New Estimation of Error in the Hadamard Inequality Pertaining to Coordinated Convex Functions in Quantum Calculus

Abstract

Convex bodies are symmetric in nature. Between the two variables of symmetry and convexity, a correlation connection is also perceptible. Due to the interchangeable analogous properties, the application on either of them has been practicable in these modern years. The current analysis sheds insight on a general new identity involving a number of parameters for a twice partial quantum differentiable function. We find several unique quantum integral inequalities by using the new identity and a twice partial quantum differentiable function whose absolute value is coordinated convex. In addition, we present several novel and interesting error estimation-like results related to the well-known quantum Hermite–Hadamard inequality. Some examples are provided at the end to support and demonstrate the effectiveness of the new outcomes.

1. Introduction

One of mathematics’ most fundamental and important ideas is convexity. Beginning with Minkowski’s ground-breaking work at the beginning of the 20th century, convexity theory has been thoroughly and methodically developing ever since. There are currently a number of subjects in this field. It can be generally classified into three key areas: convex geometry, convex analysis, and discrete or combinatorial convexity. This classification is based on the concepts, techniques, and tools used. Measures of symmetry are frequently found in the literature on convex geometry, which effectively illustrates the seeming diversity of the concept of convexity. A multitude of clearly calculable examples for measures of symmetry are provided by the literature on convex bodies of constant width, which is constantly expanding (see, for instance, [1]). With this motivation, we explore some new inequalities due to twice partial quantum differentiable functions, which gives some more applications about means of real number. We first collect some fundamental concepts.

Throughout the study, we consider a double interval (precisely rectangle) $\mathrm{Y}:=\left[m,k\right]\times \left[n,\ell \right],$ where $m<k,n<\ell$ in the plane ${\mathbb{R}}^2.$

Now, we recall that the inequality

$$\mathcal{G}\left(\frac{m+k}{2}\right)\le \frac{1}{k-m}{\int }_m^k\mathcal{G}\left(\nu \right)d\nu \le \frac{\mathcal{G}\left(m\right)+\mathcal{G}\left(k\right)}{2}$$

(1)

holds true if $\mathcal{G}:[m,k]\to \mathbb{R}$ is a convex function.

In 1883, Hermite (1) proved the inequality [2]. Hadamard [3] rediscovered it in 1893. As a result, both scholars are equally credited with discovering the inequality, which is referred to as the Hermite–Hadamard inequaltiy in the literary work.

In 2001, Dragomir [4] introduced the notion of coordinated convex functions as follows:

Definition 1. 

A function $\mathcal{G}:\mathrm{Y}\to \mathbb{R}$ is called convex on the coordinates on Y or coordinated convex function if the inequality

$$\begin{array}{ccc}\hfill \mathcal{G}\left(u{\lambda }_1+(1-u){\lambda }_2,v{\mu }_1+(1-v){\mu }_2\right)& \le & uv\mathcal{G}({\lambda }_1,{\mu }_1)+u(1-v)\mathcal{G}({\lambda }_1,{\mu }_2)\hfill \\ & & +(1-u)v\mathcal{G}({\lambda }_2,{\mu }_1)+(1-u)(1-v)\mathcal{G}({\lambda }_2,{\mu }_2),\hfill \end{array}$$

(2)

holds for all $u,v\in [0,1]$ and $({\lambda }_1,{\mu }_1),({\lambda }_1,{\mu }_2),({\lambda }_2,{\mu }_1),({\lambda }_2,{\mu }_2)\in \mathrm{Y}.$

Dragomir [4] proved the following sharp inequality by adopting the concept of coordinated convex functions.

Theorem 1. 

Suppose that $\mathcal{G}:\mathrm{Y}\to \mathbb{R}$ is a convex function on the coordinates on Y and $\mathcal{G}\in L_1\left(\mathrm{Y}\right)$. Then one has the inequalities:

$$\begin{array}{ccc}\hfill & & \mathcal{G}\left(\frac{m+k}{2},\frac{n+\ell }{2}\right)\hfill \\ & \le & \frac{1}{2}\left[\frac{1}{k-m}\underset{m}{\overset{k}{\int }}\mathcal{G}\left(x,\frac{n+\ell }{2}\right)dx+\frac{1}{\ell -n}\underset{n}{\overset{\ell }{\int }}\mathcal{G}\left(\frac{m+k}{2},y\right)dy\right]\hfill \\ & \le & \frac{1}{\left(k-m\right)\left(\ell -n\right)}\underset{m}{\overset{k}{\int }}\underset{n}{\overset{\ell }{\int }}\mathcal{G}\left(x,y\right)dydx\hfill \\ & \le & \frac{1}{4(k-m)}\underset{m}{\overset{k}{\int }}\left[\mathcal{G}\left(x,n\right)+\mathcal{G}\left(x,\ell \right)\right]dx+\frac{1}{4(\ell -n)}\underset{n}{\overset{\ell }{\int }}\left[\mathcal{G}\left(m,y\right)+\mathcal{G}\left(k,y\right)\right]dy\hfill \\ & \le & \frac{\mathcal{G}\left(m,n\right)+\mathcal{G}\left(m,\ell \right)+\mathcal{G}\left(k,n\right)+\mathcal{G}\left(k,\ell \right)}{4}.\hfill \end{array}$$

(3)

In 2010, Sarikaya et al. [5] acquired the error estimates for the third and fourth inequality in (3).

Theorem 2. 

Let $\mathcal{G}:\mathrm{Y}\to \mathbb{R}$ be a partial differentiable mapping on $\mathrm{Y}.$

(a) If $\left|\frac{{\partial }^2\mathcal{G}}{\partial u\partial v}\right|$ is convex on the coordinates on $\mathrm{Y},$ then the following inequality holds:

$$\begin{array}{ccc}\hfill & & \left|\frac{1}{(k-m)(\ell -n)}{\int }_m^k{\int }_n^{\ell }\mathcal{G}(u,v)dvdu+\frac{\mathcal{G}\left(m,n\right)+\mathcal{G}\left(m,\ell \right)+\mathcal{G}\left(k,n\right)+\mathcal{G}\left(k,\ell \right)}{4}-\Phi \right|\hfill \\ & \le & \frac{(k-m)(\ell -n)}{16}\left[\frac{\left|\frac{{\partial }^2\mathcal{G}(m,n)}{\partial u\partial v}\right|+\left|\frac{{\partial }^2\mathcal{G}(m,\ell )}{\partial u\partial v}\right|+\left|\frac{{\partial }^2\mathcal{G}(k,n)}{\partial u\partial v}\right|+\left|\frac{{\partial }^2\mathcal{G}(k,\ell )}{\partial u\partial v}\right|}{4}\right];\hfill \end{array}$$

(4)

(b) If ${\left|\frac{{\partial }^2\mathcal{G}}{\partial u\partial v}\right|}^{\lambda }$ is convex on the coordinates on Y and $\lambda \ge 1,$ then the following inequality holds:

$$\begin{array}{ccc}\hfill & & \left|\frac{1}{(k-m)(\ell -n)}{\int }_m^k{\int }_n^{\ell }\mathcal{G}(u,v)dvdu+\frac{\mathcal{G}\left(m,n\right)+\mathcal{G}\left(m,\ell \right)+\mathcal{G}\left(k,n\right)+\mathcal{G}\left(k,\ell \right)}{4}-\Phi \right|\hfill \\ & \le & \frac{(k-m)(\ell -n)}{16}{\left[\frac{{\left|\frac{{\partial }^2\mathcal{G}(m,n)}{\partial u\partial v}\right|}^{\lambda }+{\left|\frac{{\partial }^2\mathcal{G}(m,\ell )}{\partial u\partial v}\right|}^{\lambda }+{\left|\frac{{\partial }^2\mathcal{G}(k,n)}{\partial u\partial v}\right|}^{\lambda }+{\left|\frac{{\partial }^2\mathcal{G}(k,\ell )}{\partial u\partial v}\right|}^{\lambda }}{4}\right]}^{\frac{1}{\lambda }},\hfill \end{array}$$

(5)

where

$$\Phi =\frac{1}{2(k-m)}\underset{m}{\overset{k}{\int }}\left[\mathcal{G}\left(x,n\right)+\mathcal{G}\left(x,\ell \right)\right]dx+\frac{1}{2(\ell -n)}\underset{n}{\overset{\ell }{\int }}\left[\mathcal{G}\left(m,y\right)+\mathcal{G}\left(k,y\right)\right]dy.$$

(6)

In 2012, with the same motivation, Latif and Dragomir [6] developed error estimates for the first and second inequality in (3).

Theorem 3. 

Let $\mathcal{G}:\mathrm{Y}\to \mathbb{R}$ be a partial differentiable mapping on $\mathrm{Y}.$

(a) If $\left|\frac{{\partial }^2\mathcal{G}}{\partial u\partial v}\right|$ is convex on the coordinates on $\mathrm{Y},$ then the following inequality holds:

$$\begin{array}{ccc}\hfill & & \left|\frac{1}{(k-m)(\ell -n)}{\int }_m^k{\int }_n^{\ell }\mathcal{G}(u,v)dvdu+\mathcal{G}\left(\frac{m+k}{2},\frac{n+\ell }{2}\right)-\Psi \right|\hfill \\ & \le & \frac{(k-m)(\ell -n)}{16}\left[\frac{\left|\frac{{\partial }^2\mathcal{G}(m,n)}{\partial u\partial v}\right|+\left|\frac{{\partial }^2\mathcal{G}(m,\ell )}{\partial u\partial v}\right|+\left|\frac{{\partial }^2\mathcal{G}(k,n)}{\partial u\partial v}\right|+\left|\frac{{\partial }^2\mathcal{G}(k,\ell )}{\partial u\partial v}\right|}{4}\right];\hfill \end{array}$$

(7)

(b) If ${\left|\frac{{\partial }^2\mathcal{G}}{\partial u\partial v}\right|}^{\lambda }$ is convex on the coordinates on Y and $\lambda \ge 1,$ then the following inequality holds:

$$\begin{array}{ccc}\hfill & & \left|\frac{1}{(k-m)(\ell -n)}{\int }_m^k{\int }_n^{\ell }\mathcal{G}(u,v)dvdu+\mathcal{G}\left(\frac{m+k}{2},\frac{n+\ell }{2}\right)-\Psi \right|\hfill \\ & \le & \frac{(k-m)(\ell -n)}{16}{\left[\frac{{\left|\frac{{\partial }^2\mathcal{G}(m,n)}{\partial u\partial v}\right|}^{\lambda }+{\left|\frac{{\partial }^2\mathcal{G}(m,\ell )}{\partial u\partial v}\right|}^{\lambda }+{\left|\frac{{\partial }^2\mathcal{G}(k,n)}{\partial u\partial v}\right|}^{\lambda }+{\left|\frac{{\partial }^2\mathcal{G}(k,\ell )}{\partial u\partial v}\right|}^{\lambda }}{4}\right]}^{\frac{1}{\lambda }},\hfill \end{array}$$

(8)

where

$$\Psi =\frac{1}{k-m}{\int }_m^k\mathcal{G}\left(x,\frac{n+\ell }{2}\right)dx+\frac{1}{k-m}{\int }_n^{\ell }\mathcal{G}\left(\frac{m+k}{2},y\right)dy.$$

(9)

Following this commencement, the researchers put a lot of effort into extending and improving the inequality (3) by using both classical and fractional integrals. We believe interested readers should additionally take into account [7,8,9,10,11,12,13,14,15,16] and the references therein for the results that alter, refine, and generalize the inequality (3).

The last ten years have seen an increase in interest in quantum calculus among mathematicians and physicists. Calculus with non-smooth surfaces is studied in quantum calculus. A calculus without limits is what we refer to as quantum calculus. Generally speaking, the topic has many applications in several fields, including the theory of relativity, orthogonal polynomials, combinatorics, number theory, and basic hyper-geometric functions (see, for instance, [17,18,19,20,21,22,23]).

The $\hat{q}$–difference operator, which was re-introduced by Jackson [24] over the $\hat{q}$-geometric set $\mathcal{Q}$, may go back to Heine [25] or Euler, is expressed by the quotient

$$D_{\hat{q}}\mathcal{G}\left(\varpi \right)=\frac{\mathcal{G}\left(\varpi \right)-\mathcal{G}\left(\hat{q}\varpi \right)}{\varpi -\hat{q}\varpi },$$

(10)

where $\hat{q}$–geometric set $\mathcal{Q},$ is a set such that $\hat{q}u\in \mathcal{Q}$ whenever $u\in \mathcal{Q}$ and $\hat{q}$ is a fixed constant.

Jackson initiated a systematic study of this operator in [24,26,27,28,29,30,31,32,33]. The $\hat{q}$–difference operator (10) sometimes called Jackson $\hat{q}$–difference operator, Euler– Jackson $\hat{q}$–difference operator or Euler–Heine–Jackson $\hat{q}$–difference operator.

Jackson [32] presented an integral denoted by

$${\int }_0^1\mathcal{G}\left(t\right)d_{\hat{q}}t$$

as a right inverse of $\hat{q}$–derivative. It is defined by

$${\int }_m^k\mathcal{G}\left(t\right)d_{\hat{q}}t={\int }_0^k\mathcal{G}\left(t\right)d_{\hat{q}}t-{\int }_0^m\mathcal{G}\left(t\right)d_{\hat{q}}t,m,k\in \mathcal{Q},$$

(11)

where

$$\underset{0}{\overset{k}{\int }}\mathcal{G}\left(\varpi \right)d_{\hat{q}}\varpi =(1-\hat{q})k\sum _{\sigma =0}^{\infty }{\hat{q}}^{\sigma }\mathcal{G}\left(k{q}^{\sigma }\right),z\in \mathcal{Q}.$$

(12)

Kac and Cheung [23] (p.$68$) proved that if $x^{\beta }\mathcal{G}\left(x\right)$ is bounded on $[0,k]$ for some $0\le \beta <1$, then ${\int }_0^k\mathcal{G}\left(t\right)d_{\hat{q}}t$ exists for all $x\in [0,k]$. Moreover, Bromwich [34] (pp. 418–419) proved that if ${\int }_0^k\mathcal{G}\left(t\right)dt$ converges, then

$${\int }_0^k\mathcal{G}\left(t\right)dt=\underset{\hat{q}\to 1^{-}}{lim}{\int }_0^k\mathcal{G}\left(t\right)d_{\hat{q}}t.$$

In 1969, Agarwal [35] defined the $\hat{q}$–fractional derivative. Al-Salam [36,37] established a $\hat{q}$-analog of the Riemann–Liouville fractional integral in 1966–1967. A thorough analysis and development of the $\hat{q}$-fractional calculus then started. For a comprehensive overview of the topic, one should consider [38].

The notion of quantum derivatives and integrals across a finite interval is related to the area of quantum calculus that deals with convex functions and corresponding average quantum integrals. In 2013, Tariboon et al. [39,40] initiated the study of $\hat{q}$–derivatives and associated $\hat{q}$–integrals, which opened up a new horizon for the researchers. Following this, scientists started investigating well-known inequalities in the quantum framework. For example, the $\hat{q}$–analog of Hölder’s, Cauchy–Bunyakovsky–Schwarz, Grüss, Grüss–Cebyshev, Hermite–Hadamard, Ostrowski, and other related integral inequalities have been proved. The reader may further look at a few additional sobering discoveries in the field of quantum integral inequalities, particularly [41,42,43,44,45,46,47,48,49,50,51,52,53,54] and the sources given therein.

Motivated by the above results, in general, and the results given in [5,6], in particular, we aim to prove an inequality in the quantum frame work of calculus, which could extend and improve the inequalities of Theorems 2 and 3. The paper also aims to establish the error estimates of the inequality (22); that is a quantum version of the distinguished inequality (3).

2. Basic Literature and Some Recent Advancements in Quantum Math

This section reloads essential findings and terminology that are crucial to comprehending the primary findings.

In 2013–2014, Teriboon et al. [39,40] developed some results in the quantum frame work by extending the Jackson difference operator (10) and Jackson integral (12) over the finite interval as follows:

Definition 2. 

Let $\mathcal{G}:[m,k]\to \mathbb{R}$ be a continuous function and $0<\hat{q}<1.$ Then the ${\hat{q}}_m$–derivative of $\mathcal{G}$ at $\varpi \in [m,k]$ is characterized by the quotient:

$${}_m{D}_{\hat{q}}\mathcal{G}\left(\varpi \right)=\frac{\mathcal{G}\left(\varpi \right)-\mathcal{G}\left(\hat{q}\varpi +(1-\hat{q})m\right)}{(1-\hat{q})(\varpi -m)},\varpi \ne m.$$

(13)

The function $\mathcal{G}$ is called ${\hat{q}}_m$–differentiable on $[m,k]$, if ${}_m{D}_{\hat{q}}\mathcal{G}\left(\varpi \right)$ exists for all $\varpi \in [m,k]$. It is obvious that

$${}_m{D}_{\hat{q}}\mathcal{G}\left(m\right)=\underset{\varpi \to m}{lim}{}_m{D}_{\hat{q}}\mathcal{G}\left(\varpi \right).$$

(14)

If $m=0,$ then the ${\hat{q}}_m$–derivative reduces the Jackson $\hat{q}$–difference operator given by (10).

Definition 3. 

Let $\mathcal{G}:[m,k]\to \mathbb{R}$ be a continuous function and $0<\hat{q}<1.$ Then the ${\hat{q}}_m$-integral of the function $\mathcal{G}$ is defined by the series expression

$$\underset{m}{\overset{z}{\int }}\mathcal{G}\left(\varpi \right){}_m{d}_{\hat{q}}\varpi =(1-\hat{q})(z-m)\sum _{\sigma =0}^{\infty }{\hat{q}}^{\sigma }\mathcal{G}(q^{\sigma }z+(1-{\hat{q}}^{\sigma })m),z\in [m,k].$$

(15)

If $m=0,$ then the ${\hat{q}}_m$–integral coincides to the Jackson integral given by (12).

In the same paper, the following $\hat{q}$-Hölder’s inequality is proved.

Theorem 4. 

Let ${\mathcal{G}}_1,{\mathcal{G}}_2:[m,k]\to \mathbb{R}$ be two continuous functions. Then the inequality

$$\underset{m}{\overset{z}{\int }}\left|{\mathcal{G}}_1\left(\varpi \right){\mathcal{G}}_2\left(\varpi \right)\right|{}_m{d}_{\hat{q}}\varpi \le {\left(\underset{m}{\overset{z}{\int }}{\left|{\mathcal{G}}_1\left(\varpi \right)\right|}^{\gamma }{}_m{d}_{\hat{q}}\varpi \right)}^{\frac{1}{\gamma }}{\left(\underset{m}{\overset{z}{\int }}{\left|{\mathcal{G}}_2\left(\varpi \right)\right|}^{\lambda }{}_m{d}_{\hat{q}}\varpi \right)}^{\frac{1}{\lambda }}$$

(16)

holds for all $z\in [m,k]$ and $\gamma ,\lambda >1$ with ${\gamma }^{-1}+{\lambda }^{-1}=1$.

In 2017, Latif et al. [55] introduced the notion of partial $\widehat{q_1},\widehat{q_2},\widehat{q_1}\widehat{q_2}$—derivative and associated $\widehat{q_1}\widehat{q_2}$—integrals. For brevity, we use the notation $\frac{\partial \mathcal{G}}{{}_m{\partial }_{\widehat{q_1}}u}$ for partial ${\widehat{q_1}}_m$—derivative of a function $\mathcal{G}(u,v)$ with respect to u instead of $\frac{{}_m{\partial }_{\widehat{q_1}}\mathcal{G}}{{}_m{\partial }_{\widehat{q_1}}u}.$ Analogously, we use $\frac{\partial \mathcal{G}}{{}_n{\partial }_{\widehat{q_2}}v}$ instead of $\frac{{}_n{\partial }_{\widehat{q_2}}\mathcal{G}}{{}_m{\partial }_{\widehat{q_2}}v}$ for the partial ${\widehat{q_2}}_n$—derivative and $\frac{{\partial }^2\mathcal{G}}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}$ instead of $\frac{{}_{m,n}{\partial }_{\widehat{q_1}\widehat{q_2}}\mathcal{G}}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}$ for the twice partial ${\widehat{q_1}}_m{\widehat{q_2}}_n$—derivative.

Definition 4. 

Let $\mathcal{G}$ be a function of two variables and $0<\widehat{q_1}<1,0<\widehat{q_2}<1,$ then the partial ${\widehat{q_1}}_m$–derivative, ${\widehat{q_2}}_n$–derivative, and ${\widehat{q_1}}_m{\widehat{q_2}}_n$–derivative are defined at $(u,v)\in \mathrm{Y}$, respectively, as follows:

$$\frac{\partial \mathcal{G}(u,v)}{{}_m{\partial }_{\widehat{q_1}}u}=\frac{\mathcal{G}(u,v)-\mathcal{G}\left(\widehat{q_1}u+(1-\widehat{q_1})m,v\right)}{(1-\widehat{q_1})(u-m)},u\ne m,$$

(17)

$$\frac{\partial \mathcal{G}(u,v)}{{}_n{\partial }_{\widehat{q_2}}u}=\frac{\mathcal{G}(u,v)-\mathcal{G}\left(u,\widehat{q_2}v+(1-\widehat{q_2})n\right)}{(1-\widehat{q_2})(v-n)},v\ne n$$

(18)

and

$$\begin{array}{cc}\hfill \frac{\partial {\mathcal{G}}^2(u,v)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}& =\frac{{(1-\widehat{q_1})}^{-1}{(u-m)}^{-1}}{(1-\widehat{q_2})(v-n)}\left[\mathcal{G}(u,v)-\mathcal{G}\left(\widehat{q_1}u+(1-\widehat{q_1})m,v\right)-\mathcal{G}\left(u,\widehat{q_2}v+(1-\widehat{q_2})n\right)\right.\hfill \\ \hfill & +\left.\mathcal{G}\left(\widehat{q_1}u+(1-\widehat{q_1})m,\widehat{q_2}v+(1-\widehat{q_2})n\right)\right],u\ne m,v\ne n.\hfill \end{array}$$

(19)

Definition 5. 

Let $\mathcal{G}$ be a continuous function of two variables and $0<\widehat{q_1}<1,0<\widehat{q_2}<1,$ then the ${\widehat{q_1}}_m{\widehat{q_2}}_n$–integral is expressed by

$$\begin{array}{cc}\hfill & {\int }_m^u{\int }_n^v\mathcal{G}(\mu ,\omega ){}_n{d}_{\widehat{q_2}}\omega {}_m{d}_{\widehat{q_1}}\mu \hfill \\ \hfill & =(1-\widehat{q_1})(1-\widehat{q_2})(u-m)(v-n)\sum _{\sigma =0}^{\infty }\sum _{\delta =0}^{\infty }{\widehat{q_1}}^{\sigma }{\widehat{q_2}}^{\delta }\mathcal{G}({\widehat{q_1}}^{\sigma }u+(1-{\widehat{q_1}}^{\sigma })m,{\widehat{q_2}}^{\delta }v+(1-{\widehat{q_2}}^{\delta })n),\hfill \end{array}$$

(20)

for all $(u,v)\in \mathrm{Y}.$

In 2019, Kunt et al. [56] proved the following Lemma.

Lemma 1. 

If the conditions of the Definition 5 are satisfied, then

$$\begin{array}{cc}\hfill {\int }_r^x{\int }_s^y\mathcal{G}(u,v){}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u& ={\int }_m^x{\int }_s^y\mathcal{G}(u,v){}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u-{\int }_m^r{\int }_s^y\mathcal{G}(u,v){}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\hfill \\ \hfill & ={\int }_m^x{\int }_n^y\mathcal{G}(u,v){}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u-{\int }_m^x{\int }_n^s\mathcal{G}(u,v){}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\hfill \\ \hfill & -{\int }_m^r{\int }_n^y\mathcal{G}(u,v){}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u+{\int }_m^r{\int }_n^s\mathcal{G}(u,v){}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u,\hfill \end{array}$$

(21)

for all $(u,v)\in \mathrm{Y}.$

The following correct inequality, which extends inequality (3), was established in 2020 by Alp and Sarikaya [57]. It involves quantum integrals.

Theorem 5. 

Let $\mathcal{G}:\mathrm{Y}\to \mathbb{R}$ be a coordinated convex and partially differentiable function on $\mathrm{Y}.$ Then one has the inequalities:

$$\begin{array}{ccc}\hfill & & \mathcal{G}\left(\frac{\widehat{q_1}m+k}{1+\widehat{q_1}},\frac{\widehat{q_2}n+\ell }{1+\widehat{q_2}}\right)\hfill \\ & \le & \frac{1}{2}\left[\frac{1}{k-m}\underset{m}{\overset{k}{\int }}\mathcal{G}\left(x,\frac{\widehat{q_2}n+\ell }{1+\widehat{q_2}}\right){}_m{d}_{\widehat{q_1}}x+\frac{1}{\ell -n}\underset{n}{\overset{\ell }{\int }}\mathcal{G}\left(\frac{\widehat{q_1}m+k}{1+\widehat{q_1}},y\right){}_n{d}_{\widehat{q_2}}y\right]\hfill \\ & \le & \frac{1}{\left(k-m\right)\left(\ell -n\right)}{\int }_m^k{\int }_n^{\ell }\mathcal{G}\left(x,y\right){}_n{d}_{\widehat{q_2}}y{}_m{d}_{\widehat{q_1}}x\hfill \\ & \le & \frac{\widehat{q_2}}{2(1+\widehat{q_2})(k-m)}{\int }_m^k\mathcal{G}\left(x,c\right){}_m{d}_{\widehat{q_1}}x+\frac{1}{2(1+\widehat{q_2})(k-m)}{\int }_m^k\mathcal{G}\left(x,\ell \right){}_m{d}_{\widehat{q_1}}x\hfill \\ & +& \frac{\widehat{q_1}}{2(1+\widehat{q_1})(\ell -n)}{\int }_n^{\ell }\mathcal{G}\left(m,y\right){}_n{d}_{\widehat{q_2}}y+\frac{1}{2(1+\widehat{q_1})(\ell -n)}{\int }_n^{\ell }\mathcal{G}\left(k,y\right){}_n{d}_{\widehat{q_2}}y\hfill \\ & \le & \frac{\widehat{q_1}\widehat{q_2}\mathcal{G}\left(m,n\right)+\widehat{q_1}\mathcal{G}\left(m,\ell \right)+\widehat{q_2}\mathcal{G}\left(k,n\right)+\mathcal{G}\left(k,\ell \right)}{(1+\widehat{q_1})(1+\widehat{q_2})}.\hfill \end{array}$$

(22)

3. Main Results

The study’s key findings are presented in this section. This section is divided into three subsections. In the first subsection, we generate a generic multi-parameter new identity for the twice partially ${\widehat{q_1}}_m{\widehat{q_2}}_n$–differentiable function over the rectangle $\mathrm{Y}=\left[m,k\right]\times \left[n,\ell \right]\subset \mathbb{R}.$ In the second subsection, we present some very general quantum integrals that are essential to the outcomes presented in the following subsection. In the third and final subsection of this section, we first develop a highly generic error formulation for the mean value of double quantum integrals, including several parameters. For the recently corrected quantum Hadamard inequality, (22), we finally acquire some new error estimates.

3.1. New Multi-Parameter Identity for Twice Partially Quantum Differentiable Functions

Lemma 2. 

Let $\mathcal{G}:\mathrm{Y}\to \mathbb{R}$ be a twice partial ${\widehat{q_1}}_m{\widehat{q_2}}_n$–differentiable function defined on ${\mathrm{Y}}^{\circ }.$ If the partial ${\widehat{q_1}}_m{\widehat{q_2}}_n$–derivative $\frac{{\partial }^2\mathcal{G}(u,v)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}$ is continuous and ${\widehat{q_1}}_m{\widehat{q_2}}_n$–integrable over Y. Then the following identity holds:

$$\begin{array}{cc}\hfill & \frac{1}{(k-m)(\ell -n)}{\int }_m^k{\int }_n^{\ell }\mathcal{S}(u,v)\frac{{\partial }^2\mathcal{G}(u,v)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\hfill \\ \hfill & =\frac{1}{(k-m)(\ell -n)}{\int }_m^k{\int }_n^{\ell }\mathcal{G}(x,y){}_n{d}_{\widehat{q_2}}y{}_m{d}_{\widehat{q_1}}x+\mathcal{U}-\mathcal{V},\hfill \end{array}$$

(23)

where

$$\begin{array}{cc}\hfill \mathcal{U}& =\frac{1}{(k-m)(\ell -n)}\left[(\hat{B}-\hat{A})(\hat{D}-\hat{C})\mathcal{G}(d,e)+(\hat{A}-\widehat{q_1}m)(\hat{D}-\hat{C})\mathcal{G}(m,e)\right.\hfill \\ \hfill & +(\hat{C}-\widehat{q_2}n)(\hat{B}-\hat{A})\mathcal{G}(d,n)+(\ell -\hat{D}+\widehat{q_2}n-n)(\hat{B}-\hat{A})\mathcal{G}(d,\ell )\hfill \\ \hfill & +(k-\hat{B}+\widehat{q_1}m-m)(\hat{D}-\hat{C})\mathcal{G}(k,e)+(\hat{A}-\widehat{q_1}m)(\hat{C}-\widehat{q_2}n)\mathcal{G}(m,n)\hfill \\ \hfill & +(\ell -\hat{D}+\widehat{q_2}n-n)(\hat{A}-\widehat{q_1}m)\mathcal{G}(m,\ell )+(k-\hat{B}+\widehat{q_1}m-m)(\hat{C}-\widehat{q_2}n)\mathcal{G}(k,n)\hfill \\ \hfill & +\left.(\ell -\hat{D}+\widehat{q_2}n-n)(k-\hat{B}+\widehat{q_1}m-m)\mathcal{G}(k,\ell )\right]\hfill \end{array}$$

(24)

and

$$\begin{array}{cc}\hfill \mathcal{V}& =\frac{1}{k-m}{\int }_m^k\left[\frac{\hat{C}-\widehat{q_2}n}{\ell -n}\mathcal{G}(x,n)+\frac{\hat{D}-\hat{C}}{\ell -n}\mathcal{G}(x,e)+\frac{\ell -\hat{D}+\widehat{q_2}n-n}{\ell -n}\mathcal{G}(x,\ell )\right]{}_m{d}_{\widehat{q_1}}x\hfill \\ \hfill & +\frac{1}{\ell -n}{\int }_n^{\ell }\left[\frac{\hat{A}-\widehat{q_1}m}{k-m}\mathcal{G}(m,y)+\frac{\hat{B}-\hat{A}}{k-m}\mathcal{G}(d,y)+\frac{k-\hat{B}+\widehat{q_1}m-m}{k-m}\mathcal{G}(k,y)\right]{}_n{d}_{\widehat{q_2}}y.\hfill \end{array}$$

(25)

Proof. 

Using the Lemma 1, we can see right away that

$$\begin{array}{ccc}\hfill & & {\int }_m^k{\int }_n^{\ell }\mathcal{S}(u,v)\frac{{\partial }^2\mathcal{G}(u,v)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\hfill \\ & =& {\int }_m^d{\int }_n^e(\widehat{q_1}u-\hat{A})(\widehat{q_2}v-\hat{C})\frac{{\partial }^2\mathcal{G}(u,v)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\hfill \\ & & +{\int }_m^d{\int }_e^{\ell }(\widehat{q_1}u-\hat{A})(\widehat{q_2}v-\hat{D})\frac{{\partial }^2\mathcal{G}(u,v)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\hfill \\ & & +{\int }_d^k{\int }_n^e(\widehat{q_1}u-\hat{B})(\widehat{q_2}v-\hat{C})\frac{{\partial }^2\mathcal{G}(u,v)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\hfill \\ & & +{\int }_d^k{\int }_e^{\ell }(\widehat{q_1}u-\hat{B})(\widehat{q_2}v-\hat{D})\frac{{\partial }^2\mathcal{G}(u,v)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\hfill \\ & =& {\int }_m^k{\int }_n^{\ell }(\widehat{q_1}u-\hat{B})(\widehat{q_2}v-\hat{D})\frac{{\partial }^2\mathcal{G}(u,v)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\hfill \\ & & +(\hat{D}-\hat{C}){\int }_m^k{\int }_n^e(\widehat{q_1}u-\hat{B})\frac{{\partial }^2\mathcal{G}(u,v)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\hfill \\ & & +(\hat{B}-\hat{A}){\int }_m^d{\int }_n^{\ell }(\widehat{q_2}v-\hat{D})\frac{{\partial }^2\mathcal{G}(u,v)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\hfill \\ & & +(\hat{B}-\hat{A})(\hat{D}-\hat{C}){\int }_m^d{\int }_n^e\frac{{\partial }^2\mathcal{G}(u,v)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\hfill \\ & =& M_1+M_2+M_3+M_4.\hfill \end{array}$$

(26)

Now, using the partial quantum derivatives and integrals denoted, respectively, by Equations (19) and (20), we have

$$\begin{array}{ccc}\hfill M_1& =& {\int }_m^k{\int }_n^{\ell }(\widehat{q_1}u-\hat{B})(\widehat{q_2}v-\hat{D})\frac{{\partial }^2\mathcal{G}(u,v)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\hfill \\ & =& (1-\widehat{q_1})(1-\widehat{q_2})(\ell -n)(k-m)\hfill \\ & & \times \sum _{\sigma =0}^{\infty }\sum _{\delta =0}^{\infty }{\widehat{q_1}}^{\sigma }{\widehat{q_2}}^{\delta }\frac{}{}({\widehat{q_1}}^{\sigma +1}(k-m)+\widehat{q_1}m-\hat{B})({\widehat{q_2}}^{\delta +1}(\ell -n)+\widehat{q_2}n-\hat{D})\hfill \\ & & \times \frac{\left[\begin{array}{c}\mathcal{G}({\widehat{q_1}}^{\sigma }k+(1-{\widehat{q_1}}^{\sigma })m,{\widehat{q_2}}^{\delta }\ell +(1-{\widehat{q_2}}^{\delta })n)\\ -\mathcal{G}({\widehat{q_1}}^{\sigma +1}k+(1-{\widehat{q_1}}^{\sigma +1})m,{\widehat{q_2}}^{\delta }\ell +(1-{\widehat{q_2}}^{\delta })n)\\ -\mathcal{G}({\widehat{q_1}}^{\sigma }k+(1-{\widehat{q_1}}^{\sigma })m,{\widehat{q_2}}^{\delta +1}\ell +(1-{\widehat{q_2}}^{\delta +1})n)\\ +\mathcal{G}({\widehat{q_1}}^{\sigma +1}k+(1-{\widehat{q_1}}^{\sigma +1})m,{\widehat{q_2}}^{\delta +1}\ell +(1-{\widehat{q_2}}^{\delta +1})n)\end{array}\right]}{(1-\widehat{q_1})(1-\widehat{q_2})({\widehat{q_1}}^{\sigma }k+(1-{\widehat{q_1}}^{\sigma })m-m)({\widehat{q_2}}^{\delta }\ell +(1-{\widehat{q_2}}^{\delta })n-n)}\hfill \\ & =& (k-m)(\ell -n)\sum _{\sigma =0}^{\infty }\sum _{\delta =0}^{\infty }{\widehat{q_1}}^{\sigma +1}{\widehat{q_2}}^{\delta +1}\hfill \\ & & \times \left[\begin{array}{c}\mathcal{G}({\widehat{q_1}}^{\sigma }k+(1-{\widehat{q_1}}^{\sigma })m,{\widehat{q_2}}^{\delta }\ell +(1-{\widehat{q_2}}^{\delta })n)\\ -\mathcal{G}({\widehat{q_1}}^{\sigma +1}k+(1-{\widehat{q_1}}^{\sigma +1})m,{\widehat{q_2}}^{\delta }\ell +(1-{\widehat{q_2}}^{\delta })n)\\ -\mathcal{G}({\widehat{q_1}}^{\sigma }k+(1-{\widehat{q_1}}^{\sigma })m,{\widehat{q_2}}^{\delta +1}\ell +(1-{\widehat{q_2}}^{\delta +1})n)\\ +\mathcal{G}({\widehat{q_1}}^{\sigma +1}k+(1-{\widehat{q_1}}^{\sigma +1})m,{\widehat{q_2}}^{\delta +1}\ell +(1-{\widehat{q_2}}^{\delta +1})n)\end{array}\right]\hfill \\ & & +(k-m)(\widehat{q_2}n-\hat{D})\sum _{\sigma =0}^{\infty }\sum _{\delta =0}^{\infty }{\widehat{q_1}}^{\sigma +1}\hfill \\ & & \times \left[\begin{array}{c}\mathcal{G}({\widehat{q_1}}^{\sigma }k+(1-{\widehat{q_1}}^{\sigma })m,{\widehat{q_2}}^{\delta }\ell +(1-{\widehat{q_2}}^{\delta })n)\\ -\mathcal{G}({\widehat{q_1}}^{\sigma +1}k+(1-{\widehat{q_1}}^{\sigma +1})m,{\widehat{q_2}}^{\delta }\ell +(1-{\widehat{q_2}}^{\delta })n)\\ -\mathcal{G}({\widehat{q_1}}^{\sigma }k+(1-{\widehat{q_1}}^{\sigma })m,{\widehat{q_2}}^{\delta +1}\ell +(1-{\widehat{q_2}}^{\delta +1})n)\\ +\mathcal{G}({\widehat{q_1}}^{\sigma +1}k+(1-{\widehat{q_1}}^{\sigma +1})m,{\widehat{q_2}}^{\delta +1}\ell +(1-{\widehat{q_2}}^{\delta +1})n)\end{array}\right]\hfill \\ & & +(\widehat{q_1}m-\hat{B})(\ell -n)\sum _{\sigma =0}^{\infty }\sum _{\delta =0}^{\infty }{\widehat{q_2}}^{\delta +1}\hfill \\ & & \times \left[\begin{array}{c}\mathcal{G}({\widehat{q_1}}^{\sigma }k+(1-{\widehat{q_1}}^{\sigma })m,{\widehat{q_2}}^{\delta }\ell +(1-{\widehat{q_2}}^{\delta })n)\\ -\mathcal{G}({\widehat{q_1}}^{\sigma +1}k+(1-{\widehat{q_1}}^{\sigma +1})m,{\widehat{q_2}}^{\delta }\ell +(1-{\widehat{q_2}}^{\delta })n)\\ -\mathcal{G}({\widehat{q_1}}^{\sigma }k+(1-{\widehat{q_1}}^{\sigma })m,{\widehat{q_2}}^{\delta +1}\ell +(1-{\widehat{q_2}}^{\delta +1})n)\\ +\mathcal{G}({\widehat{q_1}}^{\sigma +1}k+(1-{\widehat{q_1}}^{\sigma +1})m,{\widehat{q_2}}^{\delta +1}\ell +(1-{\widehat{q_2}}^{\delta +1})n)\end{array}\right]\hfill \\ & & +(\widehat{q_1}m-\hat{B})(\widehat{q_2}n-\hat{D})\sum _{\sigma =0}^{\infty }\sum _{\delta =0}^{\infty }\left[\begin{array}{c}\mathcal{G}({\widehat{q_1}}^{\sigma }k+(1-{\widehat{q_1}}^{\sigma })m,{\widehat{q_2}}^{\delta }\ell +(1-{\widehat{q_2}}^{\delta })n)\\ -\mathcal{G}({\widehat{q_1}}^{\sigma +1}k+(1-{\widehat{q_1}}^{\sigma +1})m,{\widehat{q_2}}^{\delta }\ell +(1-{\widehat{q_2}}^{\delta })n)\\ -\mathcal{G}({\widehat{q_1}}^{\sigma }k+(1-{\widehat{q_1}}^{\sigma })m,{\widehat{q_2}}^{\delta +1}\ell +(1-{\widehat{q_2}}^{\delta +1})n)\\ +\mathcal{G}({\widehat{q_1}}^{\sigma +1}k+(1-{\widehat{q_1}}^{\sigma +1})m,{\widehat{q_2}}^{\delta +1}\ell +(1-{\widehat{q_2}}^{\delta +1})n)\end{array}\right]\hfill \\ & =& N_1+N_2+N_3+N_4.\hfill \end{array}$$

(27)

Properties of summation and application of the integral expressed in the Definition 5, we have

$$\begin{array}{ccc}\hfill N_1& =& (k-m)(\ell -n)\left[\widehat{q_1}\widehat{q_2}\sum _{\sigma =0}^{\infty }\sum _{\delta =0}^{\infty }{\widehat{q_1}}^{\sigma }{\widehat{q_2}}^{\delta }\mathcal{G}({\widehat{q_1}}^{\sigma }k+(1-{\widehat{q_1}}^{\sigma })m,{\widehat{q_2}}^{\delta }\ell +(1-{\widehat{q_2}}^{\delta })n)\right.\hfill \\ & & -\widehat{q_2}\sum _{\sigma =1}^{\infty }\sum _{\delta =0}^{\infty }{\widehat{q_1}}^{\sigma }{\widehat{q_2}}^{\delta }\mathcal{G}({\widehat{q_1}}^{\sigma }k+(1-{\widehat{q_1}}^{\sigma })m,{\widehat{q_2}}^{\delta }\ell +(1-{\widehat{q_2}}^{\delta })n)\hfill \\ & & -\widehat{q_1}\sum _{\sigma =0}^{\infty }\sum _{\delta =1}^{\infty }{\widehat{q_1}}^{\sigma }{\widehat{q_2}}^{\delta }\mathcal{G}({\widehat{q_1}}^{\sigma }k+(1-{\widehat{q_1}}^{\sigma })m,{\widehat{q_2}}^{\delta }\ell +(1-{\widehat{q_2}}^{\delta })n)\hfill \\ & & +\left.\sum _{\sigma =1}^{\infty }\sum _{\delta =1}^{\infty }{\widehat{q_1}}^{\sigma }{\widehat{q_2}}^{\delta }\mathcal{G}({\widehat{q_1}}^{\sigma }k+(1-{\widehat{q_1}}^{\sigma })m,{\widehat{q_2}}^{\delta }\ell +(1-{\widehat{q_2}}^{\delta })n)\right]\hfill \\ & =& (k-m)(\ell -n)[(1-\widehat{q_1})(1-\widehat{q_2})\sum _{\sigma =0}^{\infty }\sum _{\delta =0}^{\infty }{\widehat{q_1}}^{\sigma }{\widehat{q_2}}^{\delta }\mathcal{G}({\widehat{q_1}}^{\sigma }k+(1-{\widehat{q_1}}^{\sigma })m,{\widehat{q_2}}^{\delta }\ell +(1-{\widehat{q_2}}^{\delta })n)\hfill \\ & & -(1-\widehat{q_2})\sum _{\delta =0}^{\infty }{\widehat{q_2}}^{\delta }\mathcal{G}(k,{\widehat{q_2}}^{\delta }\ell +(1-{\widehat{q_2}}^{\delta })n)-(1-\widehat{q_1})\sum _{\sigma =0}^{\infty }{\widehat{q_1}}^{\sigma }\mathcal{G}({\widehat{q_1}}^{\sigma }k+(1-{\widehat{q_1}}^{\sigma })m,\ell )+\mathcal{G}(k,\ell )]\hfill \\ & =& {\int }_m^k{\int }_n^{\ell }\mathcal{G}(x,y){}_n{d}_{\widehat{q_2}}y{}_m{d}_{\widehat{q_1}}x-(\ell -n){\int }_m^k\mathcal{G}(x,\ell ){}_m{d}_{\widehat{q_1}}x-(k-m){\int }_n^{\ell }\mathcal{G}(k,y){}_n{d}_{\widehat{q_1}}y\hfill \\ & & +(k-m)(\ell -n)\mathcal{G}(k,\ell ).\hfill \end{array}$$

(28)

Similarly,

$$\begin{array}{ccc}\hfill N_2& =& (k-m)(\widehat{q_2}n-\hat{D})\sum _{\sigma =0}^{\infty }\sum _{\delta =0}^{\infty }{\widehat{q_1}}^{\sigma +1}\left[\begin{array}{c}\mathcal{G}({\widehat{q_1}}^{\sigma }k+(1-{\widehat{q_1}}^{\sigma })m,{\widehat{q_2}}^{\delta }\ell +(1-{\widehat{q_2}}^{\delta })n)\\ -\mathcal{G}({\widehat{q_1}}^{\sigma +1}k+(1-{\widehat{q_1}}^{\sigma +1})m,{\widehat{q_2}}^{\delta }\ell +(1-{\widehat{q_2}}^{\delta })n)\\ -\mathcal{G}({\widehat{q_1}}^{\sigma }k+(1-{\widehat{q_1}}^{\sigma })m,{\widehat{q_2}}^{\delta +1}\ell +(1-{\widehat{q_2}}^{\delta +1})n)\\ +\mathcal{G}({\widehat{q_1}}^{\sigma +1}k+(1-{\widehat{q_1}}^{\sigma +1})m,{\widehat{q_2}}^{\delta +1}\ell +(1-{\widehat{q_2}}^{\delta +1})n)\end{array}\right]\hfill \\ & =& -(\widehat{q_2}n-\hat{D}){\int }_m^k[\mathcal{G}(x,\ell )-\mathcal{G}(x,n)]{}_m{d}_{\widehat{q_1}}x+(k-m)(\widehat{q_2}n-\hat{D})\mathcal{G}(k,\ell )\hfill \\ & & -(k-m)(\widehat{q_2}n-\hat{D})\mathcal{G}(k,n),\hfill \end{array}$$

(29)

$$\begin{array}{ccc}\hfill N_3& =& (\ell -n)(\widehat{q_1}m-\hat{B})\sum _{\sigma =0}^{\infty }\sum _{\delta =0}^{\infty }{\widehat{q_2}}^{\delta +1}\left[\begin{array}{c}\mathcal{G}({\widehat{q_1}}^{\sigma }k+(1-{\widehat{q_1}}^{\sigma })m,{\widehat{q_2}}^{\delta }\ell +(1-{\widehat{q_2}}^{\delta })n)\\ -\mathcal{G}({\widehat{q_1}}^{\sigma +1}k+(1-{\widehat{q_1}}^{\sigma +1})m,{\widehat{q_2}}^{\delta }\ell +(1-{\widehat{q_2}}^{\delta })n)\\ -\mathcal{G}({\widehat{q_1}}^{\sigma }k+(1-{\widehat{q_1}}^{\sigma })m,{\widehat{q_2}}^{\delta +1}\ell +(1-{\widehat{q_2}}^{\delta +1})n)\\ +\mathcal{G}({\widehat{q_1}}^{\sigma +1}k+(1-{\widehat{q_1}}^{\sigma +1})m,{\widehat{q_2}}^{\delta +1}\ell +(1-{\widehat{q_2}}^{\delta +1})n)\end{array}\right]\hfill \\ & =& -(\widehat{q_1}m-\hat{B}){\int }_n^{\ell }[\mathcal{G}(k,y)-\mathcal{G}(m,y)]{}_n{d}_{\widehat{q_2}}y+(\ell -n)(\widehat{q_1}m-\hat{B})\mathcal{G}(k,\ell )\hfill \\ & & -(\ell -n)(\widehat{q_1}m-\hat{B})\mathcal{G}(m,\ell )\hfill \end{array}$$

(30)

and

$$\begin{array}{cc}\hfill N_4& =(\widehat{q_1}m-\hat{B})(\widehat{q_2}n-\hat{D})\sum _{\sigma =0}^{\infty }\sum _{\delta =0}^{\infty }\left[\begin{array}{c}\mathcal{G}({\widehat{q_1}}^{\sigma }k+(1-{\widehat{q_1}}^{\sigma })m,{\widehat{q_2}}^{\delta }\ell +(1-{\widehat{q_2}}^{\delta })n)\\ -\mathcal{G}({\widehat{q_1}}^{\sigma +1}k+(1-{\widehat{q_1}}^{\sigma +1})m,{\widehat{q_2}}^{\delta }\ell +(1-{\widehat{q_2}}^{\delta })n)\\ -\mathcal{G}({\widehat{q_1}}^{\sigma }k+(1-{\widehat{q_1}}^{\sigma })m,{\widehat{q_2}}^{\delta +1}\ell +(1-{\widehat{q_2}}^{\delta +1})n)\\ +\mathcal{G}({\widehat{q_1}}^{\sigma +1}k+(1-{\widehat{q_1}}^{\sigma +1})m,{\widehat{q_2}}^{\delta +1}\ell +(1-{\widehat{q_2}}^{\delta +1})n)\end{array}\right]\hfill \\ \hfill & =(\widehat{q_1}m-\hat{B})(\widehat{q_2}n-\hat{D})[\mathcal{G}(k,\ell )-\mathcal{G}(m,\ell )-\mathcal{G}(k,n)+\mathcal{G}(m,n)].\hfill \end{array}$$

(31)

Using Equations (28)–(31) in (27), we have

$$\begin{array}{ccc}\hfill M_1& =& {\int }_m^k{\int }_n^{\ell }(\widehat{q_1}u-\hat{B})(\widehat{q_2}v-\hat{D})\frac{{\partial }^2\mathcal{G}(u,v)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\hfill \\ & =& {\int }_m^k{\int }_n^{\ell }\mathcal{G}(x,y){}_n{d}_{\widehat{q_2}}y{}_m{d}_{\widehat{q_1}}x\hfill \\ & & -{\int }_n^{\ell }\left[(k-\hat{B}+\widehat{q_1}m-m)\mathcal{G}(k,y)-(\widehat{q_1}m-\hat{B})\mathcal{G}(m,y)\right]{}_n{d}_{\widehat{q_2}}y\hfill \\ & & -{\int }_m^k{\left[(\ell -\hat{D}+\widehat{q_2}n-n)\mathcal{G}(x,\ell )-(\widehat{q_2}n-\hat{D})\mathcal{G}(x,n)\right]}_m{d}_{\widehat{q_1}}x\hfill \\ & & +(k-\hat{B}+\widehat{q_1}m-m)(\ell -\hat{D}+\widehat{q_2}n-n)\mathcal{G}(k,\ell )+(\widehat{q_1}m-\hat{B})(\widehat{q_2}n-\hat{D})\mathcal{G}(m,n)\hfill \\ & & -(k-\hat{B}+\widehat{q_1}m-m)(\widehat{q_2}n-\hat{D})\mathcal{G}(k,n)-(\widehat{q_1}m-\hat{B})(\ell -\hat{D}+\widehat{q_2}n-n)\mathcal{G}(m,\ell ).\hfill \end{array}$$

(32)

The same methodology leads to:

$$\begin{array}{ccc}\hfill M_2& =& {\int }_m^k{\int }_n^e(\widehat{q_1}u-\hat{B})(\hat{D}-\hat{C})\frac{{\partial }^2\mathcal{G}(u,v)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\hfill \\ & =& (\hat{D}-\hat{C}){\int }_m^k[\mathcal{G}(x,n)-\mathcal{G}(x,e)]{}_m{d}_{\widehat{q_1}}x+(\hat{D}-\hat{C})(k-\hat{B}+\widehat{q_1}m-m)[\mathcal{G}(k,e)-\mathcal{G}(k,n)]\hfill \\ & & +(\hat{D}-\hat{C})(\widehat{q_1}m-\hat{B})[\mathcal{G}(m,n)-\mathcal{G}(m,e)],\hfill \end{array}$$

(33)

$$\begin{array}{ccc}\hfill M_3& =& {\int }_m^d{\int }_n^{\ell }(\widehat{q_2}v-\hat{D})(\hat{B}-\hat{A})\frac{{\partial }^2\mathcal{G}(u,v)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\hfill \\ & =& (\hat{B}-\hat{A}){\int }_n^{\ell }[\mathcal{G}(m,y)-\mathcal{G}(d,y)]{}_m{d}_{\widehat{q_1}}x+(\hat{B}-\hat{A})(\ell -\hat{D}+\widehat{q_2}n-n)[\mathcal{G}(d,\ell )-\mathcal{G}(m,\ell )]\hfill \\ & & +(\hat{B}-\hat{A})(\widehat{q_2}n-\hat{D})[\mathcal{G}(m,n)-\mathcal{G}(d,n)]\hfill \end{array}$$

(34)

and

$$\begin{array}{cc}\hfill M_4& ={\int }_m^d{\int }_n^e(\hat{B}-\hat{A})(\hat{D}-\hat{C})\frac{{\partial }^2\mathcal{G}(u,v)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\hfill \\ \hfill & =(\hat{B}-\hat{A})(\hat{D}-\hat{C})[\mathcal{G}(d,e)-\mathcal{G}(d,n)-\mathcal{G}(m,e)+\mathcal{G}(m,n)].\hfill \end{array}$$

(35)

Now utilizing Equations (32)–(35) in Equation (26), we have

$$\begin{array}{cc}\hfill & {\int }_m^k{\int }_n^{\ell }\mathcal{S}(u,v)\frac{{\partial }^2\mathcal{G}(u,v)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_1}}v}{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\hfill \\ \hfill & ={\int }_m^k{\int }_n^{\ell }\mathcal{G}(x,y){}_n{d}_{\widehat{q_2}}y{}_m{d}_{\widehat{q_1}}x+(\hat{B}-\hat{A})(\hat{D}-\hat{C})\mathcal{G}(d,e)+(\hat{A}-\widehat{q_1}m)(\hat{D}-\hat{C})\mathcal{G}(m,e)\hfill \\ \hfill & +(\hat{C}-\widehat{q_2}n)(\hat{B}-\hat{A})\mathcal{G}(d,n)+(\ell -\hat{D}+\widehat{q_2}n-n)(\hat{B}-\hat{A})\mathcal{G}(d,\ell )\hfill \\ \hfill & +(k-\hat{B}+\widehat{q_1}m-m)(\hat{D}-\hat{C})\mathcal{G}(k,e)+(\hat{A}-\widehat{q_1}m)(\hat{C}-\widehat{q_2}n)\mathcal{G}(m,n)\hfill \\ \hfill & +(\ell -\hat{D}+\widehat{q_2}n-n)(\hat{A}-\widehat{q_1}m)\mathcal{G}(m,\ell )+(k-\hat{B}+\widehat{q_1}m-m)(\hat{C}-\widehat{q_2}n)\mathcal{G}(k,n)\hfill \\ \hfill & +(\ell -\hat{D}+\widehat{q_2}n-n)(k-\hat{B}+\widehat{q_1}m-m)\mathcal{G}(k,\ell )\hfill \\ \hfill & -{\int }_m^k\left[(\hat{C}-\widehat{q_2}n)\mathcal{G}(x,n)+(\hat{D}-\hat{C})\mathcal{G}(x,e)+(\ell -\hat{D}+\widehat{q_2}n-n)\mathcal{G}(x,\ell )\right]{}_m{d}_{\widehat{q_1}}x\hfill \\ \hfill & -{\int }_n^{\ell }\left[(\hat{A}-\widehat{q_1}m)\mathcal{G}(m,y)+(\hat{B}-\hat{A})\mathcal{G}(d,y)+(k-\hat{B}+\widehat{q_1}m-m)\mathcal{G}(k,y)\right]{}_n{d}_{\widehat{q_2}}y.\hfill \end{array}$$

(36)

Thus, the desired identity (23) is obtained by multiplying the Equation (36) with $\frac{1}{(k-m)(\ell -n)}$. □

3.2. Useful General Quantum Integrals

In this part, we provide highly general quantum integrals that are crucial for error estimates of the Hermite–Hadamard inequality.

Let $\Re \in \mathbb{R}$ and $\hat{q}$ be any real number such that $0<\hat{q}<1.$ Assume further that $m<d<k,m,d,k\in \mathbb{R}.$ Then we have the following $\hat{q}$–integrals, which are denoted and defined by:

$\left(1\right){}_m^d{\mathcal{K}}_1(\Re ;\hat{q}):={\int }_m^d\left|\hat{q}x-\Re \right|\left(d-x\right){}_m{d}_{\hat{q}}x$

$$=\left\{\begin{array}{cc}-q\frac{{\left(d-m\right)}^2}{{\hat{q}}^3+2{\hat{q}}^2+2\hat{q}+1}\left[\Re +\Re {\hat{q}}^2-d{\hat{q}}^2-m{\hat{q}}^3+\Re \hat{q}-m\hat{q}\right],\hfill & \mathrm{if}\frac{\Re }{\hat{q}}\le m,\hfill \\ -\frac{1}{{\hat{q}}^3+2{\hat{q}}^2+2\hat{q}+1}[2{\Re }^3-2{\Re }^{2}d{\hat{q}}^2-2{\Re }^{2}d\hat{q}-2{\Re }^{2}d+2{\Re }^{2}m{\hat{q}}^2-4{\Re }^{2}m\hat{q}\hfill \\ +2{\Re }^{2}m+\Re d^2{\hat{q}}^3+\Re d^2{\hat{q}}^2+\Re d^2\hat{q}+2Adm{\hat{q}}^3+2\Re dm{\hat{q}}^2+2\Re dm\hat{q}\hfill \\ -3\Re m^2{\hat{q}}^3+3\Re m^2{\hat{q}}^2-3\Re m^2\hat{q}-d^3{\hat{q}}^3-d^{2}m{\hat{q}}^4+2d^{2}m{\hat{q}}^3-d^{2}m{\hat{q}}^2\hfill \\ -3d{m}^2{\hat{q}}^3+m^3{\hat{q}}^4+m^3{\hat{q}}^2],\hfill & \mathrm{if}m<\frac{\Re }{\hat{q}}<d,\hfill \\ \hat{q}\frac{{\left(d-m\right)}^2}{{\hat{q}}^3+2{\hat{q}}^2+2\hat{q}+1}\left[\Re +\Re {\hat{q}}^2-d{\hat{q}}^2-m{\hat{q}}^3+\Re \hat{q}-m\hat{q}\right],\hfill & \mathrm{if}d\le \frac{\Re }{\hat{q}}.\hfill \end{array}\right.$$

$\left(2\right){}_m^d{\mathcal{K}}_2(\Re ;\hat{q}):={\int }_m^d\left|\hat{q}x-\Re \right|\left(x-m\right){}_m{d}_{\hat{q}}x$

$$=\left\{\begin{array}{cc}-\frac{{\left(d-m\right)}^2}{{\hat{q}}^3+2{\hat{q}}^2+2\hat{q}+1}\left[\Re +\Re {\hat{q}}^2-d{\hat{q}}^2-m{\hat{q}}^3+\Re \hat{q}-d\hat{q}\right],\hfill & \mathrm{if}\frac{\Re }{\hat{q}}\le m,\hfill \\ -\frac{1}{{\hat{q}}^3+2{\hat{q}}^2+2\hat{q}+1}[-2{\Re }^3\hat{q}+6{\Re }^{2}m{\hat{q}}^2+\Re d^2{\hat{q}}^2+\Re d^2\hat{q}+\Re d^2\hfill \\ -2\Re dm{\hat{q}}^2-2\Re dm\hat{q}-2\Re dm-6\Re m^2{\hat{q}}^3+\Re m^2{\hat{q}}^2+\Re m^2\hat{q}\hfill \\ +\Re m^2-d^3{\hat{q}}^2-d^3\hat{q}-d^{2}m{\hat{q}}^3+2d^{2}m{\hat{q}}^2\hfill \\ +2d^{2}m\hat{q}+2d{m}^2{\hat{q}}^3-d{m}^2{\hat{q}}^2-d{m}^2\hat{q}+2m^3{\hat{q}}^4-m^3{\hat{q}}^3],\hfill & \mathrm{if}m<\frac{\Re }{\hat{q}}<d,\hfill \\ \frac{{\left(d-m\right)}^2}{{\hat{q}}^3+2{\hat{q}}^2+2\hat{q}+1}\left[\Re +\Re {\hat{q}}^2-d{\hat{q}}^2-m{\hat{q}}^3+\Re \hat{q}-d\hat{q}\right],\hfill & \mathrm{if}d\le \frac{\Re }{\hat{q}}.\hfill \end{array}\right.$$

$\left(3\right){}_d^k{\mathcal{L}}_1(\Re ;\hat{q}):={\int }_d^k\left|\hat{q}x-\Re \right|\left(k-x\right){}_m{d}_{\hat{q}}x$

$$=\left\{\begin{array}{cc}\frac{\left(d-k\right)(d-m-k\hat{q}+m\hat{q})}{{\hat{q}}^3+2{\hat{q}}^2+2\hat{q}+1}\left[\Re +\Re {\hat{q}}^2-d{\hat{q}}^2-k{\hat{q}}^2+m{\hat{q}}^2-m{\hat{q}}^3+\Re \hat{q}-d\hat{q}\right],\hfill & \mathrm{if}\frac{\Re }{\hat{q}}\le d,\hfill \\ -\frac{1}{{\hat{q}}^3+2{\hat{q}}^2+2\hat{q}+1}[2{\Re }^3-2{\Re }^{2}k{\hat{q}}^2-2{\Re }^{2}k\hat{q}-2{\Re }^{2}k+2{\Re }^{2}m{\hat{q}}^2\hfill \\ -4{\Re }^{2}m\hat{q}+2{\Re }^{2}m-\Re d^2{\hat{q}}^2-\Re d^2\hat{q}-\Re d^2+\Re dk{\hat{q}}^3+2\Re dk{\hat{q}}^2\hfill \\ +2\Re dk\hat{q}+\Re dk-\Re dm{\hat{q}}^3+\Re dm+\Re k^2{\hat{q}}^3+\Re k^2{\hat{q}}^2\hfill \\ +\Re k^2\hat{q}+\Re km{\hat{q}}^3-\Re km-2\Re m^2{\hat{q}}^3+4\Re m^2{\hat{q}}^2-2\Re m^2\hat{q}\hfill \\ +d^3{\hat{q}}^2+d^3\hat{q}-d^{2}k{\hat{q}}^3-d^{2}k{\hat{q}}^2-d^{2}k\hat{q}+2d^{2}m{\hat{q}}^3-d^{2}m{\hat{q}}^2\hfill \\ -d^{2}m\hat{q}-dkm{\hat{q}}^4+dkm\hat{q}+d{m}^2{\hat{q}}^4-2d{m}^2{\hat{q}}^3+d{m}^2{\hat{q}}^2\hfill \\ -k^3{\hat{q}}^3-k^{2}m{\hat{q}}^4+2k^{2}m{\hat{q}}^3-k^{2}m{\hat{q}}^2+k{m}^2{\hat{q}}^4-2k{m}^2{\hat{q}}^3+k{m}^2{\hat{q}}^2],\hfill & \mathrm{if}d<\frac{\Re }{\hat{q}}<k,\hfill \\ -\frac{\left(d-k\right)(d-m-k\hat{q}+m\hat{q})}{{\hat{q}}^3+2{\hat{q}}^2+2\hat{q}+1}\left[\Re +\Re {\hat{q}}^2-d{\hat{q}}^2-k{\hat{q}}^2+m{\hat{q}}^2-m{\hat{q}}^3+\Re \hat{q}-d\hat{q}\right],\hfill & \mathrm{if}k\le \frac{\Re }{\hat{q}}.\hfill \end{array}\right.$$

$\left(4\right){}_d^k{\mathcal{L}}_2(\Re ;\hat{q}):={\int }_d^k\left|\hat{q}x-\Re \right|\left(x-d\right){}_m{d}_{\hat{q}}x$

$$=\left\{\begin{array}{cc}\frac{\left(d-k\right)(k-m-\Re \hat{q}+m\hat{q})}{{\hat{q}}^3+2{\hat{q}}^2+2\hat{q}+1}\left[\Re +\Re {\hat{q}}^2-d{\hat{q}}^2-k{\hat{q}}^2+m{\hat{q}}^2-m{\hat{q}}^3+\Re \hat{q}-k\hat{q}\right],\hfill & \mathrm{if}\frac{\Re }{\hat{q}}\le d,\hfill \\ \frac{1}{{\hat{q}}^3+2{\hat{q}}^2+2\hat{q}+1}\left[2{\Re }^3-2{\Re }^{2}d{\hat{q}}^2-2{\Re }^{2}d\hat{q}-2{\Re }^{2}d+2{\Re }^{2}m{\hat{q}}^2-4{\Re }^{2}m\hat{q}+2{\Re }^{2}m\right.\hfill \\ +\Re d^2{\hat{q}}^3+\Re d^2{\hat{q}}^2+\Re d^2\hat{q}+\Re dk{\hat{q}}^3+2\Re dk{\hat{q}}^2+2\Re dk\hat{q}+\Re dk+\Re dm{\hat{q}}^3\hfill \\ -\Re dm-\Re k^2{\hat{q}}^2-\Re k^2\hat{q}-\Re k^2-\Re km{\hat{q}}^3+\Re km-2\Re m^2{\hat{q}}^3+4\Re m^2{\hat{q}}^2\hfill \\ -2\Re m^2\hat{q}-d^3{\hat{q}}^3-d^{2}m{\hat{q}}^4+2d^{2}m{\hat{q}}^3-d^{2}m{\hat{q}}^2-d{k}^2{\hat{q}}^3-d{k}^2{\hat{q}}^2\hfill \\ -d{k}^{2}q-dkm{\hat{q}}^4+dkm\hat{q}+d{m}^2{\hat{q}}^4-2d{m}^2{\hat{q}}^3+d{m}^2{\hat{q}}^2+k^3{\hat{q}}^2\hfill \\ \left.+k^3\hat{q}+2k^{2}m{\hat{q}}^3-k^{2}m{\hat{q}}^2-k^{2}m\hat{q}+k{m}^2{\hat{q}}^4-2k{m}^2{\hat{q}}^3+k{m}^2{\hat{q}}^2\right],\hfill & \mathrm{if}d<\frac{\Re }{\hat{q}}<k,\hfill \\ -\frac{\left(d-k\right)(k-m-d\hat{q}+m\hat{q})}{{\hat{q}}^3+2{\hat{q}}^2+2\hat{q}+1}\left[\Re +\Re {\hat{q}}^2-d{\hat{q}}^2-k{\hat{q}}^2+m{\hat{q}}^2-m{\hat{q}}^3+\Re \hat{q}-k\hat{q}\right],\hfill & \mathrm{if}k\le \frac{\Re }{\hat{q}}.\hfill \end{array}\right.$$

$\left(5\right){}_m^k\mathcal{M}(\Re ,\hat{q}):={\int }_m^k\left|\hat{q}u-\Re \right|{}_m{d}_{q}u$

$$=\left\{\begin{array}{cc}-\frac{k-m}{1+\hat{q}}(\Re -m{\hat{q}}^2+\Re \hat{q}-k\hat{q}),\hfill & \mathrm{if}\frac{\Re }{\hat{q}}\le m,\hfill \\ \frac{1}{1+\hat{q}}\left(2{\Re }^2\hat{q}-\Re k{\hat{q}}^2-\Re k\hat{q}-3\Re m{\hat{q}}^2+\Re m\hat{q}+k{\hat{q}}^2\right.\hfill \\ +km{\hat{q}}^3-\left.km{\hat{q}}^2+m^2{\hat{q}}^3\right)\hfill & \mathrm{if}m<\frac{\Re }{\hat{q}}<k,\hfill \\ \frac{k-m}{1+\hat{q}}(\Re -m{\hat{q}}^2+\Re \hat{q}-k\hat{q}),\hfill & \mathrm{if}k\le \frac{\Re }{\hat{q}}.\hfill \end{array}\right.$$

$\left(6\right){}_m^d\mathcal{N}\left(\hat{q}\right)={\int }_m^d(d-u){}_m{d}_{\hat{q}}u=\frac{\hat{q}{(d-m)}^2}{1+\hat{q}}.$

$\left(7\right){}_m^d\mathcal{R}\left(\hat{q}\right)={\int }_m^d(u-m){}_m{d}_{\hat{q}}u=\frac{{(d-m)}^2}{1+\hat{q}}.$

Remark 1. 

The above integrals suffices to evaluate other integrals with appropriate limits. For instance, if one needs ${}_n^{\ell }{K}_1(\hat{C};\widehat{q_2}),$ then one should replace m by n,d by e, ℜ by $\hat{C}$ and $\hat{q}$ by $\widehat{q_2}.$ All the other integrals are one step ahead by this scheme. We left the details for interested readers.

3.3. New Generalized Error Formulation Concerning Hadamard Type Inequalities and Its Applications

Theorem 6. 

Let $\mathcal{G}:\mathrm{Y}\to \mathbb{R}$ be a twice partial ${\widehat{q_1}}_m{\widehat{q_2}}_n$–differentiable function defined on ${\mathrm{Y}}^{\circ }$ such that the partial ${\widehat{q_1}}_m{\widehat{q_2}}_n$–derivative $\frac{{\partial }^2\mathcal{G}(u,v)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}$ is continuous and ${\widehat{q_1}}_m{\widehat{q_2}}_n$–integrable over Y. If $\left|\frac{{\partial }^2\mathcal{G}(u,v)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|$ is coordinated convex on $\mathrm{Y},$ then the following inequalities hold:

$$\begin{array}{ccc}\hfill & & \left|\frac{1}{(k-m)(\ell -n)}{\int }_m^k{\int }_n^{\ell }\mathcal{G}(x,y){}_n{d}_{\widehat{q_2}}y{}_m{d}_{\widehat{q_1}}x+\mathcal{U}-\mathcal{V}\right|\hfill \\ & \le & \frac{1}{(k-m)(\ell -n)}\left\{\frac{1}{(d-m)(e-n)}\left(\left|\frac{{\partial }^2\mathcal{G}\left(m,n\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|{}_m^d{\mathcal{K}}_1(\hat{A};\widehat{q_1}){}_n^e{\mathcal{K}}_1(\hat{C};\widehat{q_2})\right.\right.\hfill \\ & & +\left|\frac{{\partial }^2\mathcal{G}\left(m,e\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|{}_m^d{\mathcal{K}}_1(\hat{A};\widehat{q_1}){}_n^e{\mathcal{K}}_2(\hat{C};\widehat{q_2})+\left|\frac{{\partial }^2\mathcal{G}\left(d,n\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|{}_m^d{\mathcal{K}}_2(\hat{A};\widehat{q_1}){}_n^e{\mathcal{K}}_1(\hat{C};\widehat{q_2})\hfill \\ & & +\left.\left|\frac{{\partial }^2\mathcal{G}\left(d,e\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|{}_m^d{\mathcal{K}}_2(\hat{A};\widehat{q_1}){}_n^e{\mathcal{K}}_2(\hat{C};\widehat{q_2})\right)\hfill \\ & & +\frac{1}{(k-d)(e-n)}\left(\left|\frac{{\partial }^2\mathcal{G}\left(d,n\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|{}_d^k{\mathcal{L}}_1(\hat{B};\widehat{q_1}){}_n^e{\mathcal{K}}_1(\hat{C};\widehat{q_2})+\left|\frac{{\partial }^2\mathcal{G}\left(d,e\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|{}_d^k{\mathcal{L}}_1(\hat{B};\widehat{q_1}){}_n^e{\mathcal{K}}_2(\hat{C};\widehat{q_2})\right.\hfill \\ & & +\left|\frac{{\partial }^2\mathcal{G}\left(k,n\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|{}_d^k{\mathcal{L}}_2(\hat{B};\widehat{q_1}){}_n^e{\mathcal{K}}_1(\hat{C};\widehat{q_2})\left.+\left|\frac{{\partial }^2\mathcal{G}\left(k,e\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|{}_d^k{\mathcal{L}}_2(\hat{B};\widehat{q_1}){}_n^e{\mathcal{K}}_2(\hat{C};\widehat{q_2})\right)\hfill \\ & & +\frac{1}{(d-m)(\ell -e)}\left(\left|\frac{{\partial }^2\mathcal{G}\left(m,e\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|{}_d^k{\mathcal{K}}_1(\hat{A};\widehat{q_1}){}_e^{\ell }{\mathcal{L}}_1(\hat{D};\widehat{q_2})+\left|\frac{{\partial }^2\mathcal{G}\left(m,\ell \right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|{}_d^k{\mathcal{K}}_1(\hat{A};\widehat{q_1}){}_e^{\ell }{\mathcal{L}}_2(\hat{D};\widehat{q_2})\right.\hfill \\ & & +\left|\frac{{\partial }^2\mathcal{G}\left(d,e\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|{}_d^k{\mathcal{K}}_2(\hat{A};\widehat{q_1}){}_e^{\ell }{\mathcal{L}}_1(\hat{D};\widehat{q_2})\left.+\left|\frac{{\partial }^2\mathcal{G}\left(d,\ell \right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|{}_d^k{\mathcal{K}}_2(\hat{A};\widehat{q_1}){}_e^{\ell }{\mathcal{L}}_2(\hat{D};\widehat{q_2})\right)\hfill \\ & & +\frac{1}{(k-d)(\ell -e)}\left(\left|\frac{{\partial }^2\mathcal{G}\left(d,e\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|{}_d^k{\mathcal{L}}_1(\hat{B};\widehat{q_1}){}_e^{\ell }{\mathcal{L}}_1(\hat{D};\widehat{q_2})+\left|\frac{{\partial }^2\mathcal{G}\left(d,\ell \right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|{}_d^k{\mathcal{L}}_1(\hat{B};\widehat{q_1}){}_e^{\ell }{\mathcal{L}}_2(\hat{D};\widehat{q_2})\right.\hfill \\ & & +\left.\left|\frac{{\partial }^2\mathcal{G}\left(k,e\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|{}_d^k{\mathcal{L}}_2(\hat{B};\widehat{q_1}){}_e^{\ell }{\mathcal{L}}_1(\hat{D};\widehat{q_2})\left.+\left|\frac{{\partial }^2\mathcal{G}\left(k,\ell \right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|{}_d^k{\mathcal{L}}_2(\hat{B};\widehat{q_1}){}_e^{\ell }{\mathcal{L}}_2(\hat{D};\widehat{q_2})\right)\right\}.\hfill \end{array}$$

(37)

Proof. 

Let us consider the identity mappings,

$$\begin{array}{cc}\hfill {\mathcal{I}}_1\left(u\right):& =\left\{\begin{array}{cc}\frac{d-u}{d-m}m+\frac{u-m}{d-m}d,\hfill & \mathrm{if}m<u<d,\hfill \\ \frac{k-u}{k-d}d+\frac{u-d}{k-d}k,\hfill & \mathrm{if}d<u<k\hfill \end{array}\right.\hfill \end{array}$$

(38)

and

$$\begin{array}{cc}\hfill {\mathcal{I}}_2\left(v\right):& =\left\{\begin{array}{cc}\frac{e-v}{e-n}n+\frac{v-n}{e-n}e,\hfill & \mathrm{if}n<v<e,\hfill \\ \frac{\ell -v}{\ell -e}e+\frac{v-e}{\ell -e}\ell ,\hfill & \mathrm{if}e<v<\ell .\hfill \end{array}\right.\hfill \end{array}$$

(39)

Taking modulus on both sides of Equation (23), we have

$$\begin{array}{ccc}\hfill & & \left|\frac{1}{(k-m)(\ell -n)}{\int }_m^k{\int }_n^{\ell }\mathcal{G}(x,y){}_n{d}_{\widehat{q_2}}y{}_m{d}_{\widehat{q_1}}x+\mathcal{U}-\mathcal{V}\right|\hfill \\ & \le & \frac{1}{(k-m)(\ell -n)}{\int }_m^k{\int }_n^{\ell }\left|\mathcal{S}(u,v)\right|\left|\frac{{\partial }^2\mathcal{G}(u,v)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\hfill \\ & =& \frac{1}{(k-m)(\ell -n)}\hfill \\ & & \times \left[{\int }_m^d{\int }_n^e\left|(\widehat{q_1}u-\hat{A})(\widehat{q_2}v-\hat{C})\right|\left|\frac{{\partial }^2\mathcal{G}\left({\mathcal{I}}_1\left(u\right),{\mathcal{I}}_2\left(v\right)\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\right.\hfill \\ & & +{\int }_d^k{\int }_n^e\left|(\widehat{q_1}u-\hat{B})(\widehat{q_2}v-\hat{C})\right|\left|\frac{{\partial }^2\mathcal{G}\left({\mathcal{I}}_1\left(u\right),{\mathcal{I}}_2\left(v\right)\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\hfill \\ & & +{\int }_m^d{\int }_e^{\ell }\left|(\widehat{q_1}u-\hat{A})(\widehat{q_2}v-\hat{D})\right|\left|\frac{{\partial }^2\mathcal{G}\left({\mathcal{I}}_1\left(u\right),{\mathcal{I}}_2\left(v\right)\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\hfill \\ & & +\left.{\int }_d^k{\int }_e^{\ell }\left|(\widehat{q_1}u-\hat{B})(\widehat{q_2}v-\hat{D})\right|\left|\frac{{\partial }^2\mathcal{G}\left({\mathcal{I}}_1\left(u\right),{\mathcal{I}}_2\left(v\right)\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\right].\hfill \end{array}$$

(40)

Since $\left|\frac{{\partial }^2\mathcal{G}}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|$ is coordinated convex, so we obtain

$$\begin{array}{ccc}\hfill & & {\int }_m^d{\int }_n^e\left|(\widehat{q_1}u-\hat{A})(\widehat{q_2}v-\hat{C})\right|\left|\frac{{\partial }^2\mathcal{G}}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\left(\frac{d-u}{d-m}m+\frac{u-m}{d-m}d,\frac{e-v}{e-n}n+\frac{v-n}{e-n}e\right)\right|{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\hfill \\ & \le & \frac{1}{(d-m)(e-n)}\left[{\int }_m^d{\int }_n^e\left|(\widehat{q_1}u-\hat{A})(\widehat{q_2}v-\hat{C})\right|(d-u)(e-v)\left|\frac{{\partial }^2\mathcal{G}\left(m,n\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\right.\hfill \\ & & +{\int }_m^d{\int }_n^e\left|(\widehat{q_1}u-\hat{A})(\widehat{q_2}v-\hat{C})\right|(d-u)(v-n)\left|\frac{{\partial }^2\mathcal{G}\left(m,e\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\hfill \\ & & +{\int }_m^d{\int }_n^e\left|(\widehat{q_1}u-\hat{A})(\widehat{q_2}v-\hat{C})\right|(u-m)(e-v)\left|\frac{{\partial }^2\mathcal{G}\left(d,n\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\hfill \\ & & +\left.{\int }_m^d{\int }_n^e\left|(\widehat{q_1}u-\hat{A})(\widehat{q_2}v-\hat{C})\right|(u-m)(v-n)\left|\frac{{\partial }^2\mathcal{G}\left(d,e\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\right]\hfill \\ & =& \frac{1}{(d-m)(e-n)}\left[\left|\frac{{\partial }^2\mathcal{G}\left(m,n\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|{}_m^d{\mathcal{K}}_1(\hat{A};\widehat{q_1}){}_n^e{\mathcal{K}}_1(\hat{C};\widehat{q_2})+\left|\frac{{\partial }^2\mathcal{G}\left(m,e\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|{}_m^d{\mathcal{K}}_1(\hat{A};\widehat{q_1}){}_n^e{\mathcal{K}}_2(\hat{C};\widehat{q_2})\right.\hfill \\ & & +\left|\frac{{\partial }^2\mathcal{G}\left(d,n\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|{}_m^d{\mathcal{K}}_2(\hat{A};\widehat{q_1}){}_n^e{\mathcal{K}}_1(\hat{C};\widehat{q_2})\left.+\left|\frac{{\partial }^2\mathcal{G}\left(d,e\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|{}_m^d{\mathcal{K}}_2(\hat{A};\widehat{q_1}){}_n^e{\mathcal{K}}_2(\hat{C};\widehat{q_2})\right].\hfill \end{array}$$

(41)

Similarly,

$$\begin{array}{ccc}\hfill & & {\int }_d^k{\int }_n^e\left|(\widehat{q_1}u-\hat{B})(\widehat{q_2}v-\hat{C})\right|\left|\frac{{\partial }^2\mathcal{G}}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\left(\frac{k-u}{k-d}d+\frac{u-d}{k-d}k,\frac{e-v}{e-n}n+\frac{v-n}{e-n}e\right)\right|{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\hfill \\ & =& \frac{1}{(k-d)(e-n)}\left[\left|\frac{{\partial }^2\mathcal{G}\left(d,n\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|{}_d^k{\mathcal{L}}_1(\hat{B};\widehat{q_1}){}_n^e{\mathcal{K}}_1(\hat{C};\widehat{q_2})+\left|\frac{{\partial }^2\mathcal{G}\left(d,e\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|{}_d^k{\mathcal{L}}_1(\hat{B};\widehat{q_1}){}_n^e{\mathcal{K}}_2(\hat{C};\widehat{q_2})\right.\hfill \\ & & +\left|\frac{{\partial }^2\mathcal{G}\left(k,n\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|{}_d^k{\mathcal{L}}_2(\hat{B};\widehat{q_1}){}_n^e{\mathcal{K}}_1(\hat{C};\widehat{q_2})\left.+\left|\frac{{\partial }^2\mathcal{G}\left(k,e\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|{}_d^k{\mathcal{L}}_2(\hat{B};\widehat{q_1}){}_n^e{\mathcal{K}}_2(\hat{C};\widehat{q_2})\right],\hfill \end{array}$$

(42)

$$\begin{array}{ccc}\hfill & & {\int }_m^d{\int }_e^{\ell }\left|(\widehat{q_1}u-\hat{A})(\widehat{q_2}v-\hat{D})\right|\left|\frac{{\partial }^2\mathcal{G}}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\left(\frac{d-u}{d-m}m+\frac{u-m}{d-m}d,\frac{\ell -v}{\ell -e}e+\frac{v-e}{\ell -e}\ell \right)\right|{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\hfill \\ & =& \frac{1}{(d-m)(\ell -e)}\left[\left|\frac{{\partial }^2\mathcal{G}\left(m,e\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|{}_d^k{\mathcal{K}}_1(\hat{A};\widehat{q_1}){}_e^{\ell }{\mathcal{L}}_1(\hat{D};\widehat{q_2})+\left|\frac{{\partial }^2\mathcal{G}\left(m,\ell \right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|{}_d^k{\mathcal{K}}_1(\hat{A};\widehat{q_1}){}_e^{\ell }{\mathcal{L}}_2(\hat{D};\widehat{q_2})\right.\hfill \\ & & +\left|\frac{{\partial }^2\mathcal{G}\left(d,e\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|{}_d^k{\mathcal{K}}_2(\hat{A};\widehat{q_1}){}_e^{\ell }{\mathcal{L}}_1(\hat{D};\widehat{q_2})\left.+\left|\frac{{\partial }^2\mathcal{G}\left(d,\ell \right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|{}_d^k{\mathcal{K}}_2(\hat{A};\widehat{q_1}){}_e^{\ell }{\mathcal{L}}_2(\hat{D};\widehat{q_2})\right]\hfill \end{array}$$

(43)

and

$$\begin{array}{ccc}\hfill & & {\int }_d^k{\int }_e^{\ell }\left|(\widehat{q_1}u-\hat{B})(\widehat{q_2}v-\hat{D})\right|\left|\frac{{\partial }^2\mathcal{G}}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\left(\frac{k-u}{k-d}d+\frac{u-d}{k-d}k,\frac{\ell -v}{\ell -e}e+\frac{v-e}{\ell -e}\ell \right)\right|{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\hfill \\ & =& \frac{1}{(k-d)(\ell -e)}\left[\left|\frac{{\partial }^2\mathcal{G}\left(d,e\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|{}_d^k{\mathcal{L}}_1(\hat{B};\widehat{q_1}){}_e^{\ell }{\mathcal{L}}_1(\hat{D};\widehat{q_2})+\left|\frac{{\partial }^2\mathcal{G}\left(d,\ell \right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|{}_d^k{\mathcal{L}}_1(\hat{B};\widehat{q_1}){}_e^{\ell }{\mathcal{L}}_2(\hat{D};\widehat{q_2})\right.\hfill \\ & & +\left|\frac{{\partial }^2\mathcal{G}\left(k,e\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|{}_d^k{\mathcal{L}}_2(\hat{B};\widehat{q_1}){}_e^{\ell }{\mathcal{L}}_1(\hat{D};\widehat{q_2})\left.+\left|\frac{{\partial }^2\mathcal{G}\left(k,\ell \right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|{}_d^k{\mathcal{L}}_2(\hat{B};\widehat{q_1}){}_e^{\ell }{\mathcal{L}}_2(\hat{D};\widehat{q_2})\right].\hfill \end{array}$$

(44)

Using (41)–(44) in (40), we have the desired inequality (37). □

Corollary 1. 

If, in addition to the conditions of Theorem 6, let $\hat{A}=\widehat{q_1}m,\hat{B}=k-m+\widehat{q_1}m,\hat{C}=\widehat{q_2}n$ and $\hat{D}=\ell -n+\widehat{q_2}n,$ then the inequalities given in (37), reduce to the following inequalities:

$$\begin{array}{ccc}\hfill & & \left|\frac{1}{(k-m)(\ell -n)}{\int }_m^k{\int }_n^{\ell }\mathcal{G}(x,y){}_n{d}_{\widehat{q_2}}y{}_m{d}_{\widehat{q_1}}x+\mathcal{G}\left(\frac{\widehat{q_1}m+k}{1+\widehat{q_1}},\frac{\widehat{q_2}n+\ell }{1+\widehat{q_2}}\right)-{\mathcal{V}}_{*}\right|\hfill \\ & \le & \frac{(k-m)(\ell -n)}{{(1+\widehat{q_1})}^2{(1+\widehat{q_2})}^2(1+\widehat{q_1}+{\widehat{q_1}}^2)(1+\widehat{q_2}+{\widehat{q_2}}^2)}\hfill \\ & & \times \left[\frac{1}{(1+\widehat{q_1})(1+\widehat{q_2})}({\widehat{q_1}}^3{\widehat{q_2}}^3\left|\frac{{\partial }^2\mathcal{G}(m,n)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|+{\widehat{q_2}}^3(-{\widehat{q_1}}^2+\widehat{q_1}+1)\left|\frac{{\partial }^2\mathcal{G}(k,n)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|\right.\hfill \\ & & +\left.{\widehat{q_1}}^3(-{\widehat{q_2}}^2+\widehat{q_2}+1)\left|\frac{{\partial }^2\mathcal{G}(m,\ell )}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|+(-{\widehat{q_2}}^2+\widehat{q_2}+1)(-{\widehat{q_1}}^2+\widehat{q_1}+1)\left|\frac{{\partial }^2\mathcal{G}(k,\ell )}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|)\right]\hfill \\ & & +\frac{(k-m)(\ell -n)}{{(1+\widehat{q_1})}^2{(1+\widehat{q_2})}^2(1+\widehat{q_1}+{\widehat{q_1}}^3)(1+\widehat{q_2}+{\widehat{q_2}}^2)}\hfill \\ & & \times \left[\frac{\left({\widehat{q_1}}^3({\widehat{q_2}}^2+2\widehat{q_2}-1)\left|\frac{{\partial }^2\mathcal{G}\left(m,\frac{\widehat{q_2}n+\ell }{1+\widehat{q_2}}\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|+(-{\widehat{q_1}}^2+\widehat{q_1}+1)({\widehat{q_2}}^2+2\widehat{q_2}-1)\left|\frac{{\partial }^2\mathcal{G}\left(k,\frac{\widehat{q_2}n+\ell }{1+\widehat{q_2}}\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|\right)}{1+\widehat{q_1}}\right.\hfill \\ & & +\frac{\left({\widehat{q_2}}^3({\widehat{q_1}}^2+2\widehat{q_1}-1)\left|\frac{{\partial }^2\mathcal{G}\left(\frac{\widehat{q_1}m+k}{1+\widehat{q_1}},n\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|+(-{\widehat{q_2}}^2+\widehat{q_2}+1)({\widehat{q_1}}^2+2\widehat{q_1}-1)\left|\frac{{\partial }^2\mathcal{G}\left(\frac{\widehat{q_1}m+k}{1+\widehat{q_1}},\ell \right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|\right)}{1+\widehat{q_2}}\hfill \\ & & +\left.({\widehat{q_1}}^2+2\widehat{q_1}-1)({\widehat{q_2}}^2+2\widehat{q_2}-1)\left|\frac{{\partial }^2\mathcal{G}\left(\frac{\widehat{q_1}m+k}{1+\widehat{q_1}},\frac{\widehat{q_2}n+\ell }{1+\widehat{q_2}}\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|\right],\hfill \end{array}$$

(45)

where

$$\begin{array}{ccc}\hfill {\mathcal{V}}_{*}& =& \frac{1}{k-m}{\int }_m^k\mathcal{G}\left(u,\frac{\widehat{q_2}n+\ell }{1+\widehat{q_2}}\right){}_m{d}_{\widehat{q_1}}u+\frac{1}{\ell -n}{\int }_n^{\ell }\mathcal{G}\left(\frac{\widehat{q_1}m+k}{1+\widehat{q_1}},v\right){}_n{d}_{\widehat{q_2}}v\hfill \end{array}$$

(46)

Remark 2. 

Let $\widehat{q_1},\widehat{q_2}\to 1^{-},$ then inequality (45) leads to:

$$\begin{array}{ccc}& & \left|\frac{1}{(k-m)(\ell -n)}{\int }_m^k{\int }_n^{\ell }\mathcal{G}(x,y)dydx+\mathcal{G}\left(\frac{m+k}{2},\frac{n+\ell }{2}\right)-\Psi \right|\hfill \\ & \le & \frac{(k-m)(\ell -n)}{144}\left[\frac{\left|\frac{{\partial }^2\mathcal{G}(m,n)}{\partial u\partial v}\right|+\left|\frac{{\partial }^2\mathcal{G}(k,n)}{\partial u\partial v}\right|+\left|\frac{{\partial }^2\mathcal{G}(m,\ell )}{\partial u\partial v}\right|+\left|\frac{{\partial }^2\mathcal{G}(k,\ell )}{\partial u\partial v}\right|}{4}\right]\hfill \\ & & +\frac{(k-m)(\ell -n)}{144}\left[\left|\frac{{\partial }^2\mathcal{G}\left(m,\frac{n+\ell }{2}\right)}{\partial u\partial v}\right|+\left|\frac{{\partial }^2\mathcal{G}\left(k,\frac{n+\ell }{2}\right)}{\partial u\partial v}\right|+\left|\frac{{\partial }^2\mathcal{G}\left(\frac{m+k}{2},n\right)}{\partial u\partial v}\right|\right.\hfill \\ & & +\left|\frac{{\partial }^2\mathcal{G}\left(\frac{m+k}{2},\ell \right)}{\partial u\partial v}\right|+\left.4\left|\frac{{\partial }^2\mathcal{G}\left(\frac{m+k}{2},\frac{n+\ell }{2}\right)}{\partial u\partial v}\right|\right],\hfill \end{array}$$

(47)

where Ψ is same as given in (9). This inequality is also found in [10] (Corllary 2).

Example 1. 

If we consider $\mathcal{G}(u,v)=u^2{v}^2,$ and $m=0=n,k=\ell =1,$ then $\frac{{\partial }^2\mathcal{G}}{{}_0{\partial }_{\widehat{q_1}}u{}_0{\partial }_{\widehat{q_2}}v}(u,v)=(1+\widehat{q_1})(1+\widehat{q_2})uv.$ Now we have:

$$\begin{array}{cc}\hfill L.H.S& =\left|\frac{1}{(1-0)(1-0)}{\int }_0^1{\int }_0^1{u}^2{v}^2{}_0{d}_{\widehat{q_2}}v{}_0{d}_{\widehat{q_1}}u+\mathcal{G}\left(\frac{1}{1+\widehat{q_1}},\frac{1}{1+\widehat{q_2}}\right)\right.\hfill \\ \hfill & \left.-\frac{1}{1-0}{\int }_0^1\frac{1}{{(1+\widehat{q_2})}^2}{u}^2{}_0{d}_{\widehat{q_1}}u-\frac{1}{1-0}{\int }_0^1\frac{1}{{(1+\widehat{q_1})}^2}{v}^2{}_0{d}_{\widehat{q_2}}v\right|\hfill \\ \hfill & =\left|\frac{1}{\left(1+\widehat{q_1}+{\widehat{q_1}}^2\right)\left(1+\widehat{q_2}+{\widehat{q_2}}^2\right)}+\frac{1}{{(1+\widehat{q_1})}^2{(1+\widehat{q_2})}^2}-\frac{1}{\left(1+\widehat{q_1}+{\widehat{q_1}}^2\right){\left(1+\widehat{q_2}\right)}^2}\right.\hfill \\ \hfill & \left.-\frac{1}{{\left(1+\widehat{q_1}\right)}^2\left(1+\widehat{q_2}+{\widehat{q_2}}^2\right)}\right|.\hfill \end{array}$$

(48)

$$\begin{array}{ccc}\hfill R.H.S& =& \frac{1}{{\left(1+\widehat{q_1}\right)}^2{\left(1+\widehat{q_2}\right)}^2\left(1+\widehat{q_1}+{\widehat{q_1}}^2\right)\left(1+\widehat{q_2}+{\widehat{q_2}}^2\right)}\left[\left(-{\widehat{q_1}}^2+\widehat{q_1}+1\right)\left(-{\widehat{q_2}}^2+\widehat{q_2}+1\right)\right]\hfill \\ & & +\frac{1}{{\left(1+\widehat{q_1}\right)}^2{\left(1+\widehat{q_2}\right)}^2\left(1+\widehat{q_1}+{\widehat{q_1}}^2\right)\left(1+\widehat{q_2}+{\widehat{q_2}}^2\right)}\hfill \\ & & \times \left[\left(-{\widehat{q_1}}^2+\widehat{q_1}+1\right)\left({\widehat{q_2}}^2+2\widehat{q_2}-1\right)+\left(-{\widehat{q_2}}^2+\widehat{q_2}+1\right)\left({\widehat{q_1}}^2+2\widehat{q_1}-1\right)\right.\hfill \\ & & \left.+\left({\widehat{q_1}}^2+2\widehat{q_1}-1\right)\left({\widehat{q_2}}^2+2\widehat{q_2}-1\right)\right].\hfill \end{array}$$

(49)

The last column of the below

Table 1. Parameters $\widehat{q_1}$ and $\widehat{q_2}$ are associated to the model given in Corollary 1.

Case	$\widehat{{\mathit{q}}_1}$	$\widehat{{\mathit{q}}_2}$	L.H.S	R.H.S	L.H.S-R.H.S
1	$0.5$	$0.5$	0.079617	0.14512	$-0.065503$
2	$0.25$	$0.25$	0.014861	0.13375	$-0.11889$
3	$0.66667$	$0.66667$	0.012924	0.11632	$-0.10340$
4	$0.25$	$0.66667$	0.013859	0.12473	$-0.11087$
5	$0.66667$	$0.25$	0.013859	0.12473	$-0.11087$
6	1	1	0.0069444	0.0625	$-0.055556$

shows that the estimate given in inequality (45) is valid.

Corollary 2. 

If, in addition to the conditions of Theorem 6, $\hat{A}=\hat{B}=\frac{\widehat{q_1}(\widehat{q_1}m+k)}{1+\widehat{q_1}},d=\frac{\widehat{q_1}m+k}{1+\widehat{q_1}},\hat{C}=\hat{D}=\frac{\widehat{q_2}(\widehat{q_2}n+\ell )}{1+\widehat{q_2}}$ and $e=\frac{\widehat{q_2}n+\ell }{1+\widehat{q_2}},$ then the inequality given in (37), reduces to the following inequality:

$$\begin{array}{ccc}\hfill & & \left|\frac{1}{(k-m)(\ell -n)}{\int }_m^k{\int }_n^{\ell }\mathcal{G}(x,y){}_n{d}_{\widehat{q_2}}y{}_m{d}_{\widehat{q_1}}x\right.\hfill \\ & & \left.+\frac{\widehat{q_1}\widehat{q_2}\mathcal{G}\left(m,n\right)+\widehat{q_1}\mathcal{G}\left(m,\ell \right)+\widehat{q_2}\mathcal{G}\left(k,n\right)+\mathcal{G}\left(k,\ell \right)}{(1+\widehat{q_1})(1+\widehat{q_2})}-{\mathcal{V}}_{**}\right|\hfill \\ & \le & \frac{(k-m)(\ell -n)}{{(1+\widehat{q_1})}^3{(1+\widehat{q_2})}^3(1+\widehat{q_1}+{\widehat{q_1}}^2)(1+\widehat{q_2}+{\widehat{q_2}}^2)}\hfill \\ & & \times \left[{\widehat{q_1}}^2{\widehat{q_2}}^2(1+{\widehat{q_1}}^2)(1+{\widehat{q_2}}^2)\left|\frac{{\partial }^2\mathcal{G}(m,n)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|+2{\widehat{q_1}}^2{\widehat{q_2}}^2(1+{\widehat{q_2}}^2)\left|\frac{{\partial }^2\mathcal{G}(k,n)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|\right.\hfill \\ & & +\left.2{\widehat{q_1}}^2{\widehat{q_2}}^2(1+{\widehat{q_1}}^2)\left|\frac{{\partial }^2\mathcal{G}(m,\ell )}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|+4{\widehat{q_1}}^2{\widehat{q_2}}^2\left|\frac{{\partial }^2\mathcal{G}(k,\ell )}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|\right]\hfill \\ & & +\frac{(k-m)(\ell -n)}{{(1+\widehat{q_1})}^3{(1+\widehat{q_2})}^3(1+\widehat{q_1}+{\widehat{q_1}}^2)(1+\widehat{q_2}+{\widehat{q_2}}^2)}\hfill \\ & & \times \left[{\widehat{q_1}}^2{\widehat{q_2}}^2(1+{\widehat{q_1}}^2)({\widehat{q_2}}^2+2\widehat{q_2}-1)\left|\frac{{\partial }^2\mathcal{G}\left(m,\frac{\widehat{q_2}n+\ell }{1+\widehat{q_2}}\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|+2{\widehat{q_1}}^2{\widehat{q_2}}^2({\widehat{q_2}}^2+2\widehat{q_2}-1)\left|\frac{{\partial }^2\mathcal{G}\left(k,\frac{\widehat{q_2}n+\ell }{1+\widehat{q_2}}\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|\right.\hfill \\ & & +{\widehat{q_1}}^2{\widehat{q_2}}^2(1+{\widehat{q_2}}^2)({\widehat{q_1}}^2+2\widehat{q_1}-1)\left|\frac{{\partial }^2\mathcal{G}\left(\frac{\widehat{q_1}m+k}{1+\widehat{q_1}},n\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|+2{\widehat{q_1}}^2{\widehat{q_2}}^2({\widehat{q_1}}^2+2\widehat{q_1}-1)\left|\frac{{\partial }^2\mathcal{G}\left(\frac{\widehat{q_1}m+k}{1+\widehat{q_1}},\ell \right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|\hfill \\ & & +\left.{\widehat{q_1}}^2{\widehat{q_2}}^2({\widehat{q_1}}^2+2\widehat{q_1}-1)({\widehat{q_2}}^2+2\widehat{q_2}-1)\left|\frac{{\partial }^2\mathcal{G}\left(\frac{\widehat{q_1}m+k}{1+\widehat{q_1}},\frac{\widehat{q_2}n+\ell }{1+\widehat{q_2}}\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|\right],\hfill \end{array}$$

(50)

where

$$\begin{array}{ccc}\hfill {\mathcal{V}}_{**}& =& \frac{\widehat{q_2}}{(1+\widehat{q_2})(k-m)}{\int }_m^k\mathcal{G}\left(u,n\right){}_m{d}_{\widehat{q_1}}u+\frac{1}{(1+\widehat{q_2})(k-m)}{\int }_m^k\mathcal{G}\left(u,\ell \right){}_m{d}_{\widehat{q_1}}u\hfill \\ & & +\frac{\widehat{q_1}}{(1+\widehat{q_1})(\ell -n)}{\int }_n^{\ell }\mathcal{G}\left(m,v\right){}_n{d}_{\widehat{q_2}}v+\frac{1}{(1+\widehat{q_1})(\ell -n)}{\int }_n^{\ell }\mathcal{G}\left(k,v\right){}_n{d}_{\widehat{q_2}}v.\hfill \end{array}$$

(51)

Remark 3. 

If $\widehat{q_1},\widehat{q_2}\to 1^{-},$ then inequality (50) reduces to:

$$\begin{array}{ccc}& & \left|\frac{1}{(k-m)(\ell -n)}{\int }_m^k{\int }_n^{\ell }\mathcal{G}(x,y)dydx+\frac{\mathcal{G}\left(m,n\right)+\mathcal{G}\left(m,\ell \right)+\mathcal{G}\left(k,n\right)+\mathcal{G}\left(k,\ell \right)}{4}-\Phi \right|\hfill \\ & \le & \frac{(k-m)(\ell -n)}{36}\left[\frac{\left|\frac{{\partial }^2\mathcal{G}(m,n)}{\partial u\partial v}\right|+\left|\frac{{\partial }^2\mathcal{G}(k,n)}{\partial u\partial v}\right|+\left|\frac{{\partial }^2\mathcal{G}(m,\ell )}{\partial u\partial v}\right|+\left|\frac{{\partial }^2\mathcal{G}(k,\ell )}{\partial u\partial v}\right|}{4}\right]\hfill \\ & & +\frac{(k-m)(\ell -n)}{144}\left[\left|\frac{{\partial }^2\mathcal{G}\left(\frac{m+k}{2},n\right)}{\partial u\partial v}\right|+\left|\frac{{\partial }^2\mathcal{G}\left(\frac{m+k}{2},\ell \right)}{\partial u\partial v}\right|+\left|\frac{{\partial }^2\mathcal{G}\left(m,\frac{n+\ell }{2}\right)}{\partial u\partial v}\right|\right.\hfill \\ & & +\left|\frac{{\partial }^2\mathcal{G}\left(k,\frac{n+\ell }{2}\right)}{\partial u\partial v}\right|+\left.\left|\frac{{\partial }^2\mathcal{G}\left(\frac{m+k}{2},\frac{n+\ell }{2}\right)}{\partial u\partial v}\right|\right],\hfill \end{array}$$

(52)

where Φ is the same as given in (6). This inequality gives a new estimate to the inequality (4) and is also found in [10] (Corollary 4).

Example 2. 

If we consider $\mathcal{G}(u,v)=u^2{v}^2,$ and $m=0=n,k=\ell =1.$ Then $\frac{{\partial }^2\mathcal{G}}{{}_0{\partial }_{\widehat{q_1}}u{}_0{\partial }_{\widehat{q_2}}v}(u,v)=(1+\widehat{q_1})(1+\widehat{q_2})uv.$ From Corollary 2, we have

$$\begin{array}{ccc}\hfill L.H.S& =& \left|{\int }_0^1{\int }_0^1{u}^2{v}^2{}_0{d}_{\widehat{q_2}}v{}_0{d}_{\widehat{q_1}}u+1-\frac{1}{(1-0)(1+\widehat{q_2})}{\int }_0^1{u}^2{}_0{d}_{\widehat{q_1}}u\right.\hfill \\ & & -\left.\frac{1}{(1-0)(1+\widehat{q_1})}{\int }_0^1{v}^2{}_0{d}_{\widehat{q_2}}v\right|\hfill \\ & =& \left|\frac{1}{\left(1+\widehat{q_1}+{\widehat{q_1}}^2\right)\left(1+\widehat{q_2}+{\widehat{q_2}}^2\right)}+\frac{1}{(1+\widehat{q_1})(1+\widehat{q_2})}-\frac{1}{\left(1+\widehat{q_1}+{\widehat{q_1}}^2\right)\left(1+\widehat{q_2}\right)}\right.\hfill \\ & & \left.-\frac{1}{\left(1+\widehat{q_1}\right)\left(1+\widehat{q_2}+{\widehat{q_2}}^2\right)}\right|.\hfill \end{array}$$

(53)

Similarly,

$$\begin{array}{ccc}\hfill R.H.S& =& \frac{4{\widehat{q_1}}^2{\widehat{q_2}}^2}{{\left(1+\widehat{q_1}\right)}^2{\left(1+\widehat{q_2}\right)}^2\left(1+\widehat{q_1}+{\widehat{q_1}}^2\right)\left(1+\widehat{q_2}+{\widehat{q_2}}^2\right)}\hfill \\ & & +\frac{{\left(1+\widehat{q_2}\right)}^{-3}}{{\left(1+\widehat{q_1}\right)}^3\left(1+\widehat{q_1}+{\widehat{q_1}}^2\right)\left(1+\widehat{q_2}+{\widehat{q_2}}^2\right)}\left(\begin{array}{c}2{\widehat{q_1}}^2{\widehat{q_2}}^2\left({\widehat{q_2}}^2+2\widehat{q_2}-1\right)\left(1+\widehat{q_1}\right)\\ +2{\widehat{q_1}}^2{\widehat{q_2}}^2\left({\widehat{q_1}}^2+2\widehat{q_1}-1\right)\left(1+\widehat{q_2}\right)\\ +{\widehat{q_1}}^2{\widehat{q_2}}^2\left({\widehat{q_1}}^2+2\widehat{q_1}-1\right)\left({\widehat{q_2}}^2+2\widehat{q_2}-1\right)\end{array}\right).\hfill \end{array}$$

By assigning several values to the parameters “$\widehat{q_1}$ and $\widehat{q_2}$”, we create the table below to examine the accuracy of our estimate.

From the

Table 2. Parameters $\widehat{q_1}$ and $\widehat{q_2}$ are associated to the model given in Corollary 2.

Case	$\widehat{{\mathit{q}}_1}$	$\widehat{{\mathit{q}}_2}$	L.H.S	R.H.S	L.H.S-R.H.S
1	$0.25$	$0.25$	0.0014512	0.0025874	$-0.0011362$
2	$0.5$	$0.5$	0.0090703	0.018924	$-0.0098537$
3	$0.66667$	$0.66667$	0.015956	0.034949	$-0.018993$
4	$0.25$	$0.66667$	0.004812	0.01294	$-0.008128$
5	$0.66667$	$0.25$	0.004812	0.01294	$-0.008128$
6	1	1	0.027778	0.0625	$-0.034722$

, it is clear that the trapezoidal type estimate given in inequality (50) is valid.

Theorem 7. 

Let $\mathcal{G}:\mathrm{Y}\to \mathbb{R}$ be a twice partial ${\widehat{q_1}}_m{\widehat{q_2}}_n$–differentiable function defined on ${\mathrm{Y}}^{\circ }$ such that the partial ${\widehat{q_1}}_m{\widehat{q_2}}_n$–derivative $\frac{{\partial }^2\mathcal{G}(u,v)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}$ is continuous and ${\widehat{q_1}}_m{\widehat{q_2}}_n$–integrable over Y. If ${\left|\frac{{\partial }^2\mathcal{G}(u,v)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }$ is coordinated convex on Y for some $\lambda \ge 1,$ then the following inequalities hold:

$$\begin{array}{cc}\hfill \left|\frac{1}{(k-m)(\ell -n)}{\int }_m^k{\int }_n^{\ell }\mathcal{G}(x,y){}_n{d}_{\widehat{q_2}}y{}_m{d}_{\widehat{q_1}}x+\mathcal{U}-\mathcal{V}\right|& \le \frac{1}{(k-m)(\ell -n)}E(B,D,\widehat{q_1},\widehat{q_2}),\hfill \end{array}$$

(54)

where

$$\begin{array}{cc}\hfill & E(B,D,\widehat{q_1},\widehat{q_2})={\left({}_m^k\mathcal{M}(B,\widehat{q_1}){}_n^{\ell }\mathcal{M}(D,\widehat{q_2})\right)}^{1-\frac{1}{\lambda }}\hfill \\ \hfill & \times \left(\frac{1}{(k-m)(\ell -n)}\left[{}_m^k{\mathcal{K}}_1{(\hat{B};\widehat{q_1})}_n^{\ell }{\mathcal{K}}_1(\hat{D};\widehat{q_2}){\left|\frac{{\partial }^2\mathcal{G}\left(m,n\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }{+}_m^k{\mathcal{K}}_1{(\hat{B};\widehat{q_1})}_n^{\ell }{\mathcal{K}}_2(\hat{D};\widehat{q_2}){\left|\frac{{\partial }^2\mathcal{G}\left(m,\ell \right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }\right.\right.\hfill \\ \hfill & +{\left.{}_m^k{\mathcal{K}}_2{(\hat{B};\widehat{q_1})}_n^{\ell }{\mathcal{K}}_1(\hat{D};\widehat{q_2}){\left|\frac{{\partial }^2\mathcal{G}\left(k,n\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }+\left.{}_m^k{\mathcal{K}}_2{(\hat{B};\widehat{q_1})}_n^{\ell }{\mathcal{K}}_2(\hat{D};\widehat{q_2}){\left|\frac{{\partial }^2\mathcal{G}\left(k,\ell \right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }\right]\right)}^{\frac{1}{\lambda }}\hfill \\ \hfill & +\left(\hat{D}-\hat{C}\right){\left((e-n){}_m^k\mathcal{M}(B,\widehat{q_1})\right)}^{1-\frac{1}{\lambda }}\hfill \\ \hfill & \times \left(\frac{1}{(k-m)(e-n)}\left[{}_m^k{\mathcal{K}}_1{(B,\widehat{q_1})}_n^e\mathcal{N}\left(\widehat{q_2}\right){\left|\frac{{\partial }^2\mathcal{G}\left(m,n\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }\right.\right.{+}_m^k{\mathcal{K}}_1{(B,\widehat{q_1})}_n^e\mathcal{R}\left(\widehat{q_2}\right){\left|\frac{{\partial }^2\mathcal{G}\left(m,e\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }\hfill \\ \hfill & {+}_m^k{\mathcal{K}}_2{(B,\widehat{q_1})}_n^e\mathcal{N}\left(\widehat{q_2}\right){\left|\frac{{\partial }^2\mathcal{G}\left(k,n\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }{\left.\left.{+}_m^k{\mathcal{K}}_2{(B,\widehat{q_1})}_n^e\mathcal{R}\left(\widehat{q_2}\right){\left|\frac{{\partial }^2\mathcal{G}\left(k,e\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }\right]\right)}^{\frac{1}{\lambda }}\hfill \\ \hfill & +\left(\hat{B}-\hat{A}\right){\left((d-m){}_n^{\ell }\mathcal{M}(D,\widehat{q_2})\right)}^{1-\frac{1}{\lambda }}\hfill \\ \hfill & \times \left(\frac{1}{(d-m)(\ell -n)}\left[{}_m^d\mathcal{N}\left(\widehat{q_1}\right){}_n^{\ell }{\mathcal{K}}_1(D,\widehat{q_2}){\left|\frac{{\partial }^2\mathcal{G}\left(m,n\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }\right.\right.{+}_m^d\mathcal{N}\left(\widehat{q_1}\right){}_n^{\ell }{\mathcal{K}}_2(D,\widehat{q_2}){\left|\frac{{\partial }^2\mathcal{G}\left(m,\ell \right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }\hfill \\ \hfill & {+}_m^d\mathcal{R}\left(\widehat{q_1}\right){}_n^{\ell }{\mathcal{K}}_1(D,\widehat{q_2}){\left|\frac{{\partial }^2\mathcal{G}\left(d,n\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }{\left.\left.{+}_m^d\mathcal{R}\left(\widehat{q_1}\right){}_n^{\ell }{\mathcal{K}}_2(D,\widehat{q_2}){\left|\frac{{\partial }^2\mathcal{G}\left(d,\ell \right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }\right]\right)}^{\frac{1}{\lambda }}\hfill \\ \hfill & +\left(\hat{B}-\hat{A}\right)\left(\hat{D}-\hat{C}\right){\left((d-m)(e-n)\right)}^{1-\frac{1}{\lambda }}\hfill \\ \hfill & \times \left(\frac{1}{(d-m)(e-n)}\left[{}_m^d\mathcal{N}\left(\widehat{q_1}\right){}_n^e\mathcal{N}\left(\widehat{q_2}\right){\left|\frac{{\partial }^2\mathcal{G}\left(m,n\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }\right.\right.{+}_m^d\mathcal{N}\left(\widehat{q_1}\right){}_n^e\mathcal{R}\left(\widehat{q_2}\right){\left|\frac{{\partial }^2\mathcal{G}\left(m,e\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }\hfill \\ \hfill & {+}_m^d\mathcal{R}\left(\widehat{q_1}\right){}_n^e\mathcal{N}\left(\widehat{q_2}\right){\left|\frac{{\partial }^2\mathcal{G}\left(d,n\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }{\left.\left.{+}_m^d\mathcal{R}\left(\widehat{q_1}\right){}_n^e\mathcal{R}\left(\widehat{q_2}\right){\left|\frac{{\partial }^2\mathcal{G}\left(d,e\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }\right]\right)}^{\frac{1}{\lambda }}.\hfill \end{array}$$

(55)

Proof. 

First we consider the identity mappings

$$\begin{array}{cc}\hfill {\mathcal{I}}_3\left(u\right):& =\left\{\begin{array}{cc}\frac{d-u}{d-m}m+\frac{u-m}{d-m}d,\hfill & \mathrm{if}m<u<d,\hfill \\ \frac{k-u}{k-m}m+\frac{u-m}{k-m}k,\hfill & \mathrm{if}m<u<k\hfill \end{array}\right.\hfill \end{array}$$

(56)

and

$$\begin{array}{cc}\hfill {\mathcal{I}}_4\left(v\right):& =\left\{\begin{array}{cc}\frac{e-v}{e-n}n+\frac{v-n}{e-n}e,\hfill & \mathrm{if}n<v<e,\hfill \\ \frac{\ell -v}{\ell -n}n+\frac{v-n}{\ell -n}\ell ,\hfill & \mathrm{if}n<v<\ell .\hfill \end{array}\right.\hfill \end{array}$$

(57)

By considering the Equation (26) and applying modulus, we have

$$\begin{array}{cc}\hfill & \left|\frac{1}{(k-m)(\ell -n)}{\int }_m^k{\int }_n^{\ell }\mathcal{G}(x,y){}_n{d}_{\widehat{q_2}}y{}_m{d}_{\widehat{q_1}}x+\mathcal{U}-\mathcal{V}\right|\hfill \\ \hfill & \le \frac{1}{(k-m)(\ell -n)}\left[{\int }_m^k{\int }_n^{\ell }\left|(\widehat{q_1}u-\hat{B})(\widehat{q_2}v-\hat{D})\frac{{\partial }^2\mathcal{G}\left({\mathcal{I}}_3\left(u\right),{\mathcal{I}}_4\left(v\right)\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\right.\hfill \\ \hfill & +(\hat{D}-\hat{C}){\int }_m^k{\int }_n^e\left|(\widehat{q_1}u-\hat{B})\frac{{\partial }^2\mathcal{G}\left({\mathcal{I}}_3\left(u\right),{\mathcal{I}}_4\left(v\right)\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\hfill \\ \hfill & +(\hat{B}-\hat{A}){\int }_m^d{\int }_n^{\ell }\left|(\widehat{q_2}v-\hat{D})\frac{{\partial }^2\mathcal{G}\left({\mathcal{I}}_3\left(u\right),{\mathcal{I}}_4\left(v\right)\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\hfill \\ \hfill & +\left.(\hat{B}-\hat{A})(\hat{D}-\hat{C}){\int }_m^d{\int }_n^e\left|\frac{{\partial }^2\mathcal{G}\left({\mathcal{I}}_3\left(u\right),{\mathcal{I}}_4\left(v\right)\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\right].\hfill \end{array}$$

(58)

Now by the application of power mean inequality and coordinated convexity, we have

$$\begin{array}{cc}\hfill & {\int }_m^k{\int }_n^{\ell }\left|\widehat{q_1}u-\hat{B}\right|\left|\widehat{q_2}v-\hat{D}\right|\left|\frac{{\partial }^2\mathcal{G}({\mathcal{I}}_3\left(u\right),{\mathcal{I}}_4\left(v\right))}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\hfill \\ \hfill & ={\int }_m^k{\int }_n^{\ell }\left|\widehat{q_1}u-\hat{B}\right|\left|\widehat{q_2}v-\hat{D}\right|\left|\frac{{\partial }^2\mathcal{G}}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\left(\frac{k-u}{k-m}m+\frac{u-m}{k-m}k,\frac{\ell -v}{\ell -n}n+\frac{v-n}{\ell -n}\ell \right)\right|{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\hfill \\ \hfill & \le {\left({\int }_m^k{\int }_n^{\ell }\left|\widehat{q_1}u-\hat{B}\right|\left|\widehat{q_2}v-\hat{D}\right|{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\right)}^{1-\frac{1}{\lambda }}\hfill \\ \hfill & \times \left(\frac{1}{(k-m)(\ell -n)}\left[{\int }_m^k{\int }_n^{\ell }\left|\widehat{q_1}u-\hat{B}\right|\left|\widehat{q_2}v-\hat{D}\right|\left(k-u\right)\left(\ell -v\right){\left|\frac{{\partial }^2\mathcal{G}\left(m,n\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\right.\right.\hfill \\ \hfill & +{\int }_m^k{\int }_n^{\ell }\left|\widehat{q_1}u-\hat{B}\right|\left|\widehat{q_2}v-\hat{D}\right|\left(k-u\right)\left(v-n\right){\left|\frac{{\partial }^2\mathcal{G}\left(m,\ell \right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\hfill \\ \hfill & +{\int }_m^k{\int }_n^{\ell }\left|\widehat{q_1}u-\hat{B}\right|\left|\widehat{q_2}v-\hat{D}\right|\left(u-m\right)\left(\ell -v\right){\left|\frac{{\partial }^2\mathcal{G}\left(k,n\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\hfill \\ \hfill & +{\left.\left.{\int }_m^k{\int }_n^{\ell }\left|\widehat{q_1}u-\hat{B}\right|\left|\widehat{q_2}v-\hat{D}\right|\left(u-m\right)\left(v-n\right){\left|\frac{{\partial }^2\mathcal{G}\left(k,\ell \right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\right]\right)}^{\frac{1}{\lambda }}\hfill \\ \hfill & \le {\left({}_m^k\mathcal{M}(B,\widehat{q_1}){}_n^{\ell }\mathcal{M}(D,\widehat{q_2})\right)}^{1-\frac{1}{\lambda }}\hfill \\ \hfill & \times \left(\frac{1}{(k-m)(\ell -n)}\left[{}_m^k{\mathcal{K}}_1{(\hat{B};\widehat{q_1})}_n^{\ell }{\mathcal{K}}_1(\hat{D};\widehat{q_2}){\left|\frac{{\partial }^2\mathcal{G}\left(m,n\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }{+}_m^k{\mathcal{K}}_1{(\hat{B};\widehat{q_1})}_n^{\ell }{\mathcal{K}}_2(\hat{D};\widehat{q_2}){\left|\frac{{\partial }^2\mathcal{G}\left(m,\ell \right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }\right.\right.\hfill \\ \hfill & +{\left.{}_m^k{\mathcal{K}}_2{(\hat{B};\widehat{q_1})}_n^{\ell }{\mathcal{K}}_1(\hat{D};\widehat{q_2}){\left|\frac{{\partial }^2\mathcal{G}\left(k,n\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }+\left.{}_m^k{\mathcal{K}}_2{(\hat{B};\widehat{q_1})}_n^{\ell }{\mathcal{K}}_2(\hat{D};\widehat{q_2}){\left|\frac{{\partial }^2\mathcal{G}\left(k,\ell \right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }\right]\right)}^{\frac{1}{\lambda }}.\hfill \end{array}$$

(59)

Similarly,

$$\begin{array}{cc}\hfill & {\int }_m^k{\int }_n^e(\widehat{q_1}u-\hat{B})\frac{{\partial }^2\mathcal{G}({\mathcal{I}}_3\left(u\right),{\mathcal{I}}_4\left(v\right))}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\hfill \\ \hfill & \le {\left({\int }_m^k{\int }_n^e\left|\widehat{q_1}u-\hat{B}\right|{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\right)}^{1-\frac{1}{\lambda }}\hfill \\ \hfill & \times {\left({\int }_m^k{\int }_n^e\left|\widehat{q_1}u-\hat{B}\right|{\left|\frac{{\partial }^2\mathcal{G}}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\left(\frac{k-u}{k-m}m+\frac{u-m}{k-m}k,\frac{e-v}{e-n}n+\frac{v-n}{e-n}e\right)\right|}^{\lambda }{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\right)}^{\frac{1}{\lambda }}\hfill \\ \hfill & \le {\left((e-n){}_m^k\mathcal{M}(B,\widehat{q_1})\right)}^{1-\frac{1}{\lambda }}\hfill \\ \hfill & \times \left(\frac{1}{(k-m)(e-n)}\left[{}_m^k{\mathcal{K}}_1{(B,\widehat{q_1})}_n^e\mathcal{N}\left(\widehat{q_2}\right){\left|\frac{{\partial }^2\mathcal{G}\left(m,n\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }\right.\right.{+}_m^k{\mathcal{K}}_1{(B,\widehat{q_1})}_n^e\mathcal{R}\left(\widehat{q_2}\right){\left|\frac{{\partial }^2\mathcal{G}\left(m,e\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }\hfill \\ \hfill & {+}_m^k{\mathcal{K}}_2{(B,\widehat{q_1})}_n^e\mathcal{N}\left(\widehat{q_2}\right){\left|\frac{{\partial }^2\mathcal{G}\left(k,n\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }{\left.\left.{+}_m^k{\mathcal{K}}_2{(B,\widehat{q_1})}_n^e\mathcal{R}\left(\widehat{q_2}\right){\left|\frac{{\partial }^2\mathcal{G}\left(k,e\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }\right]\right)}^{\frac{1}{\lambda }},\hfill \end{array}$$

(60)

$$\begin{array}{cc}\hfill & {\int }_m^d{\int }_n^{\ell }(\widehat{q_2}v-\hat{D})\frac{{\partial }^2\mathcal{G}({\mathcal{I}}_3\left(u\right),{\mathcal{I}}_4\left(v\right))}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\hfill \\ \hfill & \le {\left({\int }_m^d{\int }_n^{\ell }\left|\widehat{q_2}v-\hat{D}\right|{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\right)}^{1-\frac{1}{\lambda }}\hfill \\ \hfill & \times {\left({\int }_m^d{\int }_n^{\ell }\left|\widehat{q_2}v-\hat{D}\right|{\left|\frac{{\partial }^2\mathcal{G}(u,v)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\right)}^{\frac{1}{\lambda }}\hfill \\ \hfill & \le {\left((d-m){}_n^{\ell }\mathcal{M}(D,\widehat{q_2})\right)}^{1-\frac{1}{\lambda }}\hfill \\ \hfill & \times \left(\frac{1}{(d-m)(\ell -n)}\left[{}_m^d\mathcal{N}\left(\widehat{q_1}\right){}_n^{\ell }{\mathcal{K}}_1(D,\widehat{q_2}){\left|\frac{{\partial }^2\mathcal{G}\left(m,n\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }\right.\right.{+}_m^d\mathcal{N}\left(\widehat{q_1}\right){}_n^{\ell }{\mathcal{K}}_2(D,\widehat{q_2}){\left|\frac{{\partial }^2\mathcal{G}\left(m,\ell \right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }\hfill \\ \hfill & {+}_m^d\mathcal{R}\left(\widehat{q_1}\right){}_n^{\ell }{\mathcal{K}}_1(D,\widehat{q_2}){\left|\frac{{\partial }^2\mathcal{G}\left(d,n\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }{\left.\left.{+}_m^d\mathcal{R}\left(\widehat{q_1}\right){}_n^{\ell }{\mathcal{K}}_2(D,\widehat{q_2}){\left|\frac{{\partial }^2\mathcal{G}\left(d,\ell \right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }\right]\right)}^{\frac{1}{\lambda }}\hfill \end{array}$$

(61)

and

$$\begin{array}{cc}\hfill & {\int }_m^d{\int }_n^e\frac{{\partial }^2\mathcal{G}({\mathcal{I}}_3\left(u\right),{\mathcal{I}}_4\left(v\right))}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\hfill \\ \hfill & \le {\left({\int }_m^d{\int }_n^{e}1{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\right)}^{1-\frac{1}{\lambda }}\hfill \\ \hfill & \times {\left({\int }_m^d{\int }_n^e{\left|\frac{{\partial }^2\mathcal{G}(u,v)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }{}_n{d}_{\widehat{q_2}}v{}_m{d}_{\widehat{q_1}}u\right)}^{\frac{1}{\lambda }}\hfill \\ \hfill & \le {\left((d-m)(e-n)\right)}^{1-\frac{1}{\lambda }}\hfill \\ \hfill & \times \left(\frac{1}{(d-m)(e-n)}\left[{}_m^d\mathcal{N}\left(\widehat{q_1}\right){}_n^e\mathcal{N}\left(\widehat{q_2}\right){\left|\frac{{\partial }^2\mathcal{G}\left(m,n\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }\right.\right.{+}_m^d\mathcal{N}\left(\widehat{q_1}\right){}_n^e\mathcal{R}\left(\widehat{q_2}\right){\left|\frac{{\partial }^2\mathcal{G}\left(m,e\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }\hfill \\ \hfill & {+}_m^d\mathcal{R}\left(\widehat{q_1}\right){}_n^e\mathcal{N}\left(\widehat{q_2}\right){\left|\frac{{\partial }^2\mathcal{G}\left(d,n\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }{\left.\left.{+}_m^d\mathcal{R}\left(\widehat{q_1}\right){}_n^e\mathcal{R}\left(\widehat{q_2}\right){\left|\frac{{\partial }^2\mathcal{G}\left(d,e\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }\right]\right)}^{\frac{1}{\lambda }}.\hfill \end{array}$$

(62)

We achieve the desired inequality “(54)”, by using the inequalities (59)–(62) in (58). □

Corollary 3. 

If, in addition to the requirements of Theorem 7, $\hat{A}=\widehat{q_1}m,\hat{B}=k-m+\widehat{q_1}m,d=\frac{\widehat{q_1}m+k}{1+\widehat{q_1}},\hat{C}=\widehat{q_2}n,\hat{D}=\ell -n+\widehat{q_2}n$ and $e=\frac{\widehat{q_2}n+\ell }{1+\widehat{q_2}},$ then

$$\begin{array}{ccc}\hfill & & \left|\frac{1}{(k-m)(\ell -n)}{\int }_m^k{\int }_n^{\ell }\mathcal{G}(x,y){}_n{d}_{\widehat{q_2}}y{}_m{d}_{\widehat{q_1}}x+\mathcal{G}\left(\frac{\widehat{q_1}m+k}{1+\widehat{q_1}},\frac{\widehat{q_1}n+\ell }{1+\widehat{q_2}}\right)-{\mathcal{V}}_{*}\right|\hfill \\ & \le & \frac{(k-m)(\ell -n)}{(1+\widehat{q_1})(1+\widehat{q_2})\sqrt[\lambda ]{(1+\widehat{q_1})(1+\widehat{q_2})(1+\widehat{q_1}+{\widehat{q_1}}^2)(1+\widehat{q_2}+{\widehat{q_2}}^2)}}\hfill \\ & & \times \left\{\sqrt[\lambda ]{(1+\widehat{q_1})(1+\widehat{q_2})}\left(\widehat{q_1}\widehat{q_2}(1+\widehat{q_1})(1+\widehat{q_2}){\left|\frac{{\partial }^2\mathcal{G}\left(m,n\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }+\widehat{q_1}(1+\widehat{q_1}){\left|\frac{{\partial }^2\mathcal{G}\left(m,\ell \right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }\right.\right.\hfill \\ & & +{\left.\widehat{q_2}(1+\widehat{q_2}){\left|\frac{{\partial }^2\mathcal{G}\left(k,n\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }+{\left|\frac{{\partial }^2\mathcal{G}\left(k,\ell \right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }\right)}^{\frac{1}{\lambda }}\hfill \\ & & +\sqrt[\lambda ]{(1+\widehat{q_1})(1+\widehat{q_2}+{\widehat{q_2}}^2)}\left(\widehat{q_1}\widehat{q_2}(1+\widehat{q_1}){\left|\frac{{\partial }^2\mathcal{G}\left(m,n\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }+\widehat{q_1}(1+\widehat{q_1}){\left|\frac{{\partial }^2\mathcal{G}\left(m,\frac{\widehat{q_2}n+\ell }{2}\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }\right.\hfill \\ & & {\left.+\widehat{q_2}{\left|\frac{{\partial }^2\mathcal{G}\left(k,n\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }+{\left|\frac{{\partial }^2\mathcal{G}\left(k,\frac{\widehat{q_2}n+\ell }{2}\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }\right)}^{\frac{1}{\lambda }}\hfill \\ & & +\sqrt[\lambda ]{(1+\widehat{q_1}+{\widehat{q_1}}^2)(1+\widehat{q_2})}\left(\widehat{q_1}\widehat{q_2}(1+\widehat{q_2}){\left|\frac{{\partial }^2\mathcal{G}\left(m,n\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }\right.+\widehat{q_1}{\left|\frac{{\partial }^2\mathcal{G}\left(m,\ell \right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }\hfill \\ & & +\widehat{q_2}(1+\widehat{q_2}){\left|\frac{{\partial }^2\mathcal{G}\left(\frac{\widehat{q_1}m+k}{1+\widehat{q_1}},n\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }{\left.+{\left|\frac{{\partial }^2\mathcal{G}\left(\frac{\widehat{q_1}m+k}{1+\widehat{q_1}},\ell \right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }\right)}^{\frac{1}{\lambda }}\hfill \\ & & +\sqrt[\lambda ]{(1+\widehat{q_1}+{\widehat{q_1}}^2)(1+\widehat{q_2}+{\widehat{q_2}}^2)}\left(\widehat{q_1}\widehat{q_2}{\left|\frac{{\partial }^2\mathcal{G}\left(m,n\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }\right.+\widehat{q_1}{\left|\frac{{\partial }^2\mathcal{G}\left(m,\frac{\widehat{q_2}n+\ell }{1+\widehat{q_2}}\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }\hfill \\ & & +\widehat{q_2}{\left|\frac{{\partial }^2\mathcal{G}\left(\frac{\widehat{q_1}m+k}{1+\widehat{q_1}},n\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }\left.{\left.+{\left|\frac{{\partial }^2\mathcal{G}\left(\frac{\widehat{q_1}m+k}{1+\widehat{q_1}},\frac{\widehat{q_2}n+\ell }{1+\widehat{q_2}}\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }\right)}^{\frac{1}{\lambda }}\right\},\hfill \end{array}$$

(63)

where ${\mathcal{V}}_{*}$ is given by (46).

This inequality gives error in the second and third term of (22).

Corollary 4. 

If in addition to the conditions of Corollary 3, $\widehat{q_1},\widehat{q_2}\to 1^{-},$ then

$$\begin{array}{cc}\hfill & \left|\frac{1}{(k-m)(\ell -n)}{\int }_m^k{\int }_n^{\ell }\mathcal{G}(x,y)dydx+\mathcal{G}\left(\frac{m+k}{2},\frac{n+\ell }{2}\right)-\Psi \right|\hfill \\ \hfill & \le \frac{(k-m)(\ell -n)}{4\sqrt[\lambda ]{36}}E(B,D;1,1),\hfill \end{array}$$

(64)

where

$$\begin{array}{cc}\hfill & E(B,D,1,1)=\sqrt[\lambda ]{4}{\left(4{\left|\frac{{\partial }^2\mathcal{G}\left(m,n\right)}{\partial u\partial v}\right|}^{\lambda }+2{\left|\frac{{\partial }^2\mathcal{G}\left(m,\ell \right)}{\partial u\partial v}\right|}^{\lambda }+2{\left|\frac{{\partial }^2\mathcal{G}\left(k,n\right)}{\partial u\partial v}\right|}^{\lambda }+{\left|\frac{{\partial }^2\mathcal{G}\left(k,\ell \right)}{\partial u\partial v}\right|}^{\lambda }\right)}^{\frac{1}{\lambda }}\hfill \\ \hfill & +\sqrt[\lambda ]{6}{\left(2{\left|\frac{{\partial }^2\mathcal{G}\left(m,n\right)}{\partial u\partial v}\right|}^{\lambda }+2{\left|\frac{{\partial }^2\mathcal{G}\left(m,\frac{n+\ell }{2}\right)}{\partial u\partial v}\right|}^{\lambda }+{\left|\frac{{\partial }^2\mathcal{G}\left(k,n\right)}{\partial u\partial v}\right|}^{\lambda }+{\left|\frac{{\partial }^2\mathcal{G}\left(k,\frac{n+\ell }{2}\right)}{\partial u\partial v}\right|}^{\lambda }\right)}^{\frac{1}{\lambda }}\hfill \\ \hfill & +\sqrt[\lambda ]{6}{\left(2{\left|\frac{{\partial }^2\mathcal{G}\left(m,n\right)}{\partial u\partial v}\right|}^{\lambda }+{\left|\frac{{\partial }^2\mathcal{G}\left(m,\ell \right)}{\partial u\partial v}\right|}^{\lambda }+2{\left|\frac{{\partial }^2\mathcal{G}\left(\frac{m+k}{2},n\right)}{\partial u\partial v}\right|}^{\lambda }+{\left|\frac{{\partial }^2\mathcal{G}\left(\frac{m+k}{2},\ell \right)}{\partial u\partial v}\right|}^{\lambda }\right)}^{\frac{1}{\lambda }}\hfill \\ \hfill & +\sqrt[\lambda ]{9}{\left({\left|\frac{{\partial }^2\mathcal{G}\left(m,n\right)}{\partial u\partial v}\right|}^{\lambda }+{\left|\frac{{\partial }^2\mathcal{G}\left(m,\frac{n+\ell }{2}\right)}{\partial u\partial v}\right|}^{\lambda }+{\left|\frac{{\partial }^2\mathcal{G}\left(\frac{m+k}{2},n\right)}{\partial u\partial v}\right|}^{\lambda }+{\left|\frac{{\partial }^2\mathcal{G}\left(\frac{m+k}{2},\frac{n+\ell }{2}\right)}{\partial u\partial v}\right|}^{\lambda }\right)}^{\frac{1}{\lambda }},\hfill \end{array}$$

(65)

and Ψ is given by the Equation (9).

Remark 4. 

If we compare the estimate to the estimate provided in (4), we discover that (64) represents a fresh estimate.

We now provide an illustration to support the conclusion drawn in Corollary 3.

Example 3. 

If we consider $\mathcal{G}(u,v)=u^2{v}^2,$ and $m=0=n,k=\ell =1,\lambda =2$. Then we have $\frac{{\partial }^2\mathcal{G}}{{}_0{\partial }_{\widehat{q_1}}u{}_0{\partial }_{\widehat{q_2}}v}(u,v)=(1+\widehat{q_1})(1+\widehat{q_2})uv.$ We now calculate the left and right side in the inequality (63):

$$\begin{array}{ccc}\hfill L.H.S& =& \left|\frac{1}{(1-0)(1-0)}{\int }_0^1{\int }_0^1{u}^2{v}^2{}_0{d}_{\widehat{q_2}}v{}_0{d}_{\widehat{q_1}}u+\mathcal{G}\left(\frac{1}{1+\widehat{q_1}},\frac{1}{1+\widehat{q_2}}\right)\right.\hfill \\ & & \left.-\frac{1}{1-0}{\int }_0^1\frac{1}{{(1+\widehat{q_2})}^2}{u}^2{}_0{d}_{\widehat{q_1}}u-\frac{1}{1-0}{\int }_0^1\frac{1}{{(1+\widehat{q_1})}^2}{v}^2{}_0{d}_{\widehat{q_2}}v\right|\hfill \\ & =& \left|\frac{1}{\left(1+\widehat{q_1}+{\widehat{q_1}}^2\right)\left(1+\widehat{q_2}+{\widehat{q_2}}^2\right)}+\frac{1}{{(1+\widehat{q_1})}^2{(1+\widehat{q_2})}^2}-\frac{1}{\left(1+\widehat{q_1}+{\widehat{q_1}}^2\right){\left(1+\widehat{q_2}\right)}^2}\right.\hfill \\ & & \left.-\frac{1}{{\left(1+\widehat{q_1}\right)}^2\left(1+\widehat{q_2}+{\widehat{q_2}}^2\right)}\right|.\hfill \end{array}$$

(66)

$$\begin{array}{ccc}\hfill R.H.S& =& \frac{1}{\left(1+\widehat{q_1}\right)\left(1+\widehat{q_2}\right)\sqrt{\left(1+\widehat{q_1}\right)\left(1+\widehat{q_2}\right)\left(1+\widehat{q_1}+{\widehat{q_1}}^2\right)\left(1+\widehat{q_2}+{\widehat{q_2}}^2\right)}}\hfill \\ & & \times \left(\begin{array}{c}\sqrt{\left(1+\widehat{q_1}\right)\left(1+\widehat{q_2}\right)}\left(\left(1+\widehat{q_1}\right)\left(1+\widehat{q_2}\right)\right)\\ +\sqrt{\left(1+\widehat{q_1}\right)\left(1+\widehat{q_2}+{\widehat{q_2}}^2\right)}\left(1+\widehat{q_1}\right)\\ +\sqrt{\left(1+\widehat{q_2}\right)\left(1+\widehat{q_1}+{\widehat{q_1}}^2\right)}\left(1+\widehat{q_2}\right)\\ +\sqrt{\left(1+\widehat{q_2}+{\widehat{q_2}}^2\right)\left(1+\widehat{q_1}+{\widehat{q_1}}^2\right)}\end{array}\right).\hfill \end{array}$$

(67)

The

Table 3. Parameters $\widehat{q_1}$ and $\widehat{q_2}$ are associated to the model given in Corollary 3.

Case	$\widehat{{\mathit{q}}_1}$	$\widehat{{\mathit{q}}_2}$	L.H.S	R.H.S	L.H.S-R.H.S
1	$0.5$	$0.5$	0.079617	1.6907	$-1.6111$
2	$0.25$	$0.25$	0.014861	2.5231	$-2.5082$
3	$0.66667$	$0.66667$	0.012924	1.3294	$-1.3165$
4	$0.25$	$0.66667$	0.013859	1.2689	$-1.255$
5	$0.66667$	$0.25$	0.013859	1.8314	$-1.255$
6	1	1	0.0069444	0.86658	$-0.85964$

makes it obvious that the estimate provided by the inequality “(63)” is accurate.

Corollary 5. 

If, in addition to the conditions of Theorem 7, $\hat{A}=\hat{B}=\frac{\widehat{q_1}(\widehat{q_1}m+k)}{1+\widehat{q_1}}$$d=\frac{\widehat{q_1}m+k}{1+\widehat{q_1}},$$\hat{C}=\hat{D}=\frac{\widehat{q_2}(\widehat{q_2}n+\ell )}{1+\widehat{q_2}}$ and $e=\frac{\widehat{q_2}n+\ell }{1+\widehat{q_2}},$ then

$$\begin{array}{ccc}\hfill & & \left|\frac{1}{(k-m)(\ell -n)}{\int }_m^k{\int }_n^{\ell }\mathcal{G}(x,y){}_n{d}_{\widehat{q_2}}y{}_m{d}_{\widehat{q_1}}x\right.\hfill \\ & & \left.+\frac{\widehat{q_1}\widehat{q_2}\mathcal{G}\left(m,n\right)+\widehat{q_1}\mathcal{G}\left(m,\ell \right)+\widehat{q_2}\mathcal{G}\left(k,n\right)+\mathcal{G}\left(k,\ell \right)}{(1+\widehat{q_1})(1+\widehat{q_2})}-{\mathcal{V}}_{**}\right|\hfill \\ & \le & \frac{4\widehat{q_1}\widehat{q_2}(k-m)(\ell -n)}{{(1+\widehat{q_1})}^3{(1+\widehat{q_2})}^3\sqrt[\lambda ]{4(1+\widehat{q_1})(1+\widehat{q_2})(1+\widehat{q_1}+{\widehat{q_1}}^2)(1+\widehat{q_2}+{\widehat{q_2}}^2)}}\hfill \\ & & \times \left[(1+3{\widehat{q_1}}^2+2{\widehat{q_1}}^3)(1+3{\widehat{q_2}}^2+2{\widehat{q_2}}^3){\left|\frac{{\partial }^2\mathcal{G}\left(m,n\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }\right.\hfill \\ & & +(1+3{\widehat{q_1}}^2+2{\widehat{q_1}}^3)(1+4\widehat{q_2}+{\widehat{q_2}}^2){\left|\frac{{\partial }^2\mathcal{G}\left(m,\ell \right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }\hfill \\ & & +(1+4\widehat{q_1}+{\widehat{q_1}}^2)(1+3{\widehat{q_2}}^2+2{\widehat{q_2}}^3){\left|\frac{{\partial }^2\mathcal{G}\left(k,n\right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }\hfill \\ & & +{\left.(1+4\widehat{q_1}+{\widehat{q_1}}^2)(1+4\widehat{q_2}+{\widehat{q_2}}^2){\left|\frac{{\partial }^2\mathcal{G}\left(k,\ell \right)}{{}_m{\partial }_{\widehat{q_1}}u{}_n{\partial }_{\widehat{q_2}}v}\right|}^{\lambda }\right]}^{\frac{1}{\lambda }},\hfill \end{array}$$

(68)

where ${\mathcal{V}}_{**}$ is given in (51). The inequality (68) gives error in the third and fourth term of the inequality (22).

Remark 5. 

If $\widehat{q_1},\widehat{q_2}\to 1^{-1},$ and the the requirement of Corollary 3, are fulfilled, we recapture “(5)”.

Example 4. 

If we consider $\mathcal{G}(u,v)=u^2{v}^2,$ and $m=0=n,k=\ell =1.$ We have $\frac{{\partial }^2\mathcal{G}}{{}_0{\partial }_{\widehat{q_1}}u{}_0{\partial }_{\widehat{q_2}}v}(u,v)=(1+\widehat{q_1})(1+\widehat{q_2})uv.$ If $\lambda =2,$ then the left and right side of inequality (63) leads respectively to:

$$\begin{array}{ccc}\hfill L.H.S& =& \left|{\int }_0^1{\int }_0^1{u}^2{v}^2{}_0{d}_{\widehat{q_2}}v{}_0{d}_{\widehat{q_1}}u+1-\frac{1}{(1-0)(1+\widehat{q_2})}{\int }_0^1{u}^2{}_0{d}_{\widehat{q_1}}u\right.\hfill \\ & & -\left.\frac{1}{(1-0)(1+\widehat{q_1})}{\int }_0^1{v}^2{}_0{d}_{\widehat{q_2}}v\right|\hfill \\ & =& \left|\frac{1}{\left(1+\widehat{q_1}+{\widehat{q_1}}^2\right)\left(1+\widehat{q_2}+{\widehat{q_2}}^2\right)}+\frac{1}{(1+\widehat{q_1})(1+\widehat{q_2})}-\frac{1}{\left(1+\widehat{q_1}+{\widehat{q_1}}^2\right)\left(1+\widehat{q_2}\right)}\right.\hfill \\ & & \left.-\frac{1}{\left(1+\widehat{q_1}\right)\left(1+\widehat{q_2}+{\widehat{q_2}}^2\right)}\right|.\hfill \end{array}$$

(69)

$$\begin{array}{ccc}\hfill R.H.S& =& \frac{4{\widehat{q_1}}^2{\widehat{q_2}}^2(1-0)(1-0)}{{(1+\widehat{q_1})}^3{(1+\widehat{q_2})}^3\sqrt[2]{4(1+\widehat{q_1})(1+\widehat{q_2})(1+\widehat{q_1}+{\widehat{q_1}}^2)(1+\widehat{q_2}+{\widehat{q_2}}^2)}}\hfill \\ & & \times {\left[(1+4\widehat{q_1}+{\widehat{q_1}}^2)(1+4\widehat{q_2}+{\widehat{q_2}}^2){(1+\widehat{q_1})}^2{(1+\widehat{q_2})}^2\right]}^{\frac{1}{2}}.\hfill \end{array}$$

(70)

By assigning several values to the parameters $\widehat{q_1}$ and $\widehat{q_2},$ we create the table below to examine the accuracy of our estimate.

The last column of the

Table 4. Parameters $\widehat{q_1}$ and $\widehat{q_2}$ are associated to the model given in Corollary 5.

Case	$\widehat{{\mathit{q}}_1}$	$\widehat{{\mathit{q}}_2}$	L.H.S	R.H.S	L.H.S-R.H.S
1	$0.25$	$0.25$	0.00145	1.6907	$-1.6893$
2	$0.5$	$0.5$	0.00907	0.03057	$-0.0215$
3	$0.66667$	$0.66667$	0.015956	0.059823	$-0.043867$
4	$0.25$	$0.66667$	0.004812	0.015513	$-0.010701$
5	$0.66667$	$0.25$	0.004812	0.015513	$-0.010701$
6	1	1	0.027778	0.125	$-0.097222$

shows that the inequality (68) is valid.

4. Applications to Special Means

Let us recall some useful means of real numbers ${\wp }_1,{\wp }_2$. We use for extended arithmetic means, $\mathcal{A}({\mu }_1,{\mu }_2,\dots ,{\mu }_s)=\frac{1}{s}\sum _{j=0}^s{\mu }_j,{\mu }_j\in \mathbb{R},j=1,2,\dots ,s.$ For logarithmic means, we use $L({\mu }_1,{\mu }_2)=\frac{{\mu }_2-{\mu }_1}{ln|{\mu }_2|-ln|{\mu }_1|},|{\mu }_1|\ne |{\mu }_2|,{\mu }_1{\mu }_2\ne 0.$ The generalized log-mean is denoted and defined by: ${\pounds }_{ϵ}({\mu }_1,{\mu }_2)={\left(\frac{{\mu }_2^{ϵ+1}-{\mu }_1^{ϵ+1}}{(ϵ+1)({\mu }_2-{\mu }_1)}\right)}^{\frac{1}{ϵ}},ϵ\in \mathbb{R}-\left\{-1,0\right\}.$

Now we have following applications of our results in terms of above means of real numbers.

Proposition 1. 

Let $m,k,n,\ell \in \mathbb{R},m<k,n<\ell ,0\notin [m,k],0\notin [n,\ell ],$ and $p_1,p_2\in \mathbb{R},\left|p_1\right|\ge 2$ and $|p_2|\ge 2.$ Then we have

$$\begin{array}{ccc}\hfill & & \left|\left(S_1(m,k,p_1;\widehat{q_1})-2\frac{\mathcal{A}(\widehat{q_1}{m}^{p_1},k^{p_1})}{1+\widehat{q_1}}\right)\left(S_2(n,\ell ,p_2;\widehat{q_2})-2\frac{\mathcal{A}(\widehat{q_2}{n}^{p_2},{\ell }^{p_2})}{1+\widehat{q_2}}\right)\right|\hfill \\ & \le & \frac{{\widehat{q_1}}^2{\widehat{q_2}}^2{p}_1{p}_2(k-m)(\ell -n)}{{(1+\widehat{q_1})}^3{(1+\widehat{q_2})}^3(1+\widehat{q_1}+{\widehat{q_1}}^2)(1+\widehat{q_2}+{\widehat{q_2}}^2)}\left[(1+{\widehat{q_1}}^2){\left|m\right|}^{p_1-1}\right.\hfill \\ & & \left.+2{\pounds }_{p_1-1}^{p_1-1}(\widehat{q_1}(k-m)+m,k)\right]\left[(1+{\widehat{q_2}}^2){\left|n\right|}^{p_2-1}+2{\pounds }_{p_2-1}^{p_2-1}(\widehat{q_2}(\ell -n)+n,\ell )\right]\hfill \\ & & +\frac{{\widehat{q_1}}^2{\widehat{q_2}}^2(k-m)(\ell -n)}{{(1+\widehat{q_1})}^3{(1+\widehat{q_2})}^3(1+\widehat{q_1}+{\widehat{q_1}}^2)(1+\widehat{q_2}+{\widehat{q_2}}^2)}\hfill \\ & & \times \left\{({\widehat{q_2}}^2+2\widehat{q_2}-1)p_2{\pounds }_{p_2-1}^{p_2-1}\left(\frac{\widehat{q_2}\ell +n}{1+\widehat{q_2}},\frac{\widehat{q_2}n+\ell }{1+\widehat{q_2}}\right)\right.\hfill \\ & & \times [p_1(1+{\widehat{q_1}}^2){\left|m\right|}^{p_1-1}+2p_1{\pounds }_{p_1-1}^{p_1-1}(\widehat{q_1}(k-m)+m,k)\hfill \\ & & +\frac{({\widehat{q_1}}^2+2\widehat{q_1}-1)p_1}{2}{\pounds }_{p_1-1}^{p_1-1}\left(\frac{\widehat{q_1}k+m}{1+\widehat{q_1}},\frac{\widehat{q_1}m+k}{1+\widehat{q_1}}\right)]\hfill \\ & & +({\widehat{q_1}}^2+2\widehat{q_1}-1)p_1{\pounds }_{p_1-1}^{p_1-1}\left(\frac{\widehat{q_1}k+m}{1+\widehat{q_1}},\frac{\widehat{q_1}m+k}{1+\widehat{q_1}}\right)\hfill \\ & & \times [p_2(1+{\widehat{q_2}}^2){\left|n\right|}^{p_2-1}+2p_2{\pounds }_{p_2-1}^{p_2-1}(\widehat{q_2}(\ell -n)+n,\ell )\hfill \\ & & +\frac{({\widehat{q_2}}^2+2\widehat{q_2}-1)p_2}{2}{\pounds }_{p_2-1}^{p_2-1}\left(\frac{\widehat{q_2}\ell +n}{1+\widehat{q_2}},\frac{\widehat{q_2}n+\ell }{1+\widehat{q_2}}\right)\left]\right\},\hfill \end{array}$$

(71)

where

$$S_1(m,k,p_1;\widehat{q_1}):=(1-\widehat{q_1})\sum _{\delta =0}^{\infty }{\widehat{q_1}}^{\delta }{\left({\widehat{q_1}}^{\delta }(k-m)+m\right)}^{p_1},$$

(72)

and

$$S_2(n,\ell ,p_2;\widehat{q_2}):=(1-\widehat{q_2})\sum _{\delta =0}^{\infty }{\widehat{q_2}}^{\delta }{\left({\widehat{q_2}}^{\delta }(\ell -n)+n\right)}^{p_2}.$$

(73)

Proof. 

Let $f(x,y)=x^{p_1}{y}^{p_2},p_1,p_2\in \mathbb{Z},|p_1|\ge 2,|p_2|\ge 2.$ Then the desired outcome follows from Corollary 2. □

Remark 6. 

If in addition to the conditions of Proposition 1, let $\widehat{q_1},\widehat{q_2}\to 1^{-},$ then

$$\begin{array}{ccc}\hfill & & \left|\left({\pounds }_{p_1}^{p_1}(m,k)-\mathcal{A}(m^{p_1},k^{p_1})\right)\left({\pounds }_{p_2}^{p_2}(n,\ell )-\mathcal{A}(n^{p_2},{\ell }^{p_2})\right)\right|\le \frac{|p_1\left|\right|p_2|(k-m)(\ell -n)}{16}\hfill \\ & & \times \mathcal{A}\left({\left|m\right|}^{p_1-1}{\left|n\right|}^{p_2-1}{,\left|m\right|}^{p_1-1}{|\ell |}^{p_2-1}{,\left|k\right|}^{p_1-1}{\left|n\right|}^{p_2-1}{,\left|k\right|}^{p_1-1}{|\ell |}^{p_2-1},{\left|\frac{m+k}{2}\right|}^{p_1-1}{\left|n\right|}^{p_2-1}\right.\hfill \\ & & \left.,{\left|\frac{m+k}{2}\right|}^{p_1-1}{|\ell |}^{p_2-1}{,\left|m\right|}^{p_1-1}{\left|\frac{n+\ell }{2}\right|}^{p_2-1},{\left|k\right|}^{p_1-1}{\left|\frac{n+\ell }{2}\right|}^{p_2-1},{\left|\frac{m+k}{2}\right|}^{p_1-1}{\left|\frac{n+\ell }{2}\right|}^{p_2-1}\right).\hfill \end{array}$$

(74)

Proposition 2. 

Let $m,k,n,\ell \in \mathbb{R},m<k,n<\ell ,0\notin [m,k],0\notin [n,\ell ]$. Then we have

$$\begin{array}{ccc}\hfill & & \left|\left(S_3(m,k;\widehat{q_1})-2\frac{\mathcal{A}(\widehat{q_1}{m}^{-1},k^{-1})}{1+\widehat{q_1}}\right)\left(S_4(n,\ell ;\widehat{q_2})-2\frac{\mathcal{A}(\widehat{q_2}{n}^{-1},{\ell }^{-1})}{1+\widehat{q_2}}\right)\right|\hfill \\ & \le & \frac{{\widehat{q_1}}^2{\widehat{q_2}}^2(k-m)(\ell -n)}{{(1+\widehat{q_1})}^3{(1+\widehat{q_2})}^3(1+\widehat{q_1}+{\widehat{q_1}}^2)(1+\widehat{q_2}+{\widehat{q_2}}^2)}\left[\frac{1+{\widehat{q_1}}^2}{m^2}+2{\pounds }_{-2}^{-2}(\widehat{q_1}(k-m)+m,k)\right]\hfill \\ & & \times \left[\frac{1+{\widehat{q_2}}^2}{n^2}+2{\pounds }_{-2}^{-2}(\widehat{q_2}(\ell -n)+n,\ell )\right]\hfill \\ & & +\frac{{\widehat{q_1}}^2{\widehat{q_2}}^2(k-m)(\ell -n)}{{(1+\widehat{q_1})}^3{(1+\widehat{q_2})}^3(1+\widehat{q_1}+{\widehat{q_1}}^2)(1+\widehat{q_2}+{\widehat{q_2}}^2)}\hfill \\ & & \times \left\{({\widehat{q_2}}^2+2\widehat{q_2}-1){\pounds }_{-2}^{-2}\left(\frac{\widehat{q_2}\ell +n}{1+\widehat{q_2}},\frac{\widehat{q_2}n+\ell }{1+\widehat{q_2}}\right)\right.[\frac{1+{\widehat{q_1}}^2}{m^2}+2{\pounds }_{-2}^{-2}(\widehat{q_1}(k-m)+m,k)\hfill \\ & & +\frac{({\widehat{q_1}}^2+2\widehat{q_1}-1)}{2}{\pounds }_{-2}^{-2}\left(\frac{\widehat{q_1}k+m}{1+\widehat{q_1}},\frac{\widehat{q_1}m+k}{1+\widehat{q_1}}\right)]\hfill \\ & & +({\widehat{q_1}}^2+2\widehat{q_1}-1){\pounds }_{-2}^{-2}\left(\frac{\widehat{q_1}k+m}{1+\widehat{q_1}},\frac{\widehat{q_1}m+k}{1+\widehat{q_1}}\right)\times [\frac{1+{\widehat{q_2}}^2}{n^2}+2{\pounds }_{-2}^{-2}(\widehat{q_2}(\ell -n)+n,\ell )\hfill \\ & & +\frac{({\widehat{q_2}}^2+2\widehat{q_2}-1)}{2}{\pounds }_{-2}^{-2}\left(\frac{\widehat{q_2}\ell +n}{1+\widehat{q_2}},\frac{\widehat{q_2}n+\ell }{1+\widehat{q_2}}\right)\left]\right\},\hfill \end{array}$$

(75)

where

$$S_1(m,k;\widehat{q_1}):=(1-\widehat{q_1})\sum _{\delta =0}^{\infty }{\widehat{q_1}}^{\delta }{\left({\widehat{q_1}}^{\delta }(k-m)+m\right)}^{-1},$$

(76)

and

$$S_2(n,\ell ;\widehat{q_2}):=(1-\widehat{q_2})\sum _{\delta =0}^{\infty }{\widehat{q_2}}^{\delta }{\left({\widehat{q_2}}^{\delta }(\ell -n)+n\right)}^{-1}.$$

(77)

Proof. 

The result is immediate from Corollary 2, with $\mathcal{G}(x,y)=\frac{1}{xy}.$ □

Remark 7. 

If in addition to the conditions of Proposition 2, let $\widehat{q_1},\widehat{q_2}\to 1^{-},$ then

$$\begin{array}{ccc}\hfill & & \left|\left(L^{-}(m,k)-\mathcal{A}(m^{-1},k^{-1})\right)\left(L^{-}(n,\ell )-\mathcal{A}(n^{-1},{\ell }^{-1})\right)\right|\le \frac{(k-m)(\ell -n)}{16}\hfill \\ & & \times \mathcal{A}\left({\left|mn\right|}^{-2}{,\left|kn\right|}^{-2}{,|m\ell |}^{-2},{|k\ell |}^{-2},{\left|\frac{mn+kn}{2}\right|}^{-2},{\left|\frac{m\ell +k\ell }{2}\right|}^{-2},{\left|\frac{mn+m\ell }{2}\right|}^{-2}\right.\hfill \\ & & \left.,{\left|\frac{kn+k\ell }{2}\right|}^{-2},{\left|\frac{mn+m\ell +kn+k\ell }{4}\right|}^{-2}\right).\hfill \end{array}$$

(78)

Remark 8. 

Some more outcomes about special means can be obtain by applying Corollary 1, 3, 4 and 5.

5. Discussion and Concluding Remarks

This study’s main focus is on the error estimation type finding for the quantum Hermite–Hadamard inequality. We begin our examination by keeping a twice partial quantum differentiable function. We create a generic new identity for twice partially q-differentiable functions provided by (23) by asserting a general kernel defined on the partition of a rectangular domain. Additionally, by assuming that the absolute value of the twice partial quantum differentiable function is coordinated convex, we are able to derive two new generalized inequalities, denoted by (37) and (54). We make some appropriate parameter selections in order to assess the effectiveness and correctness of our outcomes. The particular choices modify our key findings and provide new, precise error bounds for the quantum Hermite–Hadamard inequality (22) that has just been discovered. We give the new special cases in the Corollary 1,2, 3, 4, and 5. The outcomes of the aforementioned corollaries give an error in the lower and upper bounds of the inequality (22). We give interesting instances to support our findings. We point out some new applications about special means of real numbers. Additionally, we note that the methodology used in this study is useful in establishing precise error bounds for the Simpson’s-type inequality and the Quantum Ostrowski inequality. In our opinion, the convex analysis, optimization, and several fields of the pure and applied sciences can all benefit from the current findings. Regardless, we believe that these findings contribute to our growing understanding of quantum calculus’s behavior, characteristics, and wide range of practical applications. We shall use the Iscan–Holder inequality in subsequent works to develop a number of fresh, intriguing inequalities for various classes of convex and pre-invex functions.