof. 

The proof is directly followed by $(p_1{p}_2,q_1{q}_2)$-calculus. We omit the details.  □

Theorem 5.

Let $h:[a,b]\times [c,d]\subset {\mathbb{R}}^2\to \mathbb{R}$ be convex on the coordinates on $[a,b]\times [c,d]$ for $0<q_i<p_i\le 1$ where $i=1,2$, then the following inequalities hold:

$$\begin{array}{c}h\left(\frac{a{q}_1+b{p}_1}{p_1+q_1},\frac{c{q}_2+d{p}_2}{p_2+q_2}\right)\le \frac{1}{2(b-a)p_1}{\int }_a^{p_{1}b+(1-p_1)a}h\left(z,\frac{c{q}_2+d{p}_2}{p_2+q_2}\right){}_a{d}_{p_1,q_1}z\hfill \\ +\frac{1}{2(d-c)p_2}{\int }_c^{p_{2}d+(1-p_2)c}h\left(\frac{a{q}_1+b{p}_1}{p_1+q_1},w\right){}_c{d}_{p_2,q_2}w\hfill \\ \le \frac{1}{p_1(b-a)p_2(d-c)}{\int }_a^{p_{1}b+(1-p_1)a}{\int }_c^{p_{2}d+(1-p_2)c}h(z,w){}_a{d}_{p_1,q_1}z{}_c{d}_{p_2,q_2}w\hfill \\ \le \frac{q_2}{2p_1(p_2+q_2)(b-a)}{\int }_a^{p_{1}b+(1-p_1)a}h(z,c){}_a{d}_{p_1,q_1}z+\frac{p_2}{2p_1(p_2+q_2)(b-a)}{\int }_a^{p_{1}b+(1-p_1)a}h(z,d){}_a{d}_{p_1,q_1}z\hfill \\ +\frac{q_1}{2p_2(p_1+q_1)(d-c)}{\int }_c^{p_{2}d+(1-p_2)c}h(a,w){}_c{d}_{p_2,q_2}w+\frac{p_1}{2p_2(p_1+q_1)(d-c)}{\int }_c^{p_{2}d+(1-p_2)c}h(b,w){}_c{d}_{p_2,q_2}w\hfill \\ \le \frac{q_1{q}_{2}h(a,c)+p_2{q}_{1}h(a,d)+p_1{q}_{2}h(b,c)+p_1{p}_{2}h(b,d)}{(p_1+q_1)(p_2+q_2)}.\hfill \end{array}$$

Proof. 

Since $h:[a,b]\times [c,d]\subset {\mathbb{R}}^2\to \mathbb{R}$ is a convex function on the coordinates, it follows that the partial mapping $h_z:[c,d]\to \mathbb{R}$ defined by: $h_z\left(w\right)=h(z,w)$ is convex on $[c,d]$ for $z\in [a,b]$, and hence, by Theorem 2:

$$h_z\left(\frac{c{q}_2+d{p}_2}{p_2+q_2}\right)\le \frac{1}{p_2(d-c)}{\int }_c^{p_{2}d+(1-p_2)c}{h}_z\left(w\right){}_c{d}_{p_2,q_2}w\le \frac{q_2{h}_z\left(c\right)+p_2{h}_z\left(d\right)}{p_2+q_2}.$$

Equivalently,

$$h\left(z,\frac{c{q}_2+d{p}_2}{p_2+q_2}\right)\le \frac{1}{p_2(d-c)}{\int }_c^{p_{2}d+(1-p_2)c}h(z,w){}_c{d}_{p_2,q_2}w\le \frac{q_{2}h(z,c)+p_{2}h(z,d)}{p_2+q_2}.$$

(9)

By $q_1$-integrating the inequality (9) on $[a,p_{1}b+(1-p_1)a]$, we have:

$$\begin{array}{c}\frac{1}{p_1(b-a)}{\int }_a^{p_{1}b+(1-p_1)a}h\left(z,\frac{c{q}_2+d{p}_2}{p_2+q_2}\right){}_a{d}_{p_1,q_1}z\hfill \\ \le \frac{1}{p_1(b-a)p_2(d-c)}{\int }_a^{p_{1}b+(1-p_1)a}{\int }_c^{p_{2}d+(1-p_2)c}h(z,w){}_c{d}_{p_2,q_2}w{}_a{d}_{p_1,q_1}z\hfill \\ \le \frac{1}{p_1(b-a)}{\int }_a^{p_{1}b+(1-p_1)a}\frac{q_{2}h(z,c)+p_{2}h(z,d)}{p_2+q_2}{}_a{d}_{p_1,q_1}z.\hfill \end{array}$$

(10)

Similarly, consider $h_w:[a,b]\to \mathbb{R}$ defined by $h_w\left(z\right)=h(z,w).$ Then, obviously, $h_w$ is a convex function. Again, the application of Theorem 2 yields the following inequality:

$$h_w\left(\frac{a{q}_1+b{p}_1}{p_1+q_1}\right)\le \frac{1}{p_1(b-a)}{\int }_a^{p_{1}b+(1-p_1)a}{h}_w\left(z\right){}_a{d}_{p_1,q_1}z\le \frac{q_1{h}_w\left(a\right)+p_1{h}_w\left(b\right)}{p_1+q_1}.$$

Equivalently,

$$h\left(\frac{a{q}_1+b{p}_1}{p_1+q_1},w\right)\le \frac{1}{p_1(b-a)}{\int }_a^{p_{1}b+(1-p_1)a}h(z,w){}_a{d}_{p_1,q_1}z\le \frac{q_1(a,w)+p_{1}h(b,w)}{p_1+q_1}.$$

(11)

By $q_2$-integrating the inequality (11) on $[c,p_{2}d+(1-p_2)c]$, we have:

$$\begin{array}{c}\frac{1}{p_2(d-c)}{\int }_c^{p_{2}d+(1-p_2)c}h\left(\frac{a{q}_1+b{p}_1}{p_1+q_1},w\right){}_c{d}_{p_2,q_2}w\hfill \\ \le \frac{1}{p_1(b-a)p_2(d-c)}{\int }_a^{p_{1}b+(1-p_1)a}{\int }_c^{p_{2}d+(1-p_2)c}h(z,w){}_c{d}_{p_2,q_2}w{}_a{d}_{p_1,q_1}z\hfill \\ \le \frac{1}{p_2(d-c)}{\int }_c^{p_{2}d+(1-p_2)c}\frac{q_{1}h(a,w)+p_{1}h(b,w)}{p_1+q_1}{}_c{d}_{p_2,q_2}w.\hfill \end{array}$$

(12)

Adding (10) and (12):

$$\begin{array}{c}\frac{1}{2p_1(b-a)}{\int }_a^{p_{1}b+(1-p_1)a}h\left(z,\frac{c{q}_2+d{p}_2}{p_2+q_2}\right){}_a{d}_{p_1,q_1}z+\frac{1}{2p_2(d-c)}{\int }_c^{p_{2}d+(1-p_2)c}h\left(\frac{a{q}_1+b{p}_1}{p_1+q_1},w\right){}_c{d}_{p_2,q_2}w\hfill \\ \le \frac{1}{p_1(b-a)p_2(d-c)}{\int }_a^{p_{1}b+(1-p_1)a}{\int }_c^{p_{2}d+(1-p_2)c}h(z,w){}_c{d}_{p_2,q_2}w{}_a{d}_{p_1,q_1}z\hfill \\ \le \frac{q_2}{2p_1(b-a)(p_2+q_2)}{\int }_a^{p_{1}b+(1-p_1)a}h(z,c){}_a{d}_{p_1,q_1}z+\frac{p_2}{2p_1(b-a)(p_2+q_2)}{\int }_a^{p_{1}b+(1-p_1)a}h(z,d){}_a{d}_{p_1,q_1}z\hfill \\ \le \frac{q_1}{2p_2(d-c)(p_1+q_1)}{\int }_c^{p_{2}d+(1-p_2)c}h(a,w){}_c{d}_{p_2,q_2}w+\frac{p_1}{2p_2(d-c)(p_1+q_1)}{\int }_c^{p_{2}d+(1-p_2)c}h(b,w){}_c{d}_{p_2,q_2}w.\hfill \end{array}$$

(13)

Setting $z\to \frac{a{q}_1+b{p}_1}{p_1+q_1}$ and $w\to \frac{c{q}_2+d{p}_2}{p_2+q_2}$ in (9) and (11) respectively:

$$\begin{array}{cc}h\left(\frac{a{q}_1+b{p}_1}{p_1+q_1},\frac{c{q}_2+d{p}_2}{p_2+q_2}\right)\hfill & \le \frac{1}{p_2(d-c)}{\int }_c^{p_{2}d+(1-p_2)c}h\left(\frac{a{q}_1+b{p}_1}{p_1+q_1},w\right){}_c{d}_{p_2,q_2}w\hfill \\ \hfill & \le \frac{q_{2}h\left(\frac{a{q}_1+b{p}_1}{p_1+q_1},c\right)+p_{2}h\left(\frac{a{q}_1+b{p}_1}{p_1+q_1},d\right)}{p_2+q_2}\hfill \end{array}$$

(14)

$$\begin{array}{c}h\left(\frac{a{q}_1+b{p}_1}{p_1+q_1},\frac{c{q}_2+d{p}_2}{p_2+q_2}\right)\le \frac{1}{p_1(b-a)}{\int }_a^{p_{1}b+(1-p_1)a}h\left(z,\frac{c{q}_2+d{p}_2}{p_2+q_2}\right){}_a{d}_{p_1,q_1}z\hfill \\ \le \frac{q_{1}h\left(a,\frac{c{q}_2+d{p}_2}{p_2+q_2}\right)+p_{1}h\left(b,\frac{c{q}_2+d{p}_2}{p_2+q_2}\right)}{p_1+q_1}\hfill \end{array}$$

(15)

and hence, by the addition of (14) and (15), we get:

$$\begin{array}{c}h\left(\frac{a{q}_1+b{p}_1}{p_1+q_1},\frac{c{q}_2+d{p}_2}{p_2+q_2}\right)\le \frac{1}{2(b-a)p_1}{\int }_a^{p_{1}b+(1-p_1)a}h\left(z,\frac{c{q}_2+d{p}_2}{p_2+q_2}\right){}_a{d}_{p_1,q_1}z\hfill \\ +\frac{1}{2(d-c)p_2}{\int }_c^{p_{2}d+(1-p_2)c}h\left(\frac{a{q}_1+b{p}_1}{p_1+q_1},w\right){}_c{d}_{p_2,q_2}w\hfill \\ \le \frac{q_{2}h(\frac{a{q}_1+b{p}_1}{p_1+q_1},c)+p_{2}h\left(\frac{a{q}_1+b{p}_1}{p_1+q_1},d\right)}{2(p_2+q_2)}+\frac{q_{1}h\left(a,\frac{c{q}_2+d{p}_2}{p_2+q_2}\right)+p_{1}h\left(b,\frac{c{q}_2+d{p}_2}{p_2+q_2}\right)}{2(p_1+q_1)}.\hfill \end{array}$$

By using Theorem 2 on the right-hand side of (13), we have:

$$\frac{q_2}{2p_1(p_2+q_2)(b-a)}{\int }_a^{p_{1}b+(1-p_1)a}h(z,c){}_a{d}_{p_1,q_1}z\le \frac{q_2}{2(p_2+q_2)}\frac{q_{1}h(a,c)+p_{1}h(b,c)}{p_1+q_1}$$

(16)

$$\frac{p_2}{2p_1(p_2+q_2)(b-a)}{\int }_a^{p_{1}b+(1-p_1)a}h(z,d){}_a{d}_{p_1,q_1}z\le \frac{p_2}{2(p_2+q_2)}\frac{q_{1}h(a,d)+p_{1}h(b,d)}{p_1+q_1}$$

(17)

$$\frac{q_1}{2p_2(p_1+q_1)(d-c)}{\int }_c^{p_{2}d+(1-p_2)c}h(a,w){}_c{d}_{p_2,q_2}w\le \frac{q_1}{2(p_1+q_1)}\frac{q_{2}h(a,c)+p_{2}h(a,d)}{p_2+q_2}$$

(18)

$$\frac{p_1}{2p_2(p_1+q_1)(d-c)}{\int }_c^{p_{2}d+(1-p_2)c}h(b,w){}_c{d}_{p_2,q_2}w\le \frac{p_1}{2(p_1+q_1)}\frac{q_{2}h(b,c)+p_{2}h(b,d)}{p_2+q_2}.$$

(19)

The addition of inequalities (16)–(19) yields the following:

$$\begin{array}{c}\frac{q_2}{2p_1(p_2+q_2)(b-a)}{\int }_a^{p_{1}b+(1-p_1)a}h(z,c){}_a{d}_{p_1,q_1}z+\frac{p_2}{2p_1(p_2+q_2)(b-a)}{\int }_a^{p_{1}b+(1-p_1)a}h(z,d){}_a{d}_{p_1,q_1}z\hfill \\ +\frac{q_1}{2p_2(p_1+q_1)(d-c)}{\int }_c^{p_{2}d+(1-p_2)c}h(a,w){}_c{d}_{p_2,q_2}w+\frac{p_1}{2p_2(p_1+q_1)(d-c)}{\int }_c^{p_{2}d+(1-p_2)c}h(b,w){}_c{d}_{p_2,q_2}w\hfill \\ \le \frac{q_1{q}_{2}h(a,c)+p_2{q}_{1}h(a,d)+p_1{q}_{2}h(b,c)+p_1{p}_{2}h(b,d)}{(p_1+q_1)(p_2+q_2)}.\hfill \end{array}$$

□

Remark 2.

• Setting $q_i\to 1^{-}$ and $p_i=1$ for $i=1,2$ then Theorem 5 reduces to Theorem 1.
• Also, if $p_i=1$ for $i=1,2$ in Theorem 5, then Theorem 5 reduces to Theorem 3.

Theorem 6

([22]). Let ${\left(x_{mn}\right)}_{m,n\in \mathbb{N}}$ and ${\left(y_{mn}\right)}_{m,n\in \mathbb{N}}$ be sequences of real (or complex) numbers. If $r_1^{-1}+r_2^{-1}=1,$ with $r_1,r_2>1$, then the following inequality holds:

$${\displaystyle \sum _{m=0}^{\infty }}{\displaystyle \sum _{n=0}^{\infty }}\left|x_{mn}{y}_{mn}\right|\le {\left({\displaystyle \sum _{m=0}^{\infty }}{\displaystyle \sum _{n=0}^{\infty }}{\left|x_{mn}\right|}^{r_1}\right)}^{\frac{1}{r_1}}{\left({\displaystyle \sum _{m=0}^{\infty }}{\displaystyle \sum _{n=0}^{\infty }}{\left|y_{mn}\right|}^{r_2}\right)}^{\frac{1}{r_2}}$$

Provided that all the sums are finite.

Theorem 7.

Let g and h be functions defined on $[a,b]\times [c,d]$ and $0<q_i<p_i\le 1$ for $i=1,2$. If $r_1^{-1}+r_2^{-1}=1,$ with $r_1,r_2>1$, then the following inequality holds:

$$\begin{array}{c}{\int }_a^b{\int }_c^d\left|g(z,w)h(z,w)\right|{}_c{d}_{p_2,q_2}w{}_a{d}_{p_1,q_1}z\le {\left({\int }_a^b{\int }_c^d{\left|g(z,w)\right|}^{r_1}{}_a{d}_{p_1,q_1}z{}_c{d}_{p_2,q_2}w\right)}^{\frac{1}{r_1}}\hfill \\ \times {\left({\int }_a^b{\int }_c^d{\left|h(z,w)\right|}^{r_2}{}_a{d}_{p_1,q_1}z{}_c{d}_{p_2,q_2}w\right)}^{\frac{1}{r_2}}.\hfill \end{array}$$

Proof. 

The proof is directly followed by $(p_1{p}_2,q_1{q}_2)$-calculus. We omit the details.  □

Lemma 1.

Let $h:▵\subset {\mathbb{R}}^2\to \mathbb{R}$ be a function such that $(p_1{p}_2,q_1{q}_2)$-derivatives exist on ${▵}^{\mathrm{o}}$ (the interior of $▵$) and $0<q_i<p_i\le 1$ for $i=1,2.$ Moreover, if $\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h(z,w)}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}$ is continuous and integrable on $[a,b]\times [c,d]\subset {▵}^{\mathrm{o}}$, then:

$$\begin{array}{c}{\mathsf{\Gamma }}_{(p_1{p}_2,q_1{q}_2)}(a,b,c,d)\left(h\right):=\frac{q_1{q}_{2}h(a,c)+p_2{q}_{1}h(a,d)+p_1{q}_{2}h(b,c)+p_1{p}_{2}h(b,d)}{(p_1+q_1)(p_2+q_2)}\hfill \\ -\frac{q_2}{p_1(p_2+q_2)(b-a)}{\int }_a^{p_{1}b+(1-p_1)a}h(s,c){}_a{d}_{p_1,q_1}s-\frac{p_2}{p_1(p_2+q_2)(b-a)}{\int }_a^{p_{1}b+(1-p_1)a}h(s,d){}_a{d}_{p_1,q_1}s\hfill \\ -\frac{q_1}{p_2(p_1+q_1)(d-c)}{\int }_c^{p_{2}d+(1-p_2)c}h(a,t){}_c{d}_{p_2,q_2}t-\frac{p_1}{p_2(p_1+q_1)(d-c)}{\int }_c^{p_{2}d+(1-p_2)c}h(b,t){}_c{d}_{p_2,q_2}t\hfill \\ +\frac{1}{p_1(b-a)p_2(d-c)}{\int }_a^{p_{1}b+(1-p_1)a}{\int }_c^{p_{2}d+(1-p_2)c}h(s,t){}_c{d}_{p_2,q_2}t{}_a{d}_{p_1,q_1}s\hfill \\ =\frac{q_1{q}_2(b-a)(d-c)}{(p_1+q_1)(p_2+q_2)}{\int }_0^1{\int }_0^1(1-(p_1+q_1)z)(1-(p_2+q_2)w)\hfill \\ \times \frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h((1-z)a+zb,(1-w)c+wd)}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_1,q_2}w}{}_0{d}_{p_1,q_1}z{}_0{d}_{p_2,q_2}w.\hfill \end{array}$$

Proof. 

By the Definition 1 of partial $(p_1{p}_2,q_1{q}_2)$-derivatives and the Definition 3 of definite $(p_1{p}_2,q_1{q}_2)$-integrals, we have:

$$\begin{array}{c}{\int }_0^1{\int }_0^1(1-(p_1+q_1)z)(1-(p_2+q_2)w)\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h((1-z)a+zb,(1-w)c+wd)}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}{}_0{d}_{p_1,q_1}z{}_0{d}_{p_2,q_2}w\hfill \\ =\frac{1}{(p_1-q_1)(p_2-q_2)(b-a)(d-c)}{\int }_0^1{\int }_0^1\frac{(1-(p_1+q_1)z)(1-(p_2+q_2)w)}{wz}\hfill \\ \times [h(z{q}_{1}b+(1-z{q}_1)a,w{q}_{2}d+(1-w{q}_2)c)-h(z{q}_{1}b+(1-z{q}_1)a,w{p}_{2}d+(1-w{p}_2)c)\hfill \\ -h(z{p}_{1}b+(1-z{p}_1)a,w{q}_{2}d+(1-w{q}_2)c)+h(z{p}_{1}b+(1-z{p}_1)a,w{p}_{2}d+(1-w{p}_2)c)]{}_0{d}_{p_1,q_1}z{}_0{d}_{p_1,q_2}w\hfill \\ =\frac{1}{(b-a)(d-c)}{\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \sum _{m=0}^{\infty }}\left(1-(p_1+q_1\right)\frac{q_1^n}{p_1^{n+1}}\left(1-(p_2+q_2)\frac{q_2^m}{p_2^{m+1}}\right)\hfill \end{array}$$

$$\begin{array}{c}\times [h\left(\frac{q_1^{n+1}}{p_1^{n+1}}b+\left(1-\frac{q_1^{n+1}}{p_1^{n+1}}\right)a,\frac{q_2^{m+1}}{p_2^{m+1}}d+\left(1-\frac{q_2^{m+1}}{p_2^{m+1}}\right)c\right)\hfill \\ -h\left(\frac{q_1^{n+1}}{p_1^{n+1}}b+\left(1-\frac{q_1^{n+1}}{p_1^{n+1}}\right)a,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right)\hfill \\ -h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,\frac{q_2^{m+1}}{p_2^{m+1}}d+\left(1-\frac{q_2^{m+1}}{p_2^{m+1}}\right)c\right)\hfill \\ +h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right)]\hfill \\ =\frac{1}{(b-a)(d-c)}{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \sum _{m=1}^{\infty }}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right)\hfill \\ -\frac{1}{(b-a)(d-c)}{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \sum _{m=0}^{\infty }}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right)\hfill \\ -\frac{1}{(b-a)(d-c)}{\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \sum _{m=1}^{\infty }}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right)\hfill \\ +\frac{1}{(d-c)(b-a)}{\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \sum _{m=0}^{\infty }}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right)\hfill \\ -\frac{(p_1+q_1)}{q_1(b-a)(d-c)}{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \sum _{m=1}^{\infty }}\frac{q_1^n}{p_1^n}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right)\hfill \\ +\frac{(p_1+q_1)}{q_1(b-a)(d-c)}{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \sum _{m=0}^{\infty }}\frac{q_1^n}{p_1^n}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right)\hfill \\ +\frac{(p_1+q_1)}{p_1(b-a)(d-c)}{\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \sum _{m=1}^{\infty }}\frac{q_1^n}{p_1^n}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right)\hfill \\ -\frac{(p_1+q_1)}{p_1(b-a)(d-c)}{\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \sum _{m=0}^{\infty }}\frac{q_1^n}{p_1^n}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right)\hfill \\ -\frac{(p_2+q_2)}{q_2(b-a)(d-c)}{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \sum _{m=1}^{\infty }}\frac{q_2^m}{p_2^m}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right)\hfill \\ +\frac{(p_2+q_2)}{p_2(b-a)(d-c)}{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \sum _{m=0}^{\infty }}\frac{q_2^m}{p_2^m}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right)\hfill \\ +\frac{(p_2+q_2)}{q_2(b-a)(d-c)}{\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \sum _{m=1}^{\infty }}\frac{q_2^m}{p_2^m}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right)\hfill \\ -\frac{(p_2+q_2)}{p_2(b-a)(d-c)}{\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \sum _{m=0}^{\infty }}\frac{q_2^m}{p_2^m}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right)\hfill \\ +\frac{(p_1+q_1)(p_2+q_2)}{q_1{q}_2(b-a)(d-c)}{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \sum _{m=1}^{\infty }}\frac{q_1^n}{p_1^n}\frac{q_2^m}{p_2^m}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right)\hfill \\ -\frac{(p_1+q_1)(p_2+q_2)}{p_2{q}_1(b-a)(d-c)}{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \sum _{m=0}^{\infty }}\frac{q_1^n}{p_1^n}\frac{q_2^m}{p_2^m}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right)\hfill \\ -\frac{(p_1+q_1)(p_1+q_2)}{p_1{q}_2(b-a)(d-c)}{\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \sum _{m=1}^{\infty }}\frac{q_1^n}{p_1^n}\frac{q_2^m}{p_2^m}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right)\hfill \\ +\frac{(p_1+q_1)(p_2+q_2)}{p_1{p}_2(b-a)(d-c)}{\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \sum _{m=0}^{\infty }}\frac{q_1^n}{p_1^n}\frac{q_2^m}{p_2^m}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right).\hfill \end{array}$$

We observe that:

$$\begin{array}{c}\frac{1}{(b-a)(d-c)}{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \sum _{m=1}^{\infty }}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right)\hfill \\ =\frac{h(a,c)}{(b-a)(d-c)}+\frac{1}{(b-a)(d-c)}{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \sum _{m=1}^{\infty }}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right),\hfill \end{array}$$

(20)

$$\begin{array}{c}-\frac{1}{(b-a)(d-c)}{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \sum _{m=0}^{\infty }}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right)\hfill \\ =-\frac{h(a,d)}{(b-a)(d-c)}-\frac{1}{(b-a)(d-c)}{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \sum _{m=1}^{\infty }}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right),\hfill \end{array}$$

(21)

$$\begin{array}{c}-\frac{1}{(b-a)(d-c)}{\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \sum _{m=1}^{\infty }}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right)\hfill \\ =-\frac{h(b,c)}{(b-a)(d-c)}-\frac{1}{(b-a)(d-c)}{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \sum _{m=1}^{\infty }}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right),\hfill \end{array}$$

(22)

$$\begin{array}{c}\frac{1}{(b-a)(d-c)}{\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \sum _{m=0}^{\infty }}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right)\hfill \\ =\frac{h(b,d)}{(b-a)(d-c)}+\frac{1}{(b-a)(d-c)}{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \sum _{m=1}^{\infty }}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right),\hfill \end{array}$$

(23)

$$\begin{array}{c}-\frac{(p_1+q_1)}{q_1(b-a)(d-c)}{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \sum _{m=1}^{\infty }}\frac{q_1^n}{p_1^n}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right)\hfill \\ =\frac{(p_1+q_1)h(b,c)}{q_1(b-a)(d-c)}-\frac{(p_1+q_1)}{q_1(b-a)(d-c)}{\displaystyle \sum _{n=0}^{\infty }}\frac{q_1^n}{p_1^n}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,c\right)\hfill \\ -\frac{(p_1+q_1)}{q_1(b-a)(d-c)}{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \sum _{m=1}^{\infty }}\frac{q_1^n}{p_1^n}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right),\hfill \end{array}$$

(24)

$$\begin{array}{c}\frac{(p_1+q_1)}{q_1(b-a)(d-c)}{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \sum _{m=0}^{\infty }}\frac{q_1^n}{p_1^n}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right)\hfill \\ =-\frac{(p_1+q_1)h(b,d)}{q_1(b-a)(d-c)}+\frac{(p_1+q_1)}{q_1(b-a)(d-c)}{\displaystyle \sum _{n=0}^{\infty }}\frac{q_1^n}{p_1^n}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,d\right)\hfill \\ +\frac{(p_1+q_1)}{q_1(b-a)(d-c)}{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \sum _{m=1}^{\infty }}\frac{q_1^n}{p_1^n}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right),\hfill \end{array}$$

(25)

$$\begin{array}{c}\frac{(p_1+q_1)}{p_1(b-a)(d-c)}{\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \sum _{m=1}^{\infty }}\frac{q_1^n}{p_1^n}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right)\hfill \\ =\frac{(p_1+q_1)}{p_1(b-a)(d-c)}[-{\displaystyle \sum _{n=0}^{\infty }}\frac{q_1^n}{p_1^n}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,d\right)\hfill \\ +{\displaystyle \sum _{n=0}^{\infty }}\frac{q_1^n}{p_1^n}h\left(\frac{q_1^n}{p_1^n}b+\left(-\frac{q_1^n}{p_1^n}\right)a,c\right)]+\frac{(p_1+q_1)}{p_1(b-a)(d-c)}\hfill \\ \times {\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \sum _{m=0}^{\infty }}\frac{q_1^n}{p_1^n}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right),\hfill \end{array}$$

(26)

$$\begin{array}{c}-\frac{(p_2+q_2)}{q_2(b-a)(d-c)}{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \sum _{m=1}^{\infty }}\frac{q_2^m}{p_2^m}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right)\hfill \\ =\frac{(p_2+q_2)h(a,d)}{q_2(b-a)(d-c)}-\frac{(p_2+q_2)}{q_2(b-a)(d-c)}{\displaystyle \sum _{m=0}^{\infty }}\frac{q_2^m}{p_2^m}h\left(a,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right)\hfill \\ -\frac{(p_2+q_2)}{q_2(b-a)(d-c)}{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \sum _{m=1}^{\infty }}\frac{q_2^m}{p_2^m}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right),\hfill \end{array}$$

(27)

$$\begin{array}{c}\frac{(p_2+q_2)}{p_2(b-a)(d-c)}{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \sum _{m=0}^{\infty }}\frac{q_2^m}{p_2^m}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right)\hfill \\ =-\frac{(p_2+q_2)}{p_2(b-a)(d-c)}[-{\displaystyle \sum _{m=0}^{\infty }}\frac{q_2^m}{p_2^m}h\left(b,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right)\hfill \\ +{\displaystyle \sum _{m=0}^{\infty }}\frac{q_2^m}{p_2^m}h\left(a,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right)]+\frac{(p_2+q_2)}{p_2(b-a)(d-c)}\hfill \\ \times {\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \sum _{m=0}^{\infty }}\frac{q_2^m}{p_2^m}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right),\hfill \end{array}$$

(28)

$$\begin{array}{c}\frac{(p_2+q_2)}{q_2(b-a)(d-c)}{\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \sum _{m=1}^{\infty }}\frac{q_2^m}{p_2^m}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right)\hfill \\ =-\frac{(p_2+q_2)h(b,d)}{q_2(b-a)(d-c)}+\frac{(p_2+q_2)}{q_2(b-a)(d-c)}{\displaystyle \sum _{m=0}^{\infty }}\frac{q_2^m}{p_2^m}h\left(b,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right)\hfill \\ +\frac{(p_2+q_2)}{q_2(b-a)(d-c)}{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \sum _{m=1}^{\infty }}\frac{q_2^m}{p_2^m}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right),\hfill \end{array}$$

(29)

$$\begin{array}{c}\frac{(p_1+q_1)(p_2+q_2)}{q_1{q}_2(b-a)(d-c)}{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \sum _{m=1}^{\infty }}\frac{q_1^n}{p_1^n}\frac{q_2^m}{p_2^m}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right)\hfill \\ =-\frac{(p_1+q_1)(p_2+q_2)h(b,d)}{q_1{q}_2(b-a)(d-c)}-\frac{(p_1+q_1)(p_2+q_2)}{q_1{q}_2(b-a)(d-c)}{\displaystyle \sum _{n=0}^{\infty }}\frac{q_1^n}{p_1^n}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,d\right)\hfill \\ -\frac{(p_1+q_1)(p_2+q_2)}{q_1{q}_2(b-a)(d-c)}{\displaystyle \sum _{m=0}^{\infty }}\frac{q_2^m}{p_2^m}h\left(b,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right)\hfill \\ +\frac{(p_1+q_1)(p_2+q_2)}{q_1{q}_2(b-a)(d-c)}{\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \sum _{m=0}^{\infty }}\frac{q_1^n}{p_1^n}\frac{q_2^m}{p_2^m}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right),\hfill \end{array}$$

(30)

$$\begin{array}{c}-\frac{(p_1+q_1)(p_2+q_2)}{p_2{q}_1(b-a)(d-c)}{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \sum _{m=0}^{\infty }}\frac{q_1^n}{p_1^n}\frac{q_2^m}{p_2^m}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right)\hfill \\ =\frac{(p_1+q_1)(p_2+q_2)}{p_2{q}_1(b-a)(d-c)}{\displaystyle \sum _{m=0}^{\infty }}\frac{q_2^m}{p_2^m}h\left(b,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right)\hfill \\ -\frac{(p_1+q_1)(p_2+q_2)}{p_2{q}_1(b-a)(d-c)}{\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \sum _{m=0}^{\infty }}\frac{q_1^n}{p_1^n}\frac{q_2^m}{p_2^m}h\left(\frac{q_1^n}{p_1^n}b+\left((1-\frac{q_1^n}{p_1^n}\right)a,\frac{q_2^m}{p_2^m}d+\left((1-\frac{q_2^m}{p_2^m}\right)c\right),\hfill \end{array}$$

(31)

$$\begin{array}{c}-\frac{(p_1+q_1)(p_2+q_2)}{p_1{q}_2(b-a)(d-c)}{\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \sum _{m=1}^{\infty }}\frac{q_1^n}{p_1^n}\frac{q_2^m}{p_2^m}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right)\hfill \\ =\frac{(p_1+q_1)(p_2+q_2)}{p_1{q}_2(b-a)(d-c)}{\displaystyle \sum _{n=0}^{\infty }}\frac{q_1^n}{p_1^n}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,d\right)\hfill \\ -\frac{(p_1+q_1)(p_2+q_2)}{p_1{q}_2(b-a)(d-c)}{\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \sum _{m=0}^{\infty }}\frac{q_1^n}{p_1^n}\frac{q_2^m}{p_2^m}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right).\hfill \end{array}$$

(32)

Using (20)–(32) and simplifying, we get:

$$\begin{array}{c}{\int }_0^1{\int }_0^1(1-(p_1+q_1)z)(1-(p_2+q_2)w)\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h((1-z)a+zb,(1-w)c+wd)}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_1,q_1}w}{}_0{d}_{p_1,q_1}z{}_0{d}_{p_2,q_2}w\hfill \\ =\frac{q_1{q}_{2}h(a,c)+p_2{q}_{1}h(a,d)+p_1{q}_{2}h(b,c)+p_1{p}_{2}h(b,d)}{q_1{q}_2(b-a)(d-c)}\hfill \end{array}$$

$$\begin{array}{c}-\frac{(p_1+q_1)(p_1-q_1)}{p_1{q}_1(b-a)(d-c)}{\displaystyle \sum _{n=0}^{\infty }}\frac{q_1^n}{p_1^n}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,c\right)\hfill \\ -\frac{p_2(p_1+q_1)(p_1-q_1)}{p_1{q}_1{q}_2(b-a)(d-c)}{\displaystyle \sum _{n=0}^{\infty }}\frac{q_1^n}{p_1^n}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,d\right)\hfill \\ -\frac{(p_2+q_2)(p_2-q_2)}{p_2{q}_2(b-a)(d-c)}{\displaystyle \sum _{m=0}^{\infty }}\frac{q_2^m}{p_2^m}h\left(a,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right)\hfill \\ -\frac{p_1(p_2+q_2)(p_2-q_2)}{p_2{q}_1{q}_2(b-a)(d-c)}{\displaystyle \sum _{m=0}^{\infty }}\frac{q_2^m}{p_2^m}h\left(b,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right)\hfill \\ +\frac{(p_1+q_1)(p_2+q_2)(p_1-q_1)(p_2-q_2)}{p_1{p}_2{q}_1{q}_2(b-a)(d-c)}\hfill \\ \times {\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \sum _{m=0}^{\infty }}\frac{q_1^n}{p_1^n}\frac{q_2^m}{p_2^m}h\left(\frac{q_1^n}{p_1^n}b+\left(1-\frac{q_1^n}{p_1^n}\right)a,\frac{q_2^m}{p_2^m}d+\left(1-\frac{q_2^m}{p_2^m}\right)c\right)\hfill \\ =\frac{q_1{q}_{2}h(a,c)+p_2{q}_{1}h(a,d)+p_1{q}_{2}h(b,c)+p_1{p}_{2}h(b,d)}{q_1{q}_2(b-a)(d-c)}\hfill \\ -\frac{(p_1+q_1)}{p_1{q}_1{(b-a)}^2(d-c)}{\int }_a^{p_{1}b+(1-p_1)a}h(s,c){}_a{d}_{p_1,q_1}s\hfill \\ -\frac{p_2(p_1+q_1)}{p_1{q}_1{q}_2{(b-a)}^2(d-c)}{\int }_a^{p_{1}b+(1-p_1)a}h(s,d){}_a{d}_{p_1,q_1}s\hfill \\ -\frac{(p_2+q_2)}{p_2{q}_2(b-a){(d-c)}^2}{\int }_c^{p_{2}d+(1-p_2)c}h(a,t){}_c{d}_{p_2,q_2}t\hfill \\ -\frac{p_1(p_2+q_2)}{p_2{q}_1{q}_2(b-a){(d-c)}^2}{\int }_c^{p_{2}d+(1-p_2)c}h(b,t){}_c{d}_{p_2,q_2}t\hfill \\ +\frac{(p_1+q_1)(p_2+q_2)}{p_1{p}_2{q}_1{q}_2{(b-a)}^2{(d-c)}^2}{\int }_a^{p_{1}b+(1-p_1)a}{\int }_c^{p_{2}d+(1-p_2)c}h(s,t){}_c{d}_{p_2,q_2}t{}_a{d}_{p_1,q_1}s.\hfill \end{array}$$

(33)

Multiplying both sides of (33) by $\frac{q_1{q}_2(b-a)(d-c)}{(p_1+q_1)(p_2+q_2)}$, we get the desired equality.

□

Remark 3.

• Setting $q_i\to 1^{-}$ and $p_i=1$ for $i=1,2$ in Lemma 1, then Lemma 1 reduces to Lemma 1 proved in [25].
• Furthermore, if $p_i=1$ for $i=1,2$ in Lemma 1, then Lemma 1 reduces to Lemma 2 proven in [22].

Now, we can introduce quantum integral inequalities for functions whose partial $(p_1{p}_2,q_1{q}_2)$-derivative on ${▵}^0$ satisfies the convexity on coordinates on $▵$.

Theorem 8.

Let $h:▵\subset {\mathbb{R}}^2\to \mathbb{R}$ be a function such that $(p_1{p}_2,q_1{q}_2)$-derivatives exist on ${▵}^0$ (the interior of $▵$) and $0<q_i<p_i\le 1$ for $i=1,2$. Moreover, if $\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}$ is continuous and integrable on $[a,b]\times [c,d]\subset {▵}^0$ and ${\left|\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}\right|}^r$ is a convex function on the coordinates on $[a,b]\times [c,d]$ for $r\ge 1$, then the given inequality holds:

$$\begin{array}{c}\left|{\mathsf{\Gamma }}_{(p_1{p}_2,q_1{q}_2)}(a,b,c,d)\left(h\right)\right|\le \frac{q_1{q}_2(b-a)(d-c)}{(p_1+q_1)(p_2+q_2)}{\left({\varphi }_{p_1,q_1}{\varphi }_{p_2,q_2}\right)}^{\frac{r-1}{r}}\hfill \\ \times ({\psi }_{p_1,q_1}{\psi }_{p_2,q_2}{\left|\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h(a,c)}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}\right|}^r+{\psi }_{p_1,q_1}{\Delta }_{p_2,q_2}{\left|\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h(a,d)}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}\right|}^r\hfill \\ +{\Delta }_{p_1,q_1}{\psi }_{p_2,q_2}{\left|\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h(b,c)}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}\right|}^r+{\Delta }_{p_1,q_1}{\Delta }_{p_2,q_2}{\left|\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h(b,d)}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}\right|}^r{)}^{\frac{1}{r}}.\hfill \end{array}$$

Proof. 

By applying Lemma 1, using the $(p_1{p}_2,q_1{q}_2)$-Hölder inequality, and the convexity of ${\left|\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}\right|}^r$ on the coordinates on $[a,b]\times [c,d]$, then we have:

$$\begin{array}{c}\left|{\mathsf{\Gamma }}_{(p_1{p}_2,q_1{q}_2)}(a,b,c,d)\left(h\right)\right|\le \frac{q_1{q}_2(b-a)(d-c)}{(p_1+q_1)(p_2+q_2)}\hfill \\ \times {\left({\int }_0^1{\int }_0^1\left|(1-(p_1+q_1)z)(1-(p_2+q_2)w)\right|{}_0{d}_{p_1,q_1}z{}_0{d}_{p_2,q_2}w\right)}^{\frac{r-1}{r}}\hfill \\ \times \left({\int }_0^1{\int }_0^1\left|(1-(p_1+q_1)z)(1-(p_2+q_2)w)\right|\right)\left[\right((1-z)(1-w){\left|\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h(a,c)}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}\right|}^r\hfill \\ +(1-z)w{\left|\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h(a,d)}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}\right|}^r(1-w)z{\left|\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h(b,c)}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}\right|}^r+wz{\left|\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h(b,d)}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}\right|}^r]{}_0{d}_{p_1,q_1}z{}_0{d}_{p_2,q_2}w{)}^{\frac{1}{r}}.\hfill \end{array}$$

By using Definition 3, we get:

$$\begin{array}{c}{\varphi }_{p,q}={\int }_0^1|(1-(p+q)z)|{}_0{d}_{p,q}z=\frac{2(p+q-1)}{{(p+q)}^2},\hfill \\ {\psi }_{p,q}={\int }_0^1(1-z)|(1-(p+q)z)|{}_0{d}_{p,q}z\hfill \\ =\frac{q[(p^3-2+2p)+(2p^2+2)q+p{q}^2]+2p^2-2p}{{(p+q)}^3(p^2+pq+q^2)},\hfill \\ {\Delta }_{p,q}={\int }_0^{1}z|(1-(p+q)z)|{}_0{d}_{p,q}z\hfill \\ =\frac{q[(5p^3-4p^2-2p+2)+(6p^2-4p-2)q+(5p-2)q^2+2q^3]}{{(p+q)}^3(p^2+pq+q^2)}\hfill \\ +\frac{2p^4-2p^3-2p^2+2p}{{(p+q)}^3(p^2+pq+q^2)}.\hfill \end{array}$$

We observe that,

$$\begin{array}{c}{\int }_0^1{\int }_0^1|(1-(p_1+q_1)z)(1-(p_2+q_2)w)|{}_0{d}_{p_1,q_1}z{}_0{d}_{p_2,q_2}w\hfill \\ =\left({\int }_0^1\left|(1-(p_1+q_1)z)\right|{}_0{d}_{p_1,q_1}z\right)\left({\int }_0^1\left|(1-(p_2+q_2)w)\right|{}_0{d}_{p_2,q_2}w\right)\hfill \\ ={\varphi }_{p_1,q_1}{\varphi }_{p_2,q_2},\hfill \\ {\int }_0^1{\int }_0^1(1-z)(1-w)\left|(1-(p_1+q_1)z)\right|\left|(1-(p_2+q_2)w)\right|{}_0{d}_{p_1,q_1}z{}_0{d}_{p_2,q_2}w)\hfill \\ =\left({\int }_0^1(1-z)\left|(1-(p_1+q_1)z)\right|{}_0{d}_{p_1,q_1}z\right)\left({\int }_0^1(1-w)\left|(1-(p_2+q_2)w)\right|{}_0{d}_{p_2,q_2}w\right)\hfill \\ ={\psi }_{p_1,q_1}{\psi }_{p_2,q_2},\hfill \\ {\int }_0^1{\int }_0^1(1-z)w\left|(1-(p_1+q_1)z)(1-(p_2+q_2)w)\right|{}_0{d}_{p_1,q_1}z{}_0{d}_{p_2,q_2}w\hfill \\ =\left({\int }_0^1(1-z)\left|(1-(p_1+q_1)z)\right|{}_0{d}_{p_1,q_1}z\right)\left({\int }_0^{1}w\left|(1-(p_2+q_2)w)\right|{}_0{d}_{p_2,q_2}w\right)\hfill \\ ={\psi }_{p_1,q_1}{\Delta }_{p_2,q_2},\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^1{\int }_0^{1}z(1-w)\left|(1-(p_1+q_1)z)(1-(p_2+q_2)w)\right|{}_0{d}_{p_1,q_1}z{}_0{d}_{p_2,q_2}w\hfill \\ =\left({\int }_0^{1}z\left|(1-(p_1+q_1)z)\right|{}_0{d}_{p_1,q_1}z\right)\left({\int }_0^1(1-w)\left|(1-(p_2+q_2)w)\right|{}_0{d}_{p_2,q_2}w\right)\hfill \\ ={\Delta }_{p_1,q_1}{\psi }_{p_2,q_2},\hfill \\ {\int }_0^1{\int }_0^{1}zw\left|(1-(p_1+q_1)z)(1-(p_2+q_2)w)\right|{}_0{d}_{p_1,q_1}z{}_0{d}_{p_2,q_2}w\hfill \\ =\left({\int }_0^{1}z\left|(1-(p_1+q_1)z)\right|{}_0{d}_{p_1,q_1}z\right)\left({\int }_0^{1}w\left|(1-(p_2+q_2)w)\right|{}_0{d}_{p_2,q_2}w\right)\hfill \\ ={\Delta }_{p_1,q_1}{\Delta }_{p_2,q_2}.\hfill \end{array}$$

Using the values of the above $(p_1{p}_2,q_1{q}_2)$-integrals, we get the desired inequality.  □

Theorem 9.

Let $h:▵\subset {\mathbb{R}}^2\to \mathbb{R}$ be a function such that $(p_1{p}_2,q_1{q}_2)$-derivatives exist on ${▵}^{\mathrm{o}}$ (the interior of $▵$) and $0<q_i<p_i\le 1$ for $i=1,2.$ Moreover, if $\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}$ is continuous and integrable on $[a,b]\times [c,d]\subset {▵}^{\mathrm{o}}$ and ${\left|\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}\right|}^{r_1}$ is a convex function on the coordinates on $[a,b]\times [c,d]$. If $r_1^{-1}+r_2^{-1}=1$ and $r_1>1$, then the given inequality holds:

$$\begin{array}{c}\left|{\mathsf{\Gamma }}_{(p_1{p}_2,q_1{q}_2)}(a,b,c,d)\left(h\right)\right|\le \frac{q_1{q}_2(b-a)(d-c)}{{\left((p_1+q_1)(p_2+q_2)\right)}^{1+\frac{1}{r_1}}}{\left(B_{p_1,q_1}\left(r_2\right)B_{p_2,q_2}\left(r_2\right)\right)}^{\frac{1}{r_2}}\hfill \\ \times ((p_1+q_1-1)(p_2+q_2-1){\left|\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h(a,c)}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}\right|}^{r_1}+(p_1+q_1-1){\left|\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h(a,d)}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}\right|}^{r_1}\hfill \\ +(p_2+q_2-1){\left|\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h(b,c)}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}\right|}^{r_1}+{\left|\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h(b,d)}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}\right|}^{r_1}{)}^{\frac{1}{r_1}},\hfill \end{array}$$

Proof. 

By applying Lemma 1, the $(p_1{p}_2,q_1{q}_2)$-Hölder inequality, and the convexity of ${\left|\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}\right|}^{r_1}$ on the coordinates on $[a,b]\times [c,d]$, then we have:

$$\begin{array}{c}\left|{\mathsf{\Gamma }}_{(p_1{p}_2,q_1{q}_2)}(a,b,c,d)\left(h\right)\right|\le \frac{q_1{q}_2(b-a)(d-c)}{(p_1+q_1)(p_2+q_2)}\hfill \\ \times {\left({\int }_0^1{\int }_0^1{\left|(1-(p_1+q_1)z)(1-(p_2+q_2)w)\right|}^{r_2}{}_0{d}_{p_1,q_1}z{}_0{d}_{p_2,q_2}w\right)}^{\frac{1}{r_2}}\hfill \\ \times \left({\int }_0^1{\int }_0^1\right[(1-z)(1-w){\left|\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h(a,c)}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}\right|}^{r_1}+(1-z)w{\left|\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h(a,d)}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}\right|}^{r_1}\hfill \\ +(1-w)z{\left|\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h(b,c)}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}\right|}^{r_1}+wz{\left|\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h(b,d)}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}\right|}^{r_1}]{}_0{d}_{p_1,q_1}z{}_0{d}_{p_2,q_2}w{)}^{\frac{1}{r_1}}.\hfill \end{array}$$

We calculate as,

$${\int }_0^1{\left|(1-(p_1+q_1)z)\right|}^{r_2}{}_0{d}_{p_1,q_1}z={\int }_0^{\frac{1}{p_1+q_1}}{(1-(p_1+q_1)z)}^{r_2}{}_0{d}_{p_1,q_1}z+{\int }_{\frac{1}{p_1+q_1}}^1{((p_1+q_1)z-1)}^{r_2}{}_0{d}_{p_1,q_1}z.$$

(34)

Considering the first $(p_1,q_1)$-integral from (34) and making use of the substitution $1-(p_1+q_1)z=w$, we obtain:

$$\begin{array}{c}{\int }_0^{\frac{1}{p_1+q_1}}{(1-(p_1+q_1)z)}^{r_2}{}_0{d}_{p_1,q_1}z=\frac{1}{p_1+q_1}{\int }_0^1{w}^{r_2}{}_0{d}_{p_1,q_1}w=\frac{p_1-q_1}{(p_1+q_1)(p_1^{r_2+1}-q_1^{r_2+1})}.\hfill \end{array}$$

Considering the second $(p_1,q_1)$-integral from (34) and making use of the substitution $(p_1+q_1)z-1=w$, we obtain:

$${\int }_{\frac{1}{p_1+q_1}}^1{((p_1+q_1)z-1)}^{r_2}{}_0{d}_{p_1,q_1}z=\frac{1}{p_1+q_1}{\int }_0^{p_1+q_1-1}{w}^{r_2}{}_0{d}_{p_1,q_1}w$$

$$=\frac{(p_1-q_1){(p_1+q_1-1)}^{r_2+1}}{(p_1+q_1)(p_1^{r_2+1}-q_1^{r_2+1})}.$$

After this calculation, we get:

$${\int }_0^1{\left|(1-(p_1+q_1)z)\right|}^{r_2}{}_0{d}_{p_1,q_1}z=\frac{(p_1-q_1)(1+{(p_1+q_1-1)}^{r_2+1})}{(p_1+q_1)(p_1^{r_2+1}-q_1^{r_2+1})}=B_{p_1,q_1}(r_2).$$

Similarly, one can have:

$${\int }_0^1{\left|(1-(p_2+q_2)w)\right|}^{r_2}{}_0{d}_{p_2,q_2}w=\frac{(p_2-q_2)(1+{(p_2+q_2-1)}^{r_2+1})}{(p_2+q_2)(p_2^{r_2+1}-q_2^{r_2+1})}=B_{p_2,q_2}(r_2).$$

Finally, we also have:

$$\begin{array}{c}{\int }_0^1(1-z){}_0{d}_{p_1,q_1}z=\frac{p_1+q_1-1}{p_1+q_1},{\int }_0^1(1-w){}_0{d}_{p_2,q_2}w=\frac{p_2+q_2-1}{p_2+q_2},{\int }_0^{1}z{}_0{d}_{p_1,q_1}z=\frac{1}{p_1+q_1},\hfill \\ {\int }_0^{1}w{}_0{d}_{p_2,q_2}w=\frac{1}{p_2+q_2}.\hfill \end{array}$$

□

Remark 4.

• Setting $q_i\to 1^{-}$ and $p_i=1$ for $i=1,2$ in Theorems 8 and 9, then Theorems 8 and 9 reduce to Theorems 4 and 5 proven in [25].
• Furthermore, if $p_i=1$ for $i=1,2$ in Theorems 8 and 9, then Theorems 8 and 9 reduce to Theorems 6 and 7 proven in [22].

Theorem 10.

Let $h:▵\subset {\mathbb{R}}^2\to \mathbb{R}$ be a function such that $(p_1{p}_2,q_1{q}_2)$-derivatives exist on ${▵}^{\mathrm{o}}$ (the interior of $▵$) and $0<q_i<p_i\le 1$ for $i=1,2$. Moreover, if $\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}$ is continuous and integrable on $[a,b]\times [c,d]\subset {▵}^{\mathrm{o}}$ and ${\left|\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}\right|}^r$ is quasi-convex on the coordinates on $[a,b]\times [c,d]$ for $r\ge 1$, then the inequality holds:

$$\begin{array}{c}\left|{\mathsf{\Gamma }}_{(p_1{p}_2,q_1{q}_2)}(a,b,c,d)\left(h\right)\right|\le \frac{q_1{q}_2(b-a)(d-c)}{(p_1+q_1)(p_2+q_2)}\left({\varphi }_{p_1,q_1}{\varphi }_{p_2,q_2}\right)\hfill \\ \times sup\left(\left|\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h(a,c)}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}\right|,\left|\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h(a,d)}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}\right|,\left|\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h(b,c)}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}\right|,\left|\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h(b,d)}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}\right|\right),\hfill \end{array}$$

where ${\varphi }_{p_1,q_1}$ and ${\varphi }_{p_2,q_2}$ are defined in Theorem 8.

Proof. 

By applying Lemma 1, using the $(p_1{p}_2,q_1{q}_2)$-Hölder inequality, and the quasi-convexity of ${\left|\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}\right|}^r$ on the coordinates on $[a,b]\times [c,d]$, then we have:

$$\begin{array}{c}\left|{\mathsf{\Gamma }}_{(p_1{p}_2,q_1{q}_2)}(a,b,c,d)\left(h\right)\right|\le \frac{q_1{q}_2(b-a)(d-c)}{(p_1+q_1)(p_2+q_2)}\hfill \\ \times {\left({\int }_0^1{\int }_0^1\left|(1-(p_1+q_1)z)(1-(p_2+q_2)w)\right|{}_0{d}_{p_1,q_1}z{}_0{d}_{p_2,q_2}w\right)}^{\frac{r-1}{r}}\hfill \\ \times ({\int }_0^1{\int }_0^1\left|(1-(p_1+q_1)z)(1-(p_2+q_2)w)\right|\hfill \\ \times {\left|\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h((1-z)a+zb,(1-w)c+wd)}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}\right|}^r{}_0{d}_{p_1,q_1}z{}_0{d}_{p_2,q_2}w{)}^{\frac{1}{r}}\hfill \end{array}$$

$$\begin{array}{c}=\frac{q_1{q}_2(b-a)(d-c)}{(p_1+q_1)(p_2+q_2)}\left({\int }_0^1{\int }_0^1\left|(1-(p_1+q_1)z)(1-(p_2+q_2)w)\right|{}_0{d}_{p_1,q_1}z{}_0{d}_{p_2,q_2}w\right)\hfill \\ \times {\left(sup\left\{{\left|\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h(a,c)}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}\right|}^r,{\left|\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h(a,d)}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}\right|}^r,{\left|\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h(b,c)}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}\right|}^r,{\left|\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h(b,d)}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}\right|}^r\right\}\right)}^{\frac{1}{r}}.\hfill \end{array}$$

(35)

Now, using the properties of the supremum and Definition 3, we get the desired result from (35).  □

Theorem 11.

Let $h:▵\subset {\mathbb{R}}^2\to \mathbb{R}$ be a function such that $(p_1{p}_2,q_1{q}_2)$-derivatives exist on ${▵}^{\mathrm{o}}$ (the interior of $▵$) and $0<q_i<p_i\le 1,$ for $i=1,2.$ Moreover, if $\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}$ is continuous and integrable on $[a,b]\times [c,d]$ and ${\left|\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}\right|}^{r_1}$ is a quasi-convex function on the coordinates on $[a,b]\times [c,d]$. If $r_1^{-1}+r_2^{-1}=1$ and $r_1>1$, then the given inequality holds:

$$\begin{array}{c}\left|{\mathsf{\Gamma }}_{(p_1{p}_2,q_1{q}_2)}(a,b,c,d)\left(h\right)\right|\le \frac{q_1{q}_2(b-a)(d-c)}{(p_1+q_1)(p_2+q_2)}{\left(B_{p_1,q_1}\left(r_2\right)B_{p_2,q_2}\left(r_2\right)\right)}^{\frac{1}{r_2}}\hfill \\ \times sup\left\{\left|\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h(a,c)}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}\right|,\left|\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h(a,d)}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}\right|,\left|\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h(b,c)}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}\right|,\left|\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h(b,d)}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}\right|\right\},\hfill \end{array}$$

where $B_{p_1,q_1}\left(r_2\right)$ and $B_{p_2,q_2}\left(r_2\right)$ are defined in Theorem 9.

Proof. 

By applying Lemma 1, using the $(p_1{p}_2,q_1{q}_2)$- Hölder inequality, and the quasi-convexity of ${\left|\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}\right|}^{r_1}$ on the coordinates on $[a,b]\times [c,d]$, then we have:

$$\begin{array}{c}\left|{\mathsf{\Gamma }}_{(p_1{p}_2,q_1{q}_2)}(a,b,c,d)\left(h\right)\right|\le \frac{q_1{q}_2(b-a)(d-c)}{(p_1+q_1)(p_2+q_2)}\hfill \\ \times {\left({\int }_0^1{\int }_0^1{\left|(1-(p_1+q_1)z)(1-(p_2+q_2)w)\right|}_1^r{}_0{d}_{p_1,q_1}z{}_0{d}_{p_2,q_2}w\right)}^{\frac{1}{r_1}}\hfill \\ \times {\left({\int }_0^1{\int }_0^1{\left|\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h((1-z)a+zb,(1-w)c+wd)}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}\right|}^{r_2}{}_0{d}_{p_1,q_1}z{}_0{d}_{p_2,q_2}w\right)}^{\frac{1}{r_2}}\hfill \end{array}$$

$$\begin{array}{c}=\frac{q_1{q}_2(b-a)(d-c)}{(p_1+q_1)(p_2+q_2)}{\left({\int }_0^1{\int }_0^1{\left|(1-(p_1+q_1)z)(1-(p_2+q_2)w)\right|}^{r_1}{}_0{d}_{p_1,q_1}z{}_0{d}_{p_2,q_2}w\right)}^{\frac{1}{r_1}}\hfill \\ \times (sup\{{\left|\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h(a,c)}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}\right|}^{r_2},{\left|\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h(a,d)}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}\right|}^{r_2},\hfill \\ {\left|\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h(b,c)}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}\right|}^{r_2},{\left|\frac{{}_{a,c}{\partial }_{p_1{p}_2,q_1{q}_2}^{2}h(b,d)}{{}_a{\partial }_{p_1,q_1}z{}_c{\partial }_{p_2,q_2}w}\right|}^{r_2}\}{)}^{\frac{1}{r_2}}.\hfill \end{array}$$

Now, using the properties of the supremum and Definition 3, we get the desired result.  □

Remark 5.

• Setting $q_i\to 1^{-}$ and $p_i=1$ for $i=1,2$ in Theorems 10 and 11, we get the corresponding results of the classical calculus of the functions of two variables.
• Furthermore, if $p_i=1$ for $i=1,2$ in Theorems 10 and 11, then Theorems 10 and 11 reduce to Theorems 8 and 9 proven in [22].

3. Conclusions

In this paper, we acquired some new results of the Hermit-Hadamard-type inequalities using the $(p,q)$-calculus of the functions of two variables that extend the Hermit-Hadamard-type inequality proven in [22]. In this work, we generalized the results given in [22] and contained the results of the Hermit-Hadamard-type inequalities of the functions of two variables using the usual calculus as special cases when $p=1$ and $q\to 1^{-}$. The concepts and techniques discussed in this article are excepted to stimulate further research in the field of the $(p,q)$-calculus of the functions of two variables.