Some Hermite–Hadamard-mercer Inequalities on the Coordinates on Post Quantum

In this paper, we develop new Hermite-Hadamard-Mercer type inequalities on coordinates via post-quantum calculus, also known as (p, q) - calculus. By introducing novel (p1, p2, q1, q2)-differentiable and (p1, p2, q1, q2)-integrable functions, we generalize classical results and extend previous inequalities under the setting of coordinate convexity. Several new identities are derived, which naturally reduce to known results when specific parameters are chosen. Numerical examples and visualizations are also provided to illustrate the utility of our results.

1. Introduction

Mathematical inequalities are fundamental in both pure and applied mathematics, offering essential tools in areas ranging from analysis to physics. Among these, the Hermite-Hadamard inequality for convex functions plays a pivotal role, which is defined as follows:

$f\left(\frac{a+b}{2}\right)\le \frac{1}{b-a}\underset{a}{\overset{b}{{\displaystyle \int }}}f\left(x\right)dx\le \frac{f\left(a\right)+f\left(b\right)}{2}.$

The Hermite-Hadamard inequality and a variety of refinements of Hermite-Hadamard inequality have been extensively studied by many researchers, including those involving Jensen and Mercer-type refinements.

Jensen inequality has been caught attention of many researshers, and many articles related to different versions of this inequality have been found in the literature. Jensen’s inequality can be given as follows:

Let $0<x_1\le x_2\le \mathrm{...}\le x_n$ and $\mu =\left({\mu }_1,{\mu }_2,\mathrm{...},{\mu }_n\right)$ be non-negative weights such that ${{\displaystyle {\sum }_{k=1}^n\mu }}_k=11,$ =1 if f is convex function on the interval [a, b], then

$f\left({\displaystyle \sum }_{k=1}^n{u}_k{x}_k\right)\le {\displaystyle \sum }_{k=1}^n{u}_{k}f\left(x_k\right),$

where $x_k\in \left[a,b\right]$ every and all ${\mu }_k\in \left[0,1\right].$

A new variant of Jensen inequality thast has been established by Mercer can be presented as follows:

In 2003, Mercer [1] proved another version of Jensen inequality, which is called Jensen-Mercer inequality and stated as follows.

Theorem 1.1

For a convex mapping $f:\left[a,b\right]\to ℝ,$ for following inequality holds for each $x_j\in \left[a,b\right]:$

$f\left(a+b-{\displaystyle \sum }_{j=1}^n{u}_j{x}_j\right)\le f\left(a\right)+f\left(b\right)-{\displaystyle \sum }_{j=1}^n{u}_{j}f\left(x_j\right),,$

where $u_j\in \left[0,1\right]$ and ${\displaystyle {\sum }_{j=1}^n{u}_j=11.}$

In 2013, Kian, et al. [2] used this new Jensen inequality and established the following new versions of Hermite-Hadamard inequality:

Theorem 1.2

For a convex mapping $f:\left[a,b\right]\to ℝ,$ for following inequality holds for each $x,y\in \left[a,b\right]$ and $x<y:$

$f\left(a+b-\frac{x+y}{2}\right)\le f\left(a\right)+f\left(b\right)-\frac{1}{y-x}\underset{x}{\overset{y}{{\displaystyle \int }}}f\left(u\right)du\le f\left(a\right)+f\left(b\right)-f\left(\frac{x+y}{2}\right)$

and

$f\left(a+b-\frac{x+y}{2}\right)\le \frac{1}{y-x}\underset{a+b-y}{\overset{a+b-x}{{\displaystyle \int }}}f\left(u\right)du$

$\le \frac{f\left(a+b-x\right)+f\left(a+b-y\right)}{2}$

$\le f\left(a\right)+f\left(b\right)-f\left(\frac{x+y}{2}\right).$

The ordinary calculus of Newton and Leibniz is well known to be investigated extensively and intensively to produce a large number of related formulas and properties as well as applications in a variety of fields ranging from natural sciences to social sciences.

Recent studies have explored these inequalities using quantum and post-quantum calculus, which extend traditional calculus by means of q and (p,q)-analogues.

Quantum calculus, which is often known as q-calculus or calculus without limits, is based on finite difference. In quantum calculus we obtain q-analogues of mathematical objects which can be recaptured by taking. The history of q-calculus can be track to Euler, who first introduced q-calculus in the track of Newton’s work on infinite series.

Then, in 1910, F. H. Jackson presented a systematic study of q-calculus and defined the q-defined integral, which is known as the q-Jackson integral. In recent years, the interest in q-calculus been arising due to high demand of mathematics in this field. The q-calculus numerous applications in various fields of mathematics and other areas such as combinatory, dynamical systems, fractals, number theory, orthogonal polynomials, special functions, mechanics and also for scientific problems in some applied areas.

In 2013, Tariboon and Ntouyas defined new q-derivatives and q-integrals of a continuous function on a finite interval. These definitions have been studied in various inequalities, for example, Hermiter-Hadamard inequalities, Ostrowski inequalities, Fejér inequalities, Simpson inequalities and Newton inequalities, and the references cited therein [3-12].

Along with the development of the theory and application of q-calculus, the theory of q-calculus based on two parameters (p-q)-integral has also presented and received more attention during the last few decades.

Recent, Ali, et al. [3] and Sitthiwirattham, et al. [13] ued new techniques to prove the following two different and new versions of Hermite-Hadamard type inequalitites:

Theorem 1.3

For a convex mapping $f:\left[a,b\right]\to ℝ,$ for following inequality holds:

$f\left(\frac{a+b}{2}\right)\le \frac{1}{b-a}\left[\underset{a}{\overset{\frac{a+b}{2}}{{\displaystyle \int }}}f{\left(x\right)}^{\frac{a+b}{2}}{d}_{q}x+\underset{\frac{a+b}{2}}{\overset{b}{{\displaystyle \int }}}f{\left(x\right)}_{\frac{a+b}{2}}{d}_{q}x\right]\le \frac{f\left(a\right)+f\left(b\right)}{2},$

$f\left(\frac{a+b}{2}\right)\le \frac{1}{b-a}\left[\underset{a}{\overset{\frac{a+b}{2}}{{\displaystyle \int }}}f{\left(x\right)}_a{d}_{q}x+\underset{\frac{a+b}{2}}{\overset{b}{{\displaystyle \int }}}f{\left(x\right)}^b{d}_{q}x\right]\le \frac{f\left(a\right)+f\left(b\right)}{2}.$

Remark

By setting the limit as q 1- in above Theorem, we recapture the traditional Hermite-Hadamard inequality.

Post-quantum calculus, also called (p,q)-calculus, is another generalization of q-calculus on the interval . The (p,q)-calculus consists of two-parameter quantum calculus (p and q-numbers) which are independent. The (p,q)- calculus was first introduced by Chakrabarti and Jagannathan in 1991. Then, the new (p,q)-deravative and (p,q)-integral of a continuous function on finite interval were by Tunc and Gov in 2016. In (p,q)-calculus, we obtain q-calculus formula for case p=1, and then get classical formula for case of . Base on (p,q)-calculus, many literatures have been published by many researchers, see [14-25] for more details and the references cited therein.

In this paper, we continue in this direction by developing Hermite-Hadamard-Mercer type inequalities in the framework of post-quantum calculus on coordinates. Our main contributions include novel identities for functions of two variables involving mixed partial (p₁; p₂; q₁; q₂) - derivatives and integrals.

2. Notation and preliminaries

The following is the brief introduction of the research of post-quantum calculus. Throughout this topic, we let p₁; p₂; q₁; q₂ be constants with $0<q_1<p_1\le 1$ and $0<q_2<p_2\le 1$ with $\left[a,b\right]\subseteq ℝ.$

Then, for any real number, the (p₁, q₁) - analogue and (p₂, q₂) - analogue of m,n is defined by

${\left[m\right]}_{p_1,q_1}=\frac{p_1^m-q_1^m}{p_1-q_1}=p_1^{m-1}+p_1^{m-2}{q}_1+\mathrm{...}+p_1{q}_1^{m-2}+q_1^{m-1}$

and

${\left[n\right]}_{p_2,q_2}=\frac{p_2^n-q_2^n}{p2-q_2}=p_2^{n-1}+p_2^{n-2}{q}_2+\mathrm{...}+p_2{q}_2^{n-2}+q_2^{n-1},$

which is generalization of the q_1-analogue such that

${\left[m\right]}_{q_1}=\frac{1-q_1^m}{1-q_1}=1+q_1+\mathrm{...}+q_1^{m-2}+q_1^{m-1}$

and

${\left[n\right]}_{q_2}=\frac{1-q_2^n}{1-q_2}=1+q_2+\mathrm{...}+q_2^{n-2}+q_2^{n-1}.$

Definition 2.1 [26]

If is a continuous function, then (p, q) - derivative of the function on [a, b] by

$D_{p,q}f\left(x\right)=\frac{f\left(px\right)-f\left(qx\right)}{p-q}, x\ne 0$

with $0<q<p\le 1.$

Definition 2.2 [27]

If $f:\left[a,b\right]\to ℝ$ is a continuous function, then (p, q)_a - derivative of the function at x is defined by

${}_a{D}_{p,q}f\left(x\right)=\frac{f\left(px+\left(1-p\right)a\right)-f\left(qx+\left(1-q\right)a\right)}{\left(p-q\right)\left(x-a\right)}, x\ne a$

with $0<q<p\le 1.$

For x = a, we state ${}_a{D}_{p,q}f\left(a\right)=\underset{x\to a}{\text{lim}} { }_a{D}_{p,q}f\left(x\right)$ if it exists and it is finite.

Definiiton 2.3 [16]

If $f:\left[a,b\right]\to ℝ$ is a continuous function, then (p, q)^b - derivative of the function at x is defined by

${}^b{D}_{p,q}f\left(x\right)=\frac{f\left(qx+\left(1-q\right)b\right)-f\left(px+\left(1-p\right)b\right)}{\left(p-q\right)\left(b-x\right)}, x\ne b$

with $0<q<p\le 1.$

For x = b, we state ${}^b{D}_{p,q}f\left(a\right)=\underset{x\to b}{\text{lim}} { }^b{D}_{p,q}f\left(x\right)$ if it exists and it is finite.

Definition 2.4 [26]

If $f:\left[a,b\right]\to ℝ$ is a continuous function and $0<a<b,$ then the (p, q) - integral is defined by

${\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)d_{p,q}x=\left(p-q\right)\left(b-a\right){\displaystyle {\sum }_{k=0}^{\infty }f}\left(\frac{q^k}{p^{k+1}}b+\left(1-\frac{q^k}{p^{k+1}}\right)a\right)}$

with $0<q<p\le 1.$

Definition 2.5 [27]

If $f:\left[a,b\right]\to ℝ$ is a continuous function and $0<a<b,$ then the (p, q)_a - integral is defined by

${\displaystyle \underset{a}{\overset{x}{\int }}f{\left(x\right)}_a{d}_{p,q}x=\left(p-q\right)\left(x-a\right){\displaystyle {\sum }_{k=0}^{\infty }f}\left(\frac{q^k}{p^{k+1}}x+\left(1-\frac{q^k}{p^{k+1}}\right)a\right)}$

with $0<q<p\le 1.$

Definition 2.6 [16]

If $f:\left[a,b\right]\to ℝ$ is a continuous function and $0<a<b,$ then the (p, q)^b - integral is defined by

${\displaystyle \underset{x}{\overset{b}{\int }}f{\left(x\right)}^b{d}_{p,q}x=\left(p-q\right)\left(b-x\right){\displaystyle {\sum }_{k=0}^{\infty }f}\left(\frac{q^k}{p^{k+1}}b+\left(1-\frac{q^k}{p^{k+1}}\right)x\right)}$

with $0<q<p\le 1.$

In [14], Ali et al. established the Hermite-Hadamard type inequlities on post quantum calculus.

Theorem 2.7

If $f:\left[a,b\right]\to ℝ$ is a convex differentiable function on [a,b], then the (p, q)^b - Hermite-Hadamard inequalities are as follows:

$f\left(\frac{pa+qb}{p+q}\right)\le \frac{1}{p\left(b-a\right)}{\displaystyle \underset{pa+\left(1-p\right)b}{\overset{b}{\int }}f{\left(x\right)}^b{d}_{p,q}x\le \frac{qf\left(a\right)+pf\left(b\right)}{p+q}}.$

In [27], Tunc and Gov extend the Holdër inequalies ion post-quantum calculus.

Theorem 2.8

If $f:\left[a,b\right]\to ℝ$ is a continuous function and r, s > 0 with $\frac{1}{r}+\frac{1}{s}=1,$ then

${\displaystyle \underset{a}{\overset{b}{\int }}{\left|f\left(x\right)g\left(x\right)\right|}_a{d}_{p,q}x\le {\left({\displaystyle \underset{a}{\overset{b}{\int }}}{\left|f\left(x\right)\right|}^r{}_a{d}_{p,q}x\right)}^{\frac{1}{r}}{\left({\displaystyle \underset{a}{\overset{b}{\int }}}{\left|g\left(x\right)\right|}^s{}_a{d}_{p,q}x\right)}^{\frac{1}{s}}.}$

In [28], H. Kalsoom, et al. introduced the following notions of post-quantum partial derivatives:

Definition 2.9

Suppose that $f:\left[a,b\right]\times \left[c,d\right]\to ℝ$ is a continuous function of two variables. Then the derivatives are given by

$\frac{{}_{a,c}{\partial }_{p_1,p_2,q_1,q_2}f\left(x,y\right)}{{}_a{\partial }_{p_1,q_1}{x}_{ c}{\partial }_{p_2,q_2}y}=\frac{1}{\begin{array}{l}\left(p_1-q_1\right)\left(p_2-q_2\right)\left(x-a\right)\left(y-c\right)\\ [f\left(q_{1}x+\left(1-q_1\right)a,q_{2}y+\left(1-q_2\right)c\right)\\ -f\left(q_{1}x+\left(1-q_1\right)a,p_{2}y+\left(1-p_2\right)c\right)\\ -f\left(p_{1}x+\left(1-p_1\right)a,q_{2}y+\left(1-q_2\right)c\right)\\ +f\left(p_{1}x+\left(1-p_1\right)a,p_{2}y+\left(1-p_2\right)c\right)],\end{array}}\times$

for $x\ne a,y\ne c.$

$\frac{{}_c^b{\partial }_{p_1,p_2,q_1,q_2}f\left(x,y\right)}{{}^b{\partial }_{p_1,q_1}{x}_{ c}{\partial }_{p_2,q_2}y}=\frac{1}{\begin{array}{l}\left(p_1-q_1\right)\left(p_2-q_2\right)\left(b-x\right)\left(y-c\right)\\ [f\left(q_{1}x+\left(1-q_1\right)b,p_{2}y+\left(1-p_2\right)c\right)\\ -f\left(p_{1}x+\left(1-p_1\right)b,p_{2}y+\left(1-p_2\right)c\right)\\ -f\left(q_{1}x+\left(1-q_1\right)b,q_{2}y+\left(1-q_2\right)c\right)\\ +f\left(p_{1}x+\left(1-p_1\right)a,q_{2}y+\left(1-q_2\right)c\right)],\end{array}}\times$

for $x\ne b,y\ne c.$

$\frac{{}_a^d{\partial }_{p_1,p_2,q_1,q_2}f\left(x,y\right)}{{}_a{\partial }_{p_1,q_1}{x}_{ }^d{\partial }_{p_2,q_2}y}=\frac{1}{\begin{array}{l}\left(p_1-q_1\right)\left(p_2-q_2\right)\left(x-a\right)\left(d-y\right)\\ [f\left(p_{1}x+\left(1-p_1\right)a,q_{2}y+\left(1-q_2\right)d\right)\\ -f\left(q_{1}x+\left(1-q_1\right)a,q_{2}y+\left(1-q_2\right)d\right)\\ -f\left(p_{1}x+\left(1-p_1\right)a,p_{2}y+\left(1-p_2\right)d\right)\\ +f\left(q_{1}x+\left(1-q_1\right)a,p_{2}y+\left(1-p_2\right)d\right)],\end{array}}\times$

for $x\ne a,y\ne d,$

and

$\frac{{}^{b,d}{\partial }_{p_1,p_2,q_1,q_2}f\left(x,y\right)}{{}^b{\partial }_{p_1,q_1}{x}_{ }^d{\partial }_{p_2,q_2}y}=\frac{1}{\begin{array}{l}\left(p_1-q_1\right)\left(p_2-q_2\right)\left(b-x\right)\left(d-y\right)\\ [f\left(q_{1}x+\left(1-q_1\right)b,q_{2}y+\left(1-q_2\right)d\right)\\ -f\left(p_{1}x+\left(1-p_1\right)b,q_{2}y+\left(1-q_2\right)d\right)\\ -f\left(q_{1}x+\left(1-q_1\right)b,p_{2}y+\left(1-p_2\right)d\right)\\ +f\left(p_{1}x+\left(1-p_1\right)b,p_{2}y+\left(1-p_2\right)d\right)],\end{array}}\times$

for $x\ne b,y\ne d.$

Definition 2.10 [28]

Suppose that $f:\left[a,b\right]\times \left[c,d\right]\to ℝ$ is a continuous function of two variables. Then the definite (p₁, p₂, q₁, q₂) - integral are given by

${\displaystyle \underset{a}{\overset{x}{\int }}\underset{c}{\overset{y}{{\displaystyle \int }}}f{\left(t,s\right)}_c{d}_{p_2,q_2}{s}_a{d}_{p_1,q_1}t}$

$=\left(p_1-q_1\right)\left(p_2-q_2\right)\left(x-a\right)\left(y-c\right)$

$\times {\displaystyle {\sum }_{n=0}^{\infty }{\displaystyle {\sum }_{m=0}^{\infty }\frac{q_1^n}{p_1^{n+1}}\frac{q_2^m}{p_2^{m+1}}f\left(\frac{q_1^n}{p_1^{n+1}}x+\left(1-\frac{q_1^n}{p_1^{n+1}}\right)a,\frac{q_2^m}{p_2^{m+1}}y+\left(1-\frac{q_2^m}{p_2^{m+1}}\right)c\right),}}$

for $\left(x,y\right)\in \left[a,b\right]\times \left[c,d\right].$

Suppose that $f:\left[a,b\right]\times \left[c,d\right]\to ℝ$ is a continuous function of two variables. Then the definite (p₁, p₂, q₁, q₂) - integral are given by

${\displaystyle \underset{x}{\overset{b}{\int }}\underset{c}{\overset{y}{{\displaystyle \int }}}f{\left(t,s\right)}_c{d}_{p_2,q_2}{s}^b{d}_{p_1,q_1}t}$

$=\left(p_1-q_1\right)\left(p_2-q_2\right)\left(b-x\right)\left(y-c\right)$

$\times {\displaystyle {\sum }_{n=0}^{\infty }{\displaystyle {\sum }_{n=0}^{\infty }\frac{q_1^n}{p_1^{n+1}}\frac{q_2^m}{p_2^{m+1}}f\left(\frac{q_1^n}{p_1^{n+1}}x+\left(1-\frac{q_1^n}{p_1^{n+1}}\right)b,\frac{q_2^m}{p_2^{m+1}}y+\left(1-\frac{q_2^m}{p_2^{m+1}}\right)c\right),}}$

for $\left(x,y\right)\in \left[a,b\right]\times \left[c,d\right].$

Suppose that $f:\left[a,b\right]\times \left[c,d\right]\to ℝ$ is a continuous function of two variables. Then the definite (p₁, p₂, q₁, q₂) - integral are given by

${\displaystyle \underset{a}{\overset{x}{\int }}\underset{y}{\overset{d}{{\displaystyle \int }}}f{\left(t,s\right)}_c{d}_{p_2,q_2}{s}_a{d}_{p_1,q_1}t}$

$=\left(p_1-q_1\right)\left(p_2-q_2\right)\left(x-a\right)\left(d-y\right)$

$\times {\displaystyle {\sum }_{n=0}^{\infty }{\displaystyle {\sum }_{n=0}^{\infty }\frac{q_1^n}{p_1^{n+1}}\frac{q_2^m}{p_2^{m+1}}f\left(\frac{q_1^n}{p_1^{n+1}}x+\left(1-\frac{q_1^n}{p_1^{n+1}}\right)a,\frac{q_2^m}{p_2^{m+1}}y+\left(1-\frac{q_2^m}{p_2^{m+1}}\right)d\right),}}$

for $\left(x,y\right)\in \left[a,b\right]\times \left[c,d\right]$

and

Suppose that $f:\left[a,b\right]\times \left[c,d\right]\to ℝ$ is a continuous function of two variables. Then the definite (p₁, p₂, q₁, q₂) -integral are given by

${\displaystyle \underset{x}{\overset{b}{\int }}\underset{y}{\overset{d}{{\displaystyle \int }}}f{\left(t,s\right)}_c{d}_{p_2,q_2}{s}_a{d}_{p_1,q_1}t}$

$=\left(p_1-q_1\right)\left(p_2-q_2\right)\left(b-x\right)\left(d-y\right)$

$\times {\displaystyle {\sum }_{n=0}^{\infty }{\displaystyle {\sum }_{m=0}^{\infty }\frac{q_1^n}{p_1^{n+1}}\frac{q_2^m}{p_2^{m+1}}f\left(\frac{q_1^n}{p_1^{n+1}}x+\left(1-\frac{q_1^n}{p_1^{n+1}}\right)b,\frac{q_2^m}{p_2^{m+1}}y+\left(1-\frac{q_2^m}{p_2^{m+1}}\right)d\right),}}$

for $\left(x,y\right)\in \left[a,b\right]\times \left[c,d\right].$

Lemma 2.11 [25] (p₁, p₂, q₁, q₂) - Hölder’s inequality for functions of two variables

Let f, g be (p₁, p₂, q₁, q₂) - integrable functions on $\left[a,b\right]\times \left[c,d\right]$ and $\frac{1}{r}+\frac{1}{s}=1$ with r, s > 1. Then, we have

${\displaystyle \underset{a}{\overset{x}{\int }}\underset{c}{\overset{y}{{\displaystyle \int }}}{\left|f\left(t,s\right)g\left(t,s\right)\right|}_c{d}_{p_2,q_2}{s}_a{d}_{p_1,q_1}t}$

$\le {\left({\displaystyle \underset{a}{\overset{x}{\int }}\underset{c}{\overset{y}{{\displaystyle \int }}}{\left|f\left(t,s\right)\right|}^r{}_c{d}_{p_2,q_2}{s}_a{d}_{p_1,q_1}t}\right)}^{\frac{1}{r}}$

$\times {\left({\displaystyle \underset{a}{\overset{x}{\int }}\underset{c}{\overset{y}{{\displaystyle \int }}}{\left|g\left(t,s\right)\right|}^s{}_c{d}_{p_2,q_2}{s}_a{d}_{p_1,q_1}t}\right)}^{\frac{1}{s}}$

for all $\left(x,y\right)\in \left[a,b\right]\times \left[c,d\right].$

Lemma 2.12 [25] (p₁, p₂, q₁, q₂) - power mean inequality for functions of two variables

Let f, g be (p₁, p₂, q₁, q₂) - integrable functions on $\left[a,b\right]\times \left[c,d\right]$ and   1. Then, we have

${\displaystyle \underset{a}{\overset{x}{\int }}\underset{c}{\overset{y}{{\displaystyle \int }}}{\left|f\left(t,s\right)g\left(t,s\right)\right|}_c{d}_{p_2,q_2}{s}_a{d}_{p_1,q_1}t}$

$\le {\left({\displaystyle \underset{a}{\overset{x}{\int }}\underset{c}{\overset{y}{{\displaystyle \int }}}{\left|f\left(t,s\right)\right|}_c{d}_{p_2,q_2}{s}_a{d}_{p_1,q_1}t}\right)}^{1-\frac{1}{\alpha }}$

$\le {\left({\displaystyle \underset{a}{\overset{x}{\int }}\underset{c}{\overset{y}{{\displaystyle \int }}}{\left|g\left(t,s\right)\right|}^{\alpha }{}_c{d}_{p_2,q_2}{s}_a{d}_{p_1,q_1}t}\right)}^{\frac{1}{\alpha }}$

for all $\left(x,y\right)\in \left[a,b\right]\times \left[c,d\right].$

3. Main results

Lemma 3.1

Let $f:\left[a,b\right]\times \left[c,d\right]\to ℝ$ be a twice partially (p₁, p₂, q₁, q₂) - differentiable function on $\left(a,b\right)\times \left(c,d\right).$ If $\frac{{}_{a,c}{\partial }_{p_1,p_2,q_1,q_2}f\left(x,y\right)}{{}_a{\partial }_{p_1,q_1}{x}_{ c}{\partial }_{p_2,q_2}y},\frac{{}_c^b{\partial }_{p_1,p_2,q_1,q_2}f\left(x,y\right)}{{}^b{\partial }_{p_1,q_1}{x}_{ c}{\partial }_{p_2,q_2}y},\frac{{}_a^d{\partial }_{p_1,p_2,q_1,q_2}f\left(x,y\right)}{{}_a{\partial }_{p_1,q_1}{x}_{ }^d{\partial }_{p_2,q_2}y},\frac{{}^{b,d}{\partial }_{p_1,p_2,q_1,q_2}f\left(x,y\right)}{{}^b{\partial }_{p_1,q_1}{x}_{ }^d{\partial }_{p_2,q_2}y}$ are continuous and (p₁, p₂, q₁, q₂) -integrable on $\left[a,b\right]\times \left[c,d\right].$ Then we have

$\frac{1}{p_1{p}_2{q}_1{q}_2{\left(n-m\right)}^2{\left(l-k\right)}^2}$

$\times [\underset{a+b-n}{\overset{a+b-\left[\left(1-p_1\right)n+p_{1}m\right]}{{\displaystyle \int }}}\underset{c+d-l}{\overset{c+d-\left[\left(1-p_2\right)l+p_{2}k\right]}{{\displaystyle \int }}}f\left(t,s\right)d_{p_2,q_2}s{d}_{p_1,q_1}t$

$+\underset{a+b-\left[\left(1-p_1\right)m+p_{1}n\right]}{\overset{a+b-m}{{\displaystyle \int }}}\underset{c+d-l}{\overset{c+d-\left[\left(1-p_2\right)l+p_{2}k\right]}{{\displaystyle \int }}}f\left(t,s\right)d_{p_2,q_2}s{d}_{p_1,q_1}t$

$+\underset{a+b-n}{\overset{a+b-\left[\left(1-p_1\right)n+p_{1}m\right]}{{\displaystyle \int }}}\underset{c+d-\left[\left(1-p_2\right)k+p_{2}l\right]}{\overset{c+d-k}{{\displaystyle \int }}}f\left(t,s\right)d_{p_2,q_2}s{d}_{p_1,q_1}t$

$\underset{a+b-\left[\left(1-p_1\right)m+p_{1}n\right]}{\overset{a+b-m}{{\displaystyle \int }}}\underset{c+d-\left[\left(1-p_2\right)k+p_{2}l\right]}{\overset{c+d-k}{{\displaystyle \int }}}f\left(t,s\right)d_{p_2,q_2}s{d}_{p_1,q_1}t]$

$-\frac{1}{p_2\left(n-m\right){\left(l-k\right)}^2}[\underset{c+d-l}{\overset{c+d-\left[\left(1-p_2\right)l+p_{2}k\right]}{{\displaystyle \int }}}f\left(a+b-m,s\right)d_{p_2,q_2}s$

$+\underset{c+d-l}{\overset{c+d-\left[\left(1-p_2\right)l+p_{2}k\right]}{{\displaystyle \int }}}f\left(a+b-n,s\right)d_{p_2,q_2}s$

$+\underset{c+d-\left[\left(1-p_2\right)k+p_{2}l\right]}{\overset{c+d-k}{{\displaystyle \int }}}f\left(a+b-m,s\right)d_{p_2,q_2}s$

$\underset{c+d-\left[\left(1-p_2\right)k+p_{2}l\right]}{\overset{c+d-k}{{\displaystyle \int }}}f\left(a+b-n,s\right)d_{p_2,q_2}s]$

$-\frac{1}{p_1{\left(n-m\right)}^2\left(l-k\right)}[\underset{a+b-n}{\overset{a+b-\left[\left(1-p_1\right)n+p_{1}m\right]}{{\displaystyle \int }}}f\left(t,c+d-k\right)d_{p_2,q_2}s{d}_{p_1,q_1}t$

$+\underset{a+b-\left[\left(1-p_1\right)m+p_{1}n\right]}{\overset{a+b-m}{{\displaystyle \int }}}f\left(t,c+d-k\right)d_{p_2,q_2}s{d}_{p_1,q_1}t$

$+\underset{a+b-n}{\overset{a+b-\left[\left(1-p_1\right)n+p_{1}m\right]}{{\displaystyle \int }}}f\left(t,c+d-l\right)d_{p_2,q_2}s{d}_{p_1,q_1}t$

$\underset{a+b-\left[\left(1-p_1\right)m+p_{1}n\right]}{\overset{a+b-m}{{\displaystyle \int }}}f\left(t,c+d-l\right)d_{p_2,q_2}s{d}_{p_1,q_1}t]$

$+\frac{1}{\left(n-m\right)\left(l-k\right)}[\left(a+b-m,c+d-k\right)+\left(a+b-n,c+d-k\right)$

$+\left(a+b-m,c+d-l\right)+\left(a+b-n,c+d-l\right)]$

${=}_{a,c}^{b,d}{J}_{p_1,p_2,q_1,q_2}f\left(t,s\right)$

where

${}_{a,c}^{b,d}{J}_{p_1,p_2,q_1,q_2}f\left(t,s\right)$

${\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1}{\int }}ts}}\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(a+b-\left[tm+\left(1-t\right)n\right],c+d-\left[sk+\left(1-s\right)l\right]\right)d_{p_2,q_2}s{d}_{p_1,q_1}t$

${\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1}{\int }}ts\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(a+b-\left[\left(1-t\right)m+tn\right],c+d-\left[sk+\left(1-s\right)l\right]\right)d_{p_2,q_2}s{d}_{p_1,q_1}t}}$

${\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1}{\int }}ts\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(a+b-\left[tm+\left(1-t\right)n\right],c+d-\left[\left(1-s\right)k+sl\right]\right)d_{p_2,q_2}s{d}_{p_1,q_1}tt}}$

${\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1}{\int }}ts\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(a+b-\left[\left(1-t\right)m+tn\right],c+d-\left[\left(1-s\right)k+sl\right]\right)d_{p_2,q_2}s{d}_{p_1,q_1}t}}$

for $m,n\in \left[a,b\right],k,l\in \left[c,d\right]$ and $m<n,k<l.$

Proof:

By the definition 2.9, we have

$\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(a+b-\left[tm+\left(1-t\right)n\right],c+d-\left[sk+\left(1-s\right)l\right]\right)$

$=\frac{1}{\begin{array}{l}\left(p_1-q_1\right)\left(p_2-q_2\right)t\left(n-m\right)s\left(l-k\right)\\ [f\left(a+b-\left[\left(1-q_{1}t\right)n+q_{1}tm,c+d-\left[\left(1-q_{2}s\right)l+q_{2}sk\right]\right]\right)\\ -f\left(a+b-\left[\left(1-q_{1}t\right)n+q_{1}tm,c+d-\left[\left(1-p_{2}s\right)l+p_{2}sk\right]\right]\right)\\ -f\left(a+b-\left[\left(1-p_{1}t\right)n+p_{1}tm,c+d-\left[\left(1-q_{2}s\right)l+q_{2}sk\right]\right]\right)\\ +f\left(a+b-\left[\left(1-p_{1}t\right)n+p_{1}tm,c+d-\left[\left(1-p_{2}s\right)l+p_{2}sk\right]\right]\right)].\end{array}}\times$

Moreover,

${\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1}{\int }}ts\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(a+b-\left[tm+\left(1-t\right)n\right],c+d-\left[sk+\left(1-s\right)l\right]\right)d_{p_2,q_2}s{d}_{p_1,q_1}t}}$

$=\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}ts\frac{1}{\left(p_1-q_1\right)\left(p_2-q_2\right)t\left(n-m\right)s\left(l-k\right)}\times$

$[f\left(a+b-\left[\left(1-q_{1}t\right)n+q_{1}tm\right],c+d-\left[\left(1-q_{2}s\right)l+q_{2}sk\right]\right)$

$-f\left(a+b-\left[\left(1-q_{1}t\right)n+q_{1}tm\right],c+d-\left[\left(1-p_{2}s\right)l+p_{2}sk\right]\right)$

$-f\left(a+b-\left[\left(1-p_{1}t\right)n+p_{1}tm\right],c+d-\left[\left(1-q_{2}s\right)l+q_{2}sk\right]\right)$

$+f\left(a+b-\left[\left(1-p_{1}t\right)n+p_{1}tm\right],c+d-\left[\left(1-p_{2}s\right)l+p_{2}sk\right]\right)]d_{p_2,q_2}s{d}_{p_1,q_1}t$

$\begin{array}{l}=\frac{1}{\begin{array}{l}\left(p_1-q_1\right)\left(p_2-q_2\right)\left(n-m\right)\left(l-k\right)\\ [f\left(a+b-\left[\left(1-q_{1}t\right)n+q_{1}tm\right],c+d-\left[\left(1-q_{2}s\right)l+q_{2}sk\right]\right)\\ -f\left(a+b-\left[\left(1-q_{1}t\right)n+q_{1}tm\right],c+d-\left[\left(1-p_{2}s\right)l+p_{2}sk\right]\right)\\ -f\left(a+b-\left[\left(1-p_{1}t\right)n+p_{1}tm\right],c+d-\left[\left(1-q_{2}s\right)l+q_{2}sk\right]\right)\\ +f\left(a+b-\left[\left(1-p_{1}t\right)n+p_{1}tm\right],c+d-\left[\left(1-p_{2}s\right)l+p_{2}sk\right]\right)]d_{p_2,q_2}s{d}_{p_1,q_1}t\end{array}}\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}\\ =\frac{1}{\left(p_1-q_1\right)\left(p_2-q_2\right)\left(n-m\right)\left(l-k\right)}\left[I_1-I_2-I_3+I_4\right].\end{array}$

By definition 2.10, we have

$I_1={\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1}{\int }}f\left(a+b-\left[\left(1-q_{1}t\right)n+q_{1}tm,\right]c+d-\left[\left(1-q_{2}s\right)l+q_{2}sk\right]\right)d_{p_2,q_2}s{d}_{p_1,q_1}t}}$

$={\displaystyle \sum }_{n=0}^{\infty }{\displaystyle \sum }_{m=0}^{\infty }\frac{q_1^n}{p_1^{n+1}}\frac{q_2^m}{p_2^{m+1}}f\left(a+b-\left[\left(1-\frac{q_1^{n+1}}{p_1^{n+1}}\right)n+\frac{q_1^{n+1}}{p_1^{n+1}}m\right],c+d-\left[\left(1-\frac{q_2^{m+1}}{p_2^{m+1}}\right)l+\frac{q_2^{m+1}}{p_2^{m+1}}k\right]\right)$

$=\frac{p_1{p}_2}{q_1{q}_2}{\displaystyle \sum }_{n=0}^{\infty }{\displaystyle \sum }_{m=0}^{\infty }\frac{q_1^n}{p_1^{n+1}}\frac{q_2^m}{p_2^{m+1}}f\left(a+b-\left[\left(1-\frac{q_1^n}{p_1^{n+1}}{p}_1\right)n+\frac{q_1^n}{p_1^{n+1}}{p}_{1}m\right],c+d-\left[\left(1-\frac{q_2^m}{p_2^{m+1}}{p}_2\right)l+\frac{q_2^m}{p_2^{m+1}}{p}_{2}k\right]\right)$

$-\frac{p_2}{q_1{q}_2}{\displaystyle \sum }_{n=0}^{\infty }\frac{q_1^n}{p_1^{n+1}}f\left(a+b-\left[\left(1-\frac{q_1^n}{p_1^{n+1}}{p}_1\right)n+\frac{q_1^n}{p_1^{n+1}}{p}_{1}m\right],c+d-k\right)$

$-\frac{p_1}{q_1{q}_2}{\displaystyle \sum }_{m=0}^{\infty }\frac{q_2^m}{p_2^{m+1}}f\left(a+b-m,c+d-\left[\left(1-\frac{q_2^m}{p_2^{m+1}}{p}_2\right)l+\frac{q_2^m}{p_2^{m+1}}{p}_{2}k\right]\right)$

$+\frac{1}{q_1{q}_2}\left(a+b-m,c+d-k\right).$

$I_2={\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1}{\int }}f\left(a+b-\left[\left(1-q_{1}t\right)n+q_{1}tm\right],c+d-\left[\left(1-p_{2}s\right)l+p_{2}sk\right]\right)d_{p_2,q_2}s{d}_{p_1,q_1}t}}$

$={\displaystyle \sum }_{n=0}^{\infty }{\displaystyle \sum }_{m=0}^{\infty }\frac{q_1^n}{p_1^{n+1}}\frac{q_2^m}{p_2^{m+1}}f\left(a+b-\left[\left(1-\frac{q_1^{n+1}}{p_1^{n+1}}\right)n+\frac{q_1^{n+1}}{p_1^{n+1}}m\right],c+d-\left[\left(1-\frac{q_2^m}{p_2^m}\right)l+\frac{q_2^m}{p_2^m}k\right]\right)$

$=\frac{p_1}{q_1}{\displaystyle \sum }_{n=0}^{\infty }{\displaystyle \sum }_{m=0}^{\infty }\frac{q_1^n}{p_1^{n+1}}\frac{q_2^m}{p_2^{m+1}}f\left(a+b-\left[\left(1-\frac{q_1^n}{p_1^{n+1}}{p}_1\right)n+\frac{q_1^n}{p_1^{n+1}}{p}_{1}m\right],c+d-\left[\left(1-\frac{q_2^m}{p_2^{m+1}}{p}_2\right)l+\frac{q_2^m}{p_2^{m+1}}{p}_{2}k\right]\right)$

$-\frac{1}{q_1}{\displaystyle \sum }_{m=0}^{\infty }\frac{q_2^m}{p_2^{m+1}}f\left(a+b-m,c+d-\left[\left(1-\frac{q_2^m}{p_2^{m+1}}{p}_2\right)l+\frac{q_2^m}{p_2^{m+1}}{p}_{2}k\right]\right).$

$I_3=\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}f\left(a+b-\left[\left(1-p_{1}t\right)n+p_{1}tm\right],c+d-\left[\left(1-q_{2}s\right)l+q_{2}sk\right]\right)d_{p_2,q_2}s{d}_{p_1,q_1}t$

$={\displaystyle \sum }_{n=0}^{\infty }{\displaystyle \sum }_{m=0}^{\infty }\frac{q_1^n}{p_1^{n+1}}\frac{q_2^m}{p_2^{m+1}}f\left(a+b-\left[\left(1-\frac{q_1^n}{p_1^n}\right)n+\frac{q_1^n}{p_1^n}m\right],c+d-\left[\left(1-\frac{q_2^{m+1}}{p_2^{m+1}}\right)l+\frac{q_2^{m+1}}{p_2^{m+1}}k\right]\right)$

$=\frac{p_2}{q_2}{\displaystyle \sum }_{n=0}^{\infty }{\displaystyle \sum }_{m=0}^{\infty }\frac{q_1^n}{p_1^{n+1}}\frac{q_2^m}{p_2^{m+1}}f\left(a+b-\left[\left(1-\frac{q_1^n}{p_1^{n+1}}{p}_1\right)n+\frac{q_1^n}{p_1^{n+1}}{p}_{1}m\right],c+d-\left[\left(1-\frac{q_2^m}{p_2^{m+1}}{p}_2\right)l+\frac{q_2^m}{p_2^{m+1}}{p}_{2}k\right]\right)$

$-\frac{1}{q_2}{\displaystyle \sum }_{m=0}^{\infty }\frac{q_2^m}{p_2^{m+1}}f\left(a+b-\left[\left(1-\frac{q_1^n}{p_1^{n+1}}{p}_1\right)n+\frac{q_1^n}{p_1^{n+1}}{p}_{1}m\right],c+d-k\right).$

$I_4=\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}f\left(a+b-\left[\left(1-p_{1}t\right)n+p_{1}tm\right],c+d-\left[\left(1-p_{2}s\right)l+p_{2}sk\right]\right)d_{p_2,q_2}s{d}_{p_1,q_1}t$

$={\displaystyle \sum }_{n=0}^{\infty }{\displaystyle \sum }_{m=0}^{\infty }\frac{q_1^n}{p_1^{n+1}}\frac{q_2^m}{p_2^{m+1}}f\left(a+b-\left[\left(1-\frac{q_1^n}{p_1^{n+1}}{p}_1\right)n+\frac{q_1^n}{p_1^{n+1}}{p}_{1}m\right],c+d-\left[\left(1-\frac{q_2^m}{p_2^{m+1}}{p}_2\right)l+\frac{q_2^m}{p_2^{m+1}}{p}_{2}k\right]\right)$

So that,

$\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}ts\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(a+b-\left[tm+\left(1-t\right)n\right],c+d-\left[sk+\left(1-s\right)l\right]\right)d_{p_2,q_2}s{d}_{p_1,q_1}t$

$=\frac{1}{\begin{array}{l}\left(p_1-q_1\right)\left(p_2-q_2\right)\left(n-m\right)\left(l-k\right)\\ [\frac{\left(p_{1-}{q}_1\right)\left(p_2-q_2\right)}{q_1{q}_2}{\displaystyle \sum }_{n=0}^{\infty }{\displaystyle \sum }_{m=0}^{\infty }\frac{q_1^n}{p_1^{n+1}}\frac{q_2^m}{p_2^{m+1}}f\left(a+b-\left[\left(1-\frac{q_1^n}{p_1^n}\right)n+\frac{q_1^n}{p_1^n}m\right],c+d-\left[\left(1-\frac{q_2^{m+1}}{p_2^{m+1}}\right)l+\frac{q_2^{m+1}}{p_2^{m+1}}k\right]\right)\\ -\frac{\left(p_2-q_2\right)}{q_1{q}_2}{\displaystyle \sum }_{n=0}^{\infty }{\displaystyle \sum }_{m=0}^{\infty }\frac{q_1^n}{p_1^{n+1}}\frac{q_2^m}{p_2^{m+1}}f\left(a+b-m,c+d-\left[\left(1-\frac{q_2^m}{p_2^{m+1}}{p}_2\right)l+\frac{q_2^m}{p_2^{m+1}}{p}_{2}k\right]\right)\\ -\frac{\left(p_{1-}{q}_1\right)}{q_1{q}_2}{\displaystyle \sum }_{n=0}^{\infty }{\displaystyle \sum }_{m=0}^{\infty }\frac{q_1^n}{p_1^{n+1}}\frac{q_2^m}{p_2^{m+1}}f\left(a+b-\left[\left(1-\frac{q_1^n}{p_1^{n+1}}{p}_1\right)n+\frac{q_1^n}{p_1^{n+1}}{p}_{1}m\right],c+d-k\right)\\ +\frac{1}{q_1{q}_2}\left(a+b-m,c+d-k\right)\end{array}}\times$

$=\frac{1}{q_1{q}_2}[\frac{1}{p_1{p}_2{\left(n-m\right)}^2{\left(l-k\right)}^2}\underset{a+b-n}{\overset{a+b-\left[\left(1-p_1\right)n+p_{1}m\right]}{{\displaystyle \int }}}\underset{c+d-l}{\overset{c+d-\left[\left(1-p_2\right)l+p_{2}k\right]}{{\displaystyle \int }}}f\left(t,s\right)d_{p_2,q_2}s{d}_{p_1,q_1}t$

$-\frac{1}{p_2\left(n-m\right){\left(l-k\right)}^2}\underset{c+d-l}{\overset{c+d-\left[\left(1-p_2\right)l+p_{2}k\right]}{{\displaystyle \int }}}f\left(a+b-m,s\right)d_{p_2,q_2}s$

$-\frac{1}{p_1{\left(n-m\right)}^2\left(l-k\right)}\underset{a+b-n}{\overset{a+b-\left[\left(1-p_1\right)n+p_{1}m\right]}{{\displaystyle \int }}}f\left(t,c+d-k\right)d_{p_2,q_2}s{d}_{p_1,q_1}t$

$+\frac{1}{\left(n-m\right)\left(l-k\right)}\left(a+b-m,c+d-k\right)].$

Similarly, by the equality, we obtain the identities

$\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}ts\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(a+b-\left[\left(1-t\right)m+tn\right],c+d-\left[sk+\left(1-s\right)l\right]\right)d_{p_2,q_2}s{d}_{p_1,q_1}t$

$=\frac{1}{q_1{q}_2}[\frac{1}{p_1{p}_2{\left(n-m\right)}^2{\left(l-k\right)}^2}\underset{a+b-\left[\left(1-p_1\right)m+p_{1}n\right]}{\overset{a+b-m}{{\displaystyle \int }}}\underset{c+d-l}{\overset{c+d-\left[\left(1-p_2\right)l+p_{2}k\right]}{{\displaystyle \int }}}f\left(t,s\right)d_{p_2,q_2}s{d}_{p_1,q_1}t$

$-\frac{1}{p_2\left(n-m\right){\left(l-k\right)}^2}\underset{c+d-l}{\overset{c+d-\left[\left(1-p_2\right)l+p_{2}k\right]}{{\displaystyle \int }}}f\left(a+b-n,s\right)d_{p_2,q_2}s$

$-\frac{1}{p_1{\left(n-m\right)}^2\left(l-k\right)}\underset{a+b-\left[\left(1-p_1\right)m+p_{1}n\right]}{\overset{a+b-m}{{\displaystyle \int }}}f\left(t,c+d-k\right)d_{p_2,q_2}s{d}_{p_1,q_1}t$

$+\frac{1}{\left(n-m\right)\left(l-k\right)}\left(a+b-n,c+d-k\right)].$

$\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}ts\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(a+b-\left[tm+\left(1-t\right)n\right],c+d-\left[\left(1-s\right)k+sl\right]\right)d_{p_2,q_2}s{d}_{p_1,q_1}t$

$=\frac{1}{q_1{q}_2}[\frac{1}{p_1{p}_2{\left(n-m\right)}^2{\left(l-k\right)}^2}\underset{a+b-n}{\overset{a+b-\left[\left(1-p_1\right)n+p_{1}m\right]}{{\displaystyle \int }}}\underset{c+d-\left[\left(1-p_2\right)k+p_{2}l\right]}{\overset{c+d-k}{{\displaystyle \int }}}f\left(t,s\right)d_{p_2,q_2}s{d}_{p_1,q_1}t$

$-\frac{1}{p_2\left(n-m\right){\left(l-k\right)}^2}\underset{c+d-\left[\left(1-p_2\right)k+p_{2}l\right]}{\overset{c+d-k}{{\displaystyle \int }}}f\left(a+b-m,s\right)d_{p_2,q_2}s$

$-\frac{1}{p_1{\left(n-m\right)}^2\left(l-k\right)}\underset{a+b-n}{\overset{a+b-\left[\left(1-p_1\right)n+p_{1}m\right]}{{\displaystyle \int }}}f\left(t,,c+d-l\right)d_{p_2,q_2}s{d}_{p_1,q_1}t$

$+\frac{1}{\left(n-m\right)\left(l-k\right)}\left(a+b-m,c+d-l\right)].$

$\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}ts\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(a+b-\left[\left(1-t\right)m+tn\right],c+d-\left[\left(1-s\right)k+sl\right]\right)d_{p_2,q_2}s{d}_{p_1,q_1}t$

$=\frac{1}{q_1{q}_2}[\frac{1}{p_1{p}_2{\left(n-m\right)}^2{\left(l-k\right)}^2}\underset{a+b-\left[\left(1-p_1\right)m+p_{1}n\right]}{\overset{a+b-m}{{\displaystyle \int }}}\underset{c+d-\left[\left(1-p_2\right)k+p_{2}l\right]}{\overset{c+d-k}{{\displaystyle \int }}}f\left(t,s\right)d_{p_2,q_2}s{d}_{p_1,q_1}t$

$-\frac{1}{p_2\left(n-m\right){\left(l-k\right)}^2}\underset{c+d-\left[\left(1-p_2\right)k+p_{2}l\right]}{\overset{c+d-k}{{\displaystyle \int }}}f\left(a+b-n,s\right)d_{p_2,q_2}s$

$-\frac{1}{p_1{\left(n-m\right)}^2\left(l-k\right)}\underset{a+b-\left[\left(1-p_1\right)m+p_{1}n\right]}{\overset{a+b-m}{{\displaystyle \int }}}f\left(t,c+d-l\right)d_{p_2,q_2}s{d}_{p_1,q_1}t$

$+\frac{1}{\left(n-m\right)\left(l-k\right)}\left(a+b-n,c+d-l\right)].$

Thus, we have

$\frac{1}{p_1{p}_2{q}_1{q}_2{\left(n-m\right)}^2{\left(l-k\right)}^2}[\underset{a+b-n}{\overset{a+b-\left[\left(1-p_1\right)n+p_{1}m\right]}{{\displaystyle \int }}}\underset{c+d-l}{\overset{c+d-\left[\left(1-p_2\right)l+p_{2}k\right]}{{\displaystyle \int }}}f\left(t,s\right)d_{p_2,q_2}s{d}_{p_1,q_1}t$

$+\underset{a+b-\left[\left(1-p_1\right)m+p_{1}n\right]}{\overset{a+b-m}{{\displaystyle \int }}}\underset{c+d-l}{\overset{c+d-\left[\left(1-p_2\right)l+p_{2}k\right]}{{\displaystyle \int }}}f\left(t,s\right)d_{p_2,q_2}s{d}_{p_1,q_1}t$

$+\underset{a+b-n}{\overset{a+b-\left[\left(1-p_1\right)n+p_{1}m\right]}{{\displaystyle \int }}}\underset{c+d-\left[\left(1-p_2\right)k+p_{2}l\right]}{\overset{c+d-k}{{\displaystyle \int }}}f\left(t,s\right)d_{p_2,q_2}s{d}_{p_1,q_1}t$

$\underset{a+b-\left[\left(1-p_1\right)m+p_{1}n\right]}{\overset{a+b-m}{{\displaystyle \int }}}\underset{c+d-\left[\left(1-p_2\right)k+p_{2}l\right]}{\overset{c+d-k}{{\displaystyle \int }}}f\left(t,s\right)d_{p_2,q_2}s{d}_{p_1,q_1}t]$

$-\frac{1}{p_2\left(n-m\right){\left(l-k\right)}^2}[\underset{c+d-l}{\overset{c+d-\left[\left(1-p_2\right)l+p_{2}k\right]}{{\displaystyle \int }}}f\left(a+b-m,s\right)d_{p_2,q_2}s$

$+\underset{c+d-l}{\overset{c+d-\left[\left(1-p_2\right)l+p_{2}k\right]}{{\displaystyle \int }}}f\left(a+b-n,s\right)d_{p_2,q_2}s$

$+\underset{c+d-\left[\left(1-p_2\right)k+p_{2}l\right]}{\overset{c+d-k}{{\displaystyle \int }}}f\left(a+b-m,s\right)d_{p_2,q_2}s$

$\underset{c+d-\left[\left(1-p_2\right)k+p_{2}l\right]}{\overset{c+d-k}{{\displaystyle \int }}}f\left(a+b-n,s\right)d_{p_2,q_2}s]$

$-\frac{1}{p_1{\left(n-m\right)}^2\left(l-k\right)}[\underset{a+b-n}{\overset{a+b-\left[\left(1-p_1\right)n+p_{1}m\right]}{{\displaystyle \int }}}f\left(t,c+d-k\right)d_{p_2,q_2}s{d}_{p_1,q_1}t$

$+\underset{a+b-\left[\left(1-p_1\right)m+p_{1}n\right]}{\overset{a+b-m}{{\displaystyle \int }}}f\left(t,c+d-k\right)d_{p_2,q_2}s{d}_{p_1,q_1}t$

$+\underset{a+b-n}{\overset{a+b-\left[\left(1-p_1\right)n+p_{1}m\right]}{{\displaystyle \int }}}f\left(t,c+d-l\right)d_{p_2,q_2}s{d}_{p_1,q_1}t$

$\underset{a+b-\left[\left(1-p_1\right)m+p_{1}n\right]}{\overset{a+b-m}{{\displaystyle \int }}}f\left(t,c+d-l\right)d_{p_2,q_2}s{d}_{p_1,q_1}t]$

$+\frac{1}{\left(n-m\right)\left(l-k\right)}[\left(a+b-m,c+d-k\right)+\left(a+b-n,c+d-k\right)$

$+\left(a+b-m,c+d-l\right)+\left(a+b-n,c+d-l\right)]$

${=}_{a,c}^{b,d}{J}_{p_1,p_2,q_1,q_2}f\left(t,s\right).$

Corollary 3.2

If we set p₁, p₂ = 1, then the Lemma 3.1 reduces to the following equality.

$\frac{4}{q_1{q}_2{\left(n-m\right)}^2{\left(l-k\right)}^2}\underset{a+b-n}{\overset{a+b-m}{{\displaystyle \int }}}\underset{c+d-l}{\overset{c+d-k}{{\displaystyle \int }}}f\left(t,s\right)d_{q_2}s{d}_{q_1}t$

$-\frac{1}{\left(n-m\right){\left(l-k\right)}^2}[2\underset{c+d-l}{\overset{c+d-k}{{\displaystyle \int }}}f\left(a+b-m,s\right)d_{q_2}s$

$+2\underset{c+d-l}{\overset{c+d-k}{{\displaystyle \int }}}f\left(a+b-n,s\right)d_{q_2}s]$

$-\frac{1}{{\left(n-m\right)}^2\left(l-k\right)}[2\underset{a+b-n}{\overset{a+b-m}{{\displaystyle \int }}}f\left(t,c+d-k\right)d_{q_2}s{d}_{q_1}t$

$+2\underset{a+b-n}{\overset{a+b-m}{{\displaystyle \int }}}f\left(t,c+d-l\right)d_{q_2}s{d}_{q_1}t]$

$+\frac{1}{\left(n-m\right)\left(l-k\right)}[\left(a+b-m,c+d-k\right)+\left(a+b-n,c+d-k\right)$

$+\left(a+b-m,c+d-l\right)+\left(a+b-n,c+d-l\right)]$

${=}_{a,c}^{b,d}{J}_{q_1,q_2}f\left(t,s\right)$

where

${}_{a,c}^{b,d}{J}_{q_1,q_2}f\left(t,s\right)$

$=\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}ts\frac{{}_{a+b-n,c+d-l}{\partial }_{q_1,q_2}}{{}_{a+b-n}{\partial }_{q_1}{t}_{ c+d-l}{\partial }_{q_2}s}f\left(a+b-\left[tm+\left(1-t\right)n\right],c+d-\left[sk+\left(1-s\right)l\right]\right)d_{q_2}s{d}_{q_1}t$

$+\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}ts\frac{{}_{c+d-l}^{a+b-m}{\partial }_{q_1,q_2}}{{}^{a+b-m}{\partial }_{q_1}{t}_{ c+d-l}{\partial }_{q_2}s}f\left(a+b-\left[\left(1-t\right)m+tn\right],c+d-\left[sk+\left(1-s\right)l\right]\right)d_{q_2}s{d}_{q_1}t$

$+\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}ts\frac{{}_{a+b-n}^{c+d-k}{\partial }_{q_1,q_2}}{{}_{a+b-n}{\partial }_{q_1}{t}^{c+d-k}{\partial }_{q_2}s}f\left(a+b-\left[tm+\left(1-t\right)n\right],c+d-\left[\left(1-s\right)k+sl\right]\right)d_{q_2}s{d}_{q_1}t$

$+\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}ts\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{q_1}{t}^{c+d-k}{\partial }_{q_2}s}f\left(a+b-\left[\left(1-t\right)m+tn\right],c+d-\left[\left(1-s\right)k+sl\right]\right)d_{q_2}s{d}_{q_1}t$

Theorem 3.3

If the conditions of Lemma 3.1 hold and

$\frac{{}_{a,c}{\partial }_{p_1,p_2,q_1,q_2}f\left(x,y\right)}{{}_a{\partial }_{p_1,q_1}{x}_{ c}{\partial }_{p_2,q_2}y},\frac{{}_c^b{\partial }_{p_1,p_2,q_1,q_2}f\left(x,y\right)}{{}^b{\partial }_{p_1,q_1}{x}_{ c}{\partial }_{p_2,q_2}y},\frac{{}_a^d{\partial }_{p_1,p_2,q_1,q_2}f\left(x,y\right)}{{}_a{\partial }_{p_1,q_1}{x}_{ }^d{\partial }_{p_2,q_2}y},\frac{{}^{b,d}{\partial }_{p_1,p_2,q_1,q_2}f\left(x,y\right)}{{}^b{\partial }_{p_1,q_1}{x}_{ }^d{\partial }_{p_2,q_2}y}$

are coordinated convex, then we have the following inequality:

$\left|{}_{a,c}^{b,d}{J}_{p_1,p_2,q_1,q_2}f\left(t,s\right)\right|$

$\le \frac{1}{{\left[2\right]}_{p_1,q_1}{\left[2\right]}_{p_2,q_2}}[\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(a,c\right)\right|+\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(a,d\right)\right|$

$+\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(b,c\right)\right|+\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(b,d\right)\right|$

$+\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(a,c\right)\right|+\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(a,d\right)\right|$

$+\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(b,c\right)\right|+\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(b,d\right)\right|$

$+\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(a,c\right)\right|+\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(a,d\right)\right|$

$+\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(b,c\right)\right|+\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(b,d\right)\right|$

$+\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(a,c\right)\right|+\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(a,d\right)\right|$

$+\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(b,c\right)\right|+\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(b,d\right)\right|]$

$-\frac{1}{{\left[3\right]}_{p_1,q_1}{\left[3\right]}_{p_2,q_2}}$

$\times [\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(m,k\right)\right|+\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(n,k\right)\right|$

$+\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(m,l\right)\right|+\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(n,l\right)\right|]$

$-\frac{{\left[3\right]}_{p_2,q_2}-{\left[2\right]}_{p_2,q_2}}{{\left[3\right]}_{p_1,q_1}{\left[2\right]}_{p_2,q_2}{\left[3\right]}_{p_2,q_2}}$

$\times [\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(m,l\right)\right|+\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(n,l\right)\right|$

$+\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(m,k\right)\right|+\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(n,k\right)\right|]$

$-\frac{{\left[3\right]}_{p_1,q_1}-{\left[2\right]}_{p_1,q_1}}{{\left[2\right]}_{p_1,q_1}{\left[3\right]}_{p_1,q_1}{\left[3\right]}_{p_2,q_2}}$

$\times [\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(n,k\right)\right|+\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(m,k\right)\right|$

$+\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(n,l\right)\right|+\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(m,l\right)\right|]$

$-\frac{\left({\left[3\right]}_{p_1,q_1}-{\left[2\right]}_{p_1,q_1}\right)\left({\left[3\right]}_{p_2,q_2}-{\left[2\right]}_{p_2,q_2}\right)}{{\left[2\right]}_{p_1,q_1}{\left[2\right]}_{p_2,q_2}{\left[3\right]}_{p_1,q_1}{\left[3\right]}_{p_2,q_2}}$

$\times [\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(n,l\right)\right|+\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(m,l\right)\right|$

$\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(n,k\right)\right|+\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(m,k\right)\right|].$

proof:

From the result of lemma 3.1 and Jensen-Mercer inequality, we have

$\left|{}_{a,c}^{b,d}{J}_{p_1,p_2,q_1,q_2}f\left(t,s\right)\right|$

$\le \underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}ts[\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(a,c\right)\right|+\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(a,d\right)\right|$

$+\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(b,c\right)\right|+\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(b,d\right)\right|$

$-ts\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(m,k\right)\right|-t\left(1-s\right)\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(m,l\right)\right|$

$-\left(1-t\right)s\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(n,k\right)\right|-\left(1-t\right)\left(1-s\right)\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(n,l\right)\right|]d_{p_2,q_2}s{d}_{p_1,q_1}t$

$+\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}ts[\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(a,c\right)\right|+\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(a,d\right)\right|$

$+\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(b,c\right)\right|+\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(b,d\right)\right|$

$-\left(1-t\right)s\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(m,k\right)\right|-\left(1-t\right)\left(1-s\right)\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(m,l\right)\right|$

$-ts\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(n,k\right)\right|-t\left(1-s\right)\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(n,l\right)\right|]d_{p_2,q_2}s{d}_{p_1,q_1}t$

$+\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}ts[\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(a,c\right)\right|+\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(a,d\right)\right|$

$+\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(b,c\right)\right|+\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(b,d\right)\right|$

$-t\left(1-s\right)\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(m,k\right)\right|-ts\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(m,l\right)\right|$

$-\left(1-t\right)\left(1-s\right)\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(n,k\right)\right|-\left(1-t\right)s\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(n,l\right)\right|]d_{p_2,q_2}s{d}_{p_1,q_1}t$

$+\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}ts[\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(a,c\right)\right|+\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(a,d\right)\right|$

$+\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(b,c\right)\right|+\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(b,d\right)\right|$

$-\left(1-t\right)\left(1-s\right)\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(m,k\right)\right|$

$-\left(1-t\right)s\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(m,l\right)\right|$

$-t\left(1-s\right)\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(n,k\right)\right|$

$-ts\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(n,l\right)\right|]d_{p_2,q_2}s{d}_{p_1,q_1}t$

$=\frac{1}{{\left[2\right]}_{p_1,q_1}{\left[2\right]}_{p_2,q_2}}[\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(a,c\right)\right|$

$+\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(a,d\right)\right|$

$+\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(b,c\right)\right|$

$+\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(b,d\right)\right|$

$+\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(a,c\right)\right|$

$+\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(a,d\right)\right|$

$+\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(b,c\right)\right|$

$+\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(b,d\right)\right|$

$+\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(a,c\right)\right|$

$+\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(a,d\right)\right|$

$+\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(b,c\right)\right|$

$+\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(b,d\right)\right|$

$+\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(a,c\right)\right|$

$+\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(a,d\right)\right|$

$+\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(b,c\right)\right|$

$+\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(b,d\right)\right|]$

$-\frac{1}{{\left[3\right]}_{p_1,q_1}{\left[3\right]}_{p_2,q_2}}[\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(m,k\right)\right|$

$+\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(n,k\right)\right|$

$+\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(m,l\right)\right|$

$+\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(n,l\right)\right|]$

$-\frac{{\left[3\right]}_{p_2,q_2}-{\left[2\right]}_{p_2,q_2}}{{\left[3\right]}_{p_1,q_1}{\left[2\right]}_{p_2,q_2}{\left[3\right]}_{p_2,q_2}}[\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(m,l\right)\right|$

$+\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(n,l\right)\right|$

$+\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(m,k\right)\right|$

$+\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(n,k\right)\right|]$

$-\frac{{\left[3\right]}_{p_1,q_1}-{\left[2\right]}_{p_1,q_1}}{{\left[2\right]}_{p_1,q_1}{\left[3\right]}_{p_1,q_1}{\left[3\right]}_{p_2,q_2}}[\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(n,k\right)\right|$

$+\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(m,k\right)\right|$

$+\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(n,l\right)\right|$

$+\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(m,l\right)\right|]$

$-\frac{\left({\left[3\right]}_{p_1,q_1}-{\left[2\right]}_{p_1,q_1}\right)\left({\left[3\right]}_{p_2,q_2}-{\left[2\right]}_{p_2,q_2}\right)}{{\left[2\right]}_{p_1,q_1}{\left[2\right]}_{p_2,q_2}{\left[3\right]}_{p_1,q_1}{\left[3\right]}_{p_2,q_2}}$

$\times [\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(n,l\right)\right|+\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(m,l\right)\right|$

$\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(n,k\right)\right|+\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(m,k\right)\right|].$

Corollary 3.4

If we set p₁, p₂ = 1, then the Theorem 3.2 reduces to the following inequality.

$\left|{}_{a,c}^{b,d}{J}_{q_1,q_2}f\left(t,s\right)\right|$

$\le \frac{1}{{\left[2\right]}_{q_1}{\left[2\right]}_{q_2}}[\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{q_1,q_2}}{{}_{a+b-n}{\partial }_{q_1}{t}_{ c+d-l}{\partial }_{q_2}s}f\left(a,c\right)\right|+\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{q_1,q_2}}{{}_{a+b-n}{\partial }_{q_1}{t}_{ c+d-l}{\partial }_{q_2}s}f\left(a,d\right)\right|$

$+\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{q_1,q_2}}{{}_{a+b-n}{\partial }_{q_1}{t}_{ c+d-l}{\partial }_{q_2}s}f\left(b,c\right)\right|+\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{q_1,q_2}}{{}_{a+b-n}{\partial }_{q_1}{t}_{ c+d-l}{\partial }_{q_2}s}f\left(b,d\right)\right|$

$+\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{q_1,q_2}}{{}_{a+b-n}{\partial }_{q_1}{t}_{ c+d-l}{\partial }_{q_2}s}f\left(a,c\right)\right|+\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{q_1,q_2}}{{}_{a+b-n}{\partial }_{q_1}{t}_{ c+d-l}{\partial }_{q_2}s}f\left(a,d\right)\right|$

$+\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{q_1,q_2}}{{}_{a+b-n}{\partial }_{q_1}{t}_{ c+d-l}{\partial }_{q_2}s}f\left(b,c\right)\right|+\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{q_1,q_2}}{{}_{a+b-n}{\partial }_{q_1}{t}_{ c+d-l}{\partial }_{q_2}s}f\left(b,d\right)\right|$

$+\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{q_1,q_2}}{{}_{a+b-n}{\partial }_{q_1}{t}^{c+d-k}{\partial }_{q_2}s}f\left(a,c\right)\right|+\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{q_1,q_2}}{{}_{a+b-n}{\partial }_{q_1}{t}^{c+d-k}{\partial }_{q_2}s}f\left(a,d\right)\right|$

$+\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{q_1,q_2}}{{}_{a+b-n}{\partial }_{q_1}{t}^{c+d-k}{\partial }_{q_2}s}f\left(b,c\right)\right|+\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{q_1,q_2}}{{}_{a+b-n}{\partial }_{q_1}{t}^{c+d-k}{\partial }_{q_2}s}f\left(b,d\right)\right|$

$+\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{q_1,q_2}}{{}^{a+b-m}{\partial }_{q_1}{t}^{c+d-k}{\partial }_{q_2}s}f\left(a,c\right)\right|+\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{q_1,q_2}}{{}^{a+b-m}{\partial }_{q_1}{t}^{c+d-k}{\partial }_{q_2}s}f\left(a,d\right)\right|$

$+\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{q_1,q_2}}{{}^{a+b-m}{\partial }_{q_1}{t}^{c+d-k}{\partial }_{q_2}s}f\left(b,c\right)\right|+\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{q_1,q_2}}{{}^{a+b-m}{\partial }_{q_1}{t}^{c+d-k}{\partial }_{q_2}s}f\left(b,d\right)\right|]$

$-\frac{1}{{\left[3\right]}_{q_1}{\left[3\right]}_{q_2}}[\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{q_1,q_2}}{{}_{a+b-n}{\partial }_{q_1}{t}_{ c+d-l}{\partial }_{q_2}s}f\left(m,k\right)\right|+\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{q_1,q_2}}{{}_{a+b-n}{\partial }_{q_1}{t}_{ c+d-l}{\partial }_{q_2}s}f\left(n,k\right)\right|$

$+\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{q_1,q_2}}{{}_{a+b-n}{\partial }_{q_1}{t}^{c+d-k}{\partial }_{q_2}s}f\left(m,l\right)\right|+\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{q_1,q_2}}{{}^{a+b-m}{\partial }_{q_1}{t}^{c+d-k}{\partial }_{q_2}s}f\left(n,l\right)\right|]$

$-\frac{{\left(q_2\right)}^2}{{\left[3\right]}_{q_1}{\left[2\right]}_{q_2}{\left[3\right]}_{q_2}}[\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{q_1,q_2}}{{}_{a+b-n}{\partial }_{q_1}{t}_{ c+d-l}{\partial }_{q_2}s}f\left(m,l\right)\right|+\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{q_1,q_2}}{{}_{a+b-n}{\partial }_{q_1}{t}_{ c+d-l}{\partial }_{q_2}s}f\left(n,l\right)\right|$

$+\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{q_1,q_2}}{{}_{a+b-n}{\partial }_{q_1}{t}^{c+d-k}{\partial }_{q_2}s}f\left(m,k\right)\right|+\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{q_1,q_2}}{{}^{a+b-m}{\partial }_{q_1}{t}^{c+d-k}{\partial }_{q_2}s}f\left(n,k\right)\right|]$

$-\frac{{\left(q_1\right)}^2}{{\left[2\right]}_{q_1}{\left[3\right]}_{q_1}{\left[3\right]}_{q_2}}[\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{q_1,q_2}}{{}_{a+b-n}{\partial }_{q_1}{t}_{ c+d-l}{\partial }_{q_2}s}f\left(n,k\right)\right|+\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{q_1,q_2}}{{}_{a+b-n}{\partial }_{q_1}{t}_{ c+d-l}{\partial }_{q_2}s}f\left(m,k\right)\right|$

$+\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{q_1,q_2}}{{}_{a+b-n}{\partial }_{q_1}{t}^{c+d-k}{\partial }_{q_2}s}f\left(n,l\right)\right|+\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{q_1,q_2}}{{}^{a+b-m}{\partial }_{q_1}{t}^{c+d-k}{\partial }_{q_2}s}f\left(m,l\right)\right|]$

$-\frac{{\left(q_1\right)}^2{\left(q_2\right)}^2}{{\left[2\right]}_{q_1}{\left[2\right]}_{q_2}{\left[3\right]}_{q_1}{\left[3\right]}_{q_2}}[\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{q_1,q_2}}{{}_{a+b-n}{\partial }_{q_1}{t}_{ c+d-l}{\partial }_{q_2}s}f\left(n,l\right)\right|+\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{q_1,q_2}}{{}_{a+b-n}{\partial }_{q_1}{t}_{ c+d-l}{\partial }_{q_2}s}f\left(m,l\right)\right|$

$\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{q_1,q_2}}{{}_{a+b-n}{\partial }_{q_1}{t}^{c+d-k}{\partial }_{q_2}s}f\left(n,k\right)\right|+\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{q_1,q_2}}{{}^{a+b-m}{\partial }_{q_1}{t}^{c+d-k}{\partial }_{q_2}s}f\left(m,k\right)\right|].$

Theorem 3.5

Suppose that tha assumptions of Lemma 3.1 are hold. If

${\left|\frac{{}_{a,c}{\partial }_{p_1,p_2,q_1,q_2}f\left(x,y\right)}{{}_a{\partial }_{p_1,q_1}{x}_{ c}{\partial }_{p_2,q_2}y}\right|}^{\beta },{\left|\frac{{}_c^b{\partial }_{p_1,p_2,q_1,q_2}f\left(x,y\right)}{{}^b{\partial }_{p_1,q_1}{x}_{ c}{\partial }_{p_2,q_2}y}\right|}^{\beta },{\left|\frac{{}_a^d{\partial }_{p_1,p_2,q_1,q_2}f\left(x,y\right)}{{}_a{\partial }_{p_1,q_1}{x}_{ }^d{\partial }_{p_2,q_2}y}\right|}^{\beta },{\left|\frac{{}^{b,d}{\partial }_{p_1,p_2,q_1,q_2}f\left(x,y\right)}{{}^b{\partial }_{p_1,q_1}{x}_{ }^d{\partial }_{p_2,q_2}y}\right|}^{\beta }$

are coordinated convex function on $\left[a,b\right]\times \left[c,d\right]$ , then we have the inequality

$\left|{}_{a,c}^{b,d}{J}_{p_1,p_2,q_1,q_2}f\left(t,s\right)\right|$

$\le {\left(\frac{1}{{\left[\alpha +1\right]}_{p_1,q_1}{\left[\alpha +1\right]}_{p_2,q_2}}\right)}^{\frac{1}{\alpha }}\{{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(a,c\right)\right|}^{\beta }+{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(a,d\right)\right|}^{\beta }$

$+{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(b,c\right)\right|}^{\beta }+{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(b,d\right)\right|}^{\beta }$

$+{\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(a,c\right)\right|}^{\beta }+{\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(a,d\right)\right|}^{\beta }$

$+{\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(b,c\right)\right|}^{\beta }+{\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(b,d\right)\right|}^{\beta }$

$+{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(a,c\right)\right|}^{\beta }+{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(a,d\right)\right|}^{\beta }$

$+{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(b,c\right)\right|}^{\beta }+{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(b,d\right)\right|}^{\beta }$

$+{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(a,c\right)\right|}^{\beta }+{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(a,d\right)\right|}^{\beta }$

$+{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(b,c\right)\right|}^{\beta }+{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(b,d\right)\right|}^{\beta }$

$-\frac{1}{{\left[\beta +1\right]}_{p_1,q_1}{\left[\beta +1\right]}_{p_2,q_2}}[{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(m,k\right)\right|}^{\beta }+{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(m,l\right)\right|}^{\beta }$

$+{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(n,k\right)\right|}^{\beta }+{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(n,l\right)\right|}^{\beta }$

$+{\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(m,k\right)\right|}^{\beta }+{\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(m,l\right)\right|}^{\beta }$

$+{\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(n,k\right)\right|}^{\beta }+{\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(n,l\right)\right|}^{\beta }$

$+{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(m,k\right)\right|}^{\beta }+{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(m,l\right)\right|}^{\beta }$

$+{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(n,k\right)\right|}^{\beta }+{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(n,l\right)\right|}^{\beta }$

$+{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(m,k\right)\right|}^{\beta }+{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(m,l\right)\right|}^{\beta }$

${+{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(n,k\right)\right|}^{\beta }+{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(n,l\right)\right|}^{\beta }]\}}^{\frac{1}{\beta }}.$

where $\frac{1}{\alpha }+\frac{1}{\beta }=1,\beta >1.$

Proof:

From the Lemma 3.1 and and Jensen-Mercer inequality by using the Hölder inequality and the convexity of ${\left|\frac{{}_{a,c}{\partial }_{p_1,p_2,q_1,q_2}f\left(x,y\right)}{{}_a{\partial }_{p_1,q_1}{x}_{ c}{\partial }_{p_2,q_2}y}\right|}^{\beta },{\left|\frac{{}_c^b{\partial }_{p_1,p_2,q_1,q_2}f\left(x,y\right)}{{}^b{\partial }_{p_1,q_1}{x}_{ c}{\partial }_{p_2,q_2}y}\right|}^{\beta },{\left|\frac{{}_a^d{\partial }_{p_1,p_2,q_1,q_2}f\left(x,y\right)}{{}_a{\partial }_{p_1,q_1}{x}_{ }^d{\partial }_{p_2,q_2}y}\right|}^{\beta },{\left|\frac{{}^{b,d}{\partial }_{p_1,p_2,q_1,q_2}f\left(x,y\right)}{{}^b{\partial }_{p_1,q_1}{x}_{ }^d{\partial }_{p_2,q_2}y}\right|}^{\beta },$ we obtain

$\left|{}_{a,c}^{b,d}{J}_{p_1,p_2,q_1,q_2}f\left(t,s\right)\right|$

$\le {\left(\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}{t}^{\alpha }{s}^{\alpha }{d}_{p_2,q_2}s{d}_{p_1,q_1}t\right)}^{\frac{1}{\alpha }}[{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(a,c\right)\right|}^{\beta }+{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(a,d\right)\right|}^{\beta }$

$+{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(b,c\right)\right|}^{\beta }+{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(b,d\right)\right|}^{\beta }$

$-{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(m,k\right)\right|}^{\beta }\left(\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}{t}^{\beta }{s}^{\beta }{d}_{p_2,q_2}s{d}_{p_1,q_1}t\right)$

$-{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(n,l\right)\right|}^{\beta }\left(\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}{\left(1-t\right)}^{\beta }{\left(1-s\right)}^{\beta }{d}_{p_2,q_2}s{d}_{p_1,q_1}t\right)]$

$-{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(n,l\right)\right|}^{\beta }\left(\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}{\left(1-t\right)}^{\beta }{\left(1-s\right)}^{\beta }{d}_{p_2,q_2}s{d}_{p_1,q_1}t\right)]$

${-{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(n,l\right)\right|}^{\beta }\left(\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}{\left(1-t\right)}^{\beta }{\left(1-s\right)}^{\beta }{d}_{p_2,q_2}s{d}_{p_1,q_1}t\right)]}^{\frac{1}{\beta }}$

$+{\left(\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}{t}^{\alpha }{s}^{\alpha }{d}_{p_2,q_2}s{d}_{p_1,q_1}t\right)}^{\frac{1}{\alpha }}[{\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(a,c\right)\right|}^{\beta }+{\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(a,d\right)\right|}^{\beta }$

$+{\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(b,c\right)\right|}^{\beta }+{\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(b,d\right)\right|}^{\beta }$

$-{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(n,l\right)\right|}^{\beta }\left(\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}{\left(1-t\right)}^{\beta }{\left(1-s\right)}^{\beta }{d}_{p_2,q_2}s{d}_{p_1,q_1}t\right)]$

$-{\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(m,l\right)\right|}^{\beta }\left(\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}{\left(1-t\right)}^{\beta }{\left(1-s\right)}^{\beta }{d}_{p_2,q_2}s{d}_{p_1,q_1}t\right)$

$-{\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(n,k\right)\right|}^{\beta }\left(\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}{t}^{\beta }{s}^{\beta }{d}_{p_2,q_2}s{d}_{p_1,q_1}t\right)$

${-{\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(n,l\right)\right|}^{\beta }\left(\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}{t}^{\beta }{\left(1-s\right)}^{\beta }{d}_{p_2,q_2}s{d}_{p_1,q_1}t\right)]}^{\frac{1}{\beta }}$

$+{\left(\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}{t}^{\alpha }{s}^{\alpha }{d}_{p_2,q_2}s{d}_{p_1,q_1}t\right)}^{\frac{1}{\alpha }}[{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(a,c\right)\right|}^{\beta }+{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(a,d\right)\right|}^{\beta }$

$+{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(b,c\right)\right|}^{\beta }+{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(b,d\right)\right|}^{\beta }$

$-{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(m,k\right)\right|}^{\beta }\left(\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}{t}^{\beta }{\left(1-s\right)}^{\beta }{d}_{p_2,q_2}s{d}_{p_1,q_1}t\right)$

$-{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(m,l\right)\right|}^{\beta }\left(\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}{t}^{\beta }{s}^{\beta }{d}_{p_2,q_2}s{d}_{p_1,q_1}t\right)$

$-{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(n,k\right)\right|}^{\beta }\left(\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}{\left(1-t\right)}^{\beta }{\left(1-s\right)}^{\beta }{d}_{p_2,q_2}s{d}_{p_1,q_1}t\right)$

${-{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(n,l\right)\right|}^{\beta }\left(\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}{\left(1-t\right)}^{\beta }{s}^{\beta }{d}_{p_2,q_2}s{d}_{p_1,q_1}t\right)]}^{\frac{1}{\beta }}$

$+{\left(\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}{t}^{\alpha }{s}^{\alpha }{d}_{p_2,q_2}s{d}_{p_1,q_1}t\right)}^{\frac{1}{\alpha }}[{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(a,c\right)\right|}^{\beta }+{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(a,d\right)\right|}^{\beta }$

$+{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(b,c\right)\right|}^{\beta }+{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(b,d\right)\right|}^{\beta }$

$-{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(m,k\right)\right|}^{\beta }\left(\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}{\left(1-t\right)}^{\beta }{\left(1-s\right)}^{\beta }{d}_{p_2,q_2}s{d}_{p_1,q_1}t\right)$

$-{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(m,l\right)\right|}^{\beta }\left(\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}{\left(1-t\right)}^{\beta }{s}^{\beta }{d}_{p_2,q_2}s{d}_{p_1,q_1}t\right)$

$-{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(n,k\right)\right|}^{\beta }\left(\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}{t}^{\beta }{\left(1-s\right)}^{\beta }{d}_{p_2,q_2}s{d}_{p_1,q_1}t\right)$

${-{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,=q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(n,l\right)\right|}^{\beta }\left(\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}{t}^{\beta }{s}^{\beta }{d}_{p_2,q_2}s{d}_{p_1,q_1}t\right)]}^{\frac{1}{\beta }}$

$={\left(\frac{1}{{\left[\alpha +1\right]}_{p_1,q_1}{\left[\alpha +1\right]}_{p_2,q_2}}\right)}^{\frac{1}{\alpha }}\{{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(a,c\right)\right|}^{\beta }+{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(a,d\right)\right|}^{\beta }$

$+{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(b,c\right)\right|}^{\beta }+{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(b,d\right)\right|}^{\beta }$

$+{\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(a,c\right)\right|}^{\beta }+{\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(a,d\right)\right|}^{\beta }$

$+{\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(b,c\right)\right|}^{\beta }+{\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(b,d\right)\right|}^{\beta }$

$+{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(a,c\right)\right|}^{\beta }+{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(a,d\right)\right|}^{\beta }$

$+{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(b,c\right)\right|}^{\beta }+{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(b,d\right)\right|}^{\beta }$

$+{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(a,c\right)\right|}^{\beta }+{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(a,d\right)\right|}^{\beta }$

$+{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(b,c\right)\right|}^{\beta }+{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(b,d\right)\right|}^{\beta }$

$-\frac{1}{{\left[\beta +1\right]}_{p_1,q_1}{\left[\beta +1\right]}_{p_2,q_2}}[{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(m,k\right)\right|}^{\beta }+{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(m,l\right)\right|}^{\beta }$

$+{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(n,k\right)\right|}^{\beta }+{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(n,l\right)\right|}^{\beta }$

$+{\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(m,k\right)\right|}^{\beta }+{\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(m,l\right)\right|}^{\beta }$

$+{\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(n,k\right)\right|}^{\beta }+{\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(n,l\right)\right|}^{\beta }$

$+{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(m,k\right)\right|}^{\beta }+{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(m,l\right)\right|}^{\beta }$

$+{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(n,k\right)\right|}^{\beta }+{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(n,l\right)\right|}^{\beta }$

$+{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(m,k\right)\right|}^{\beta }+{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(m,l\right)\right|}^{\beta }$

${+{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(n,k\right)\right|}^{\beta }+{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(n,l\right)\right|}^{\beta }]\}}^{\frac{1}{\beta }}.$

Theorem 3.6

Suppose that tha assumptions of Lemma 3.1 are hold. If

${\left|\frac{{}_{a,c}{\partial }_{p_1,p_2,q_1,q_2}f\left(x,y\right)}{{}_a{\partial }_{p_1,q_1}{x}_{ c}{\partial }_{p_2,q_2}y}\right|}^r,{\left|\frac{{}_c^b{\partial }_{p_1,p_2,q_1,q_2}f\left(x,y\right)}{{}^b{\partial }_{p_1,q_1}{x}_{ c}{\partial }_{p_2,q_2}y}\right|}^r,{\left|\frac{{}_a^d{\partial }_{p_1,p_2,q_1,q_2}f\left(x,y\right)}{{}_a{\partial }_{p_1,q_1}{x}_{ }^d{\partial }_{p_2,q_2}y}\right|}^r,{\left|\frac{{}^{b,d}{\partial }_{p_1,p_2,q_1,q_2}f\left(x,y\right)}{{}^b{\partial }_{p_1,q_1}{x}_{ }^d{\partial }_{p_2,q_2}y}\right|}^r$

are coordinated convex function on $\left[a,b\right]\times \left[c,d\right]$ for $r\ge 1,$ then we have the inequality

$\left|{}_{a,c}^{b,d}{J}_{p_1,p_2,q_1,q_2}f\left(t,s\right)\right|$

$\le {\left(\frac{1}{{\left[2\right]}_{p_1,q_1}{\left[2\right]}_{p_2,q_2}}\right)}^{1-\frac{1}{r}}\{{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(a,c\right)\right|}^r+{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(a,d\right)\right|}^r$

$+{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(b,c\right)\right|}^r+{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(b,d\right)\right|}^r$

$+{\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(a,c\right)\right|}^r+{\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(a,d\right)\right|}^r$

$+{\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(b,c\right)\right|}^r+{\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(b,d\right)\right|}^r$

$+{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(a,c\right)\right|}^r+{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(a,d\right)\right|}^r$

$+{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(b,c\right)\right|}^r+{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(b,d\right)\right|}^r$

$+{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(a,c\right)\right|}^r+{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(a,d\right)\right|}^r$

$+{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(b,c\right)\right|}^r+{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(b,d\right)\right|}^r$

$-\frac{1}{{\left[r+1\right]}_{p_1,q_1}{\left[r+1\right]}_{p_2,q_2}}[{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(m,k\right)\right|}^r+{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(m,l\right)\right|}^r$

$+{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(n,k\right)\right|}^r+{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(n,l\right)\right|}^r$

$+{\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(m,k\right)\right|}^r+{\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(m,l\right)\right|}^r$

$+{\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(n,k\right)\right|}^r+{\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(n,l\right)\right|}^r$

$+{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(m,k\right)\right|}^r+{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(m,l\right)\right|}^r$

$+{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(n,k\right)\right|}^r+{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(n,l\right)\right|}^r$

$+{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(m,k\right)\right|}^r+{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(m,l\right)\right|}^r$

${+{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(n,k\right)\right|}^r+{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(n,l\right)\right|}^r]\}}^{\frac{1}{r}}.$

Proof:

From the Lemma 3.1 and and Jensen-Mercer inequality by using the power-mean inequality and the convexity of ${\left|\frac{{}_{a,c}{\partial }_{p_1,p_2,q_1,q_2}f\left(x,y\right)}{{}_a{\partial }_{p_1,q_1}{x}_{ c}{\partial }_{p_2,q_2}y}\right|}^r,{\left|\frac{{}_c^b{\partial }_{p_1,p_2,q_1,q_2}f\left(x,y\right)}{{}^b{\partial }_{p_1,q_1}{x}_{ c}{\partial }_{p_2,q_2}y}\right|}^r,{\left|\frac{{}_a^d{\partial }_{p_1,p_2,q_1,q_2}f\left(x,y\right)}{{}_a{\partial }_{p_1,q_1}{x}_{ }^d{\partial }_{p_2,q_2}y}\right|}^r,{\left|\frac{{}^{b,d}{\partial }_{p_1,p_2,q_1,q_2}f\left(x,y\right)}{{}^b{\partial }_{p_1,q_1}{x}_{ }^d{\partial }_{p_2,q_2}y}\right|}^r,$ we obtain

$\left|{}_{a,c}^{b,d}{J}_{p_1,p_2,q_1,q_2}f\left(t,s\right)\right|$

$\le {\left(\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}ts{d}_{p_2,q_2}s{d}_{p_1,q_1}t\right)}^{1-\frac{1}{r}}[{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(a,c\right)\right|}^r+{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(a,d\right)\right|}^r$

$+{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(b,c\right)\right|}^r+{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(b,d\right)\right|}^r$

$-{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(m,k\right)\right|}^r\left(\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}{t}^r{s}^r{d}_{p_2,q_2}s{d}_{p_1,q_1}t\right)$

$-{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(m,l\right)\right|}^r\left(\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}{t}^r{\left(1-s\right)}^r{d}_{p_2,q_2}s{d}_{p_1,q_1}t\right)$

$-{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(n,k\right)\right|}^r\left(\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}{\left(1-t\right)}^r{s}^r{d}_{p_2,q_2}s{d}_{p_1,q_1}t\right)$

${-{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(n,l\right)\right|}^r\left(\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}{\left(1-t\right)}^r{\left(1-s\right)}^r{d}_{p_2,q_2}s{d}_{p_1,q_1}t\right)]}^{\frac{1}{r}}$

$+{\left(\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}ts{d}_{p_2,q_2}s{d}_{p_1,q_1}t\right)}^{1-\frac{1}{r}}[{\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(a,c\right)\right|}^r+{\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(a,d\right)\right|}^r$

$+{\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(b,c\right)\right|}^r+{\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(b,d\right)\right|}^r$

$-{\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(m,k\right)\right|}^r\left(\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}{\left(1-t\right)}^r{s}^r{d}_{p_2,q_2}s{d}_{p_1,q_1}t\right)$

$-{\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(m,l\right)\right|}^r\left(\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}{\left(1-t\right)}^r{\left(1-s\right)}^r{d}_{p_2,q_2}s{d}_{p_1,q_1}t\right)$

$-{\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(n,k\right)\right|}^r\left(\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}{t}^r{s}^r{d}_{p_2,q_2}s{d}_{p_1,q_1}t\right)$

${-{\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(n,l\right)\right|}^r\left(\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}{t}^r{\left(1-s\right)}^r{d}_{p_2,q_2}s{d}_{p_1,q_1}t\right)]}^{\frac{1}{r}}$

$+{\left(\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}ts{d}_{p_2,q_2}s{d}_{p_1,q_1}t\right)}^{1-\frac{1}{r}}[{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(a,c\right)\right|}^r+{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(a,d\right)\right|}^r$

$+{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(b,c\right)\right|}^r+{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(b,d\right)\right|}^r$

$-{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(m,k\right)\right|}^r\left(\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}{t}^r{\left(1-s\right)}^r{d}_{p_2,q_2}s{d}_{p_1,q_1}t\right)$

$-{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(m,l\right)\right|}^r\left(\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}{t}^r{s}^r{d}_{p_2,q_2}s{d}_{p_1,q_1}t\right)$

$-{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(n,k\right)\right|}^r\left(\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}{\left(1-t\right)}^r{\left(1-s\right)}^r{d}_{p_2,q_2}s{d}_{p_1,q_1}t\right)$

${-{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(n,l\right)\right|}^r\left(\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}{\left(1-t\right)}^r{s}^r{d}_{p_2,q_2}s{d}_{p_1,q_1}t\right)]}^{\frac{1}{r}}$

$+{\left(\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}ts{d}_{p_2,q_2}s{d}_{p_1,q_1}t\right)}^{1-\frac{1}{r}}[{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(a,c\right)\right|}^r+{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(a,d\right)\right|}^r$

$+{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(b,c\right)\right|}^r+{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(b,d\right)\right|}^r$

$-{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(m,k\right)\right|}^r\left(\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}{\left(1-t\right)}^r{\left(1-s\right)}^r{d}_{p_2,q_2}s{d}_{p_1,q_1}t\right)$

$-{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(m,l\right)\right|}^r\left(\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}{\left(1-t\right)}^r{s}^r{d}_{p_2,q_2}s{d}_{p_1,q_1}t\right)$

$-{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(n,k\right)\right|}^r\left(\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}{t}^r{\left(1-s\right)}^r{d}_{p_2,q_2}s{d}_{p_1,q_1}t\right)$

${-{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,=q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(n,l\right)\right|}^r\left(\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}{t}^r{s}^r{d}_{p_2,q_2}s{d}_{p_1,q_1}t\right)]}^{\frac{1}{r}}$

$={\left(\frac{1}{{\left[2\right]}_{p_1,q_1}{\left[2\right]}_{p_2,q_2}}\right)}^{1-\frac{1}{r}}\{{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(a,c\right)\right|}^r+{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(a,d\right)\right|}^r$

$+{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(b,c\right)\right|}^r+{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(b,d\right)\right|}^r$

$+{\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(a,c\right)\right|}^r+{\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(a,d\right)\right|}^r$

$+{\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(b,c\right)\right|}^r+{\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(b,d\right)\right|}^r$

$+{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(a,c\right)\right|}^r+{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(a,d\right)\right|}^r$

$+{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(b,c\right)\right|}^r+{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(b,d\right)\right|}^r$

$+{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(a,c\right)\right|}^r+{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(a,d\right)\right|}^r$

$+{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(b,c\right)\right|}^r+{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(b,d\right)\right|}^r$

$-\frac{1}{{\left[r+1\right]}_{p_1,q_1}{\left[r+1\right]}_{p_2,q_2}}[{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(m,k\right)\right|}^r+{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(m,l\right)\right|}^r$

$+{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(n,k\right)\right|}^r+{\left|\frac{{}_{a+b-n,c+d-l}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(n,l\right)\right|}^r$

$+{\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(m,k\right)\right|}^r+{\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(m,l\right)\right|}^r$

$+{\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(n,k\right)\right|}^r+{\left|\frac{{}_{c+d-l}^{a+b-m}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}_{ c+d-l}{\partial }_{p_2,q_2}s}f\left(n,l\right)\right|}^r$

$+{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(m,k\right)\right|}^r+{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(m,l\right)\right|}^r$

$+{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(n,k\right)\right|}^r+{\left|\frac{{}_{a+b-n}^{c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}_{a+b-n}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(n,l\right)\right|}^r$

$+{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(m,k\right)\right|}^r+{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(m,l\right)\right|}^r$

${+{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(n,k\right)\right|}^r+{\left|\frac{{}^{a+b-m,c+d-k}{\partial }_{p_1,p_2,q_1,q_2}}{{}^{a+b-m}{\partial }_{p_1,q_1}{t}^{c+d-k}{\partial }_{p_2,q_2}s}f\left(n,l\right)\right|}^r]\}}^{\frac{1}{r}}.$

Conclusion

In this paper, we have established several new Hermite-Hadamard-Mercer type inequalities on coordinates using post-quantum (p, q)-calculus. These results generalize existing inequalities and provide a unified framework encompassing both classical and quantum cases. Our work extends the literature by presenting double-integral identities under (p₁, p₂, q₁, q₂) - partial differentiability and integrability assumptions.Further research may explore: Higher-dimensional generalizations, Inequalities for other convexity types (e.g., s-convex, log-convex) and the applications in optimization or quantum information theory.

Data availability

All data required for this paper is included within this paper.