Raina’s Function-Based Formulations of Right-Sided Simpson’s and Newton’s Inequalities for Generalized Coordinated Convex Functions

Abstract

The main focus of this article is to derive some new counterparts to Simpson’s and Newton’s type inequalities involve a class of generalized coordinated convex mappings. This class contains several new and known classes of convexity as special cases. For further demonstration, we deploy the concept of right quantum derivatives to develop two new identities involving Raina’s function. Moreover, by implementing these auxiliary results together with generalized convexity, we acquire a Holder-type inequality. We also acquire some applications of our main findings by making use of suitable substitutions in Raina’s function.

1. Introduction

Thomas Simpson, a pioneering mathematician from 1710–1761, made significant contributions to the field of numerical integration and the estimation of definite integrals, specifically through the development of Simpson’s rule. A comparable approximation was employed by Kepler nearly a century earlier, leading to its alternate designation as Kepler’s rule. Simpson’s rule encompasses the three-point Newton–Cotes quadrature rule, giving rise to estimations that rely on a three-step quadratic kernel, which is occasionally referred to as Newton-type results.

(1) Simpson’s quadrature formula (Simpson’s 1/3 rule) is expressed as follows:

$${\int }_{{\pi }_1}^{{\pi }_2}\mathsf{Ϝ}\left(x\right)d\left(x\right)\approx \frac{{\pi }_2-{\pi }_1}{6}\left[\mathsf{Ϝ}\left({\pi }_1\right)+4\mathsf{Ϝ}\left(\frac{{\pi }_1+{\pi }_2}{2}\right)+\mathsf{Ϝ}\left({\pi }_2\right)\right].$$

(1) Simpson’s second formula, or the Newton–Cotes quadrature formula (Simpson’s 3/8 rule), is expressed as follows:

$${\int }_{{\pi }_1}^{{\pi }_2}\mathsf{Ϝ}\left(x\right)d\left(x\right)\approx \frac{{\pi }_2-{\pi }_1}{8}\left[\mathsf{Ϝ}\left({\pi }_1\right)+3\mathsf{Ϝ}\left(\frac{2{\pi }_1+{\pi }_2}{3}\right)+3\mathsf{Ϝ}\left(\frac{{\pi }_1+2{\pi }_2}{3}\right)+\mathsf{Ϝ}\left({\pi }_2\right)\right].$$

A large variety of methods and techniques for estimations with these quadrature rules are available. One of them is Simpson’s inequality, which can be stated in the following way:

Theorem 1.

Suppose that $\mathsf{Ϝ}:\left[{\pi }_1,{\pi }_2\right]\to \mathbb{R}$ is a four times continuously differentiable mapping on $\left({\pi }_1,{\pi }_2\right),$ and let ${∥{\mathsf{Ϝ}}^{\left(4\right)}∥}_{\infty }={\displaystyle \underset{x\in \left({\pi }_1,{\pi }_2\right)}{sup}}\left|{\mathsf{Ϝ}}^{\left(4\right)}\left(x\right)\right|<\infty .$ Then, one has the inequality

$$\left|\frac{1}{3}\left[\frac{\mathsf{Ϝ}\left({\pi }_1\right)+\mathsf{Ϝ}\left({\pi }_2\right)}{2}+2\mathsf{Ϝ}\left(\frac{{\pi }_1+{\pi }_2}{2}\right)\right]-\frac{1}{{\pi }_2-{\pi }_1}{\int }_{{\pi }_1}^{{\pi }_2}\mathsf{Ϝ}\left(x\right)d\left(x\right)\right|\le \frac{1}{2880}{∥{\mathsf{Ϝ}}^{\left(4\right)}∥}_{\infty }{\left({\pi }_2-{\pi }_1\right)}^4.$$

The work of numerous mathematicians has been dedicated to the advancement of Simpson- and Newton-type results for convex functions. The theory of convexity has been instrumental in addressing various problems. Such a domain involves studying numerical integration quadrature formulas, where novel Simpson’s inequalities and their far-reaching applications have been propounded by Dragomir et al. [1]. Furthermore, Alomari et al. [2] developed some new inequalities of Simpson-type s-convex functions. The classical concept of convexity also has a close connection with the concept of symmetry. In the literature, there exist numerous properties of symmetric convex sets. One fascinating aspect of this relationship is that we work on one and apply it to the other.

Mathematicians were inspired by Euler’s groundbreaking work to establish a vital connection between physics and mathematics, paving the way for the development of quantum computing, quantum calculus, and quantitative computer mathematics. The impact of these innovations is wide-ranging, with implications for numerous mathematical disciplines such as combinatorics, number theory, polynomial orthogonal, and fundamental hypergeometric functions, as well as other scientific fields including mechanics, relativity theory, and quantum theory [3,4]. Quantum calculus leverages quantum information theory to address diverse applications spanning the domains of computer science, information theory, philosophy, and cryptography, thereby constituting a compelling interdisciplinary domain [5,6]. Subsequently, Jackson introduced q-calculus, which did not rely on limit calculus [7,8]. In 2013 the ${}_{{\pi }_1}{D}_q$ difference operator and $q_{{\pi }_1}$ integral were presented by Jessada Tariboon and Sotiris Ntouyas et al. [9]. After that, the expression of ${}^{{\pi }_2}{D}_q$ for the derivative and the $q^{{\pi }_2}$-integral were established by Bermudo et al. [10] in 2020. Sadjang et al. [11] generalized quantum calculus and post-quantum calculus and introduced new notions. Tunç and Göv introduced the post-quantum variant of the ${}_{{\pi }_1}{D}_q$ difference operator and $q_{{\pi }_1}$ integral in [12]. The authors of [13] used the ${}_{{\pi }_1}{D}_q{,}^{{\pi }_2}{D}_q$ derivatives and $q_{{\pi }_1},q^{{\pi }_2}$ integrals in the proof of Hermite–Hadamard integral inequalities and left-right estimates for the coordinated convex and convex functions of these inequalities. The generalization of the results in [14] and the work on the proof of Hermite–Hadamard-type inequalities with the left estimates using a ${}_{{\pi }_1}{D}_{p,q}$ difference operator and ${\left(p,q\right)}_{{\pi }_1}$ integral was carried out by Kunt et al. In [15] Noor has given quantum estimates for Hermite-Hadamard inequalities. Nwaeze et al. [16] proved some satisfying parameterized quantum integral inequalities for generalized qūasi-convex functions. The quantum Hermite–Hadamard inequality was proven by Khan et al. by using a green function [17]. In [18,19,20,21] authors have introduced the new quantum Simpson- and quantum Newton-type inequalities for convex and coordinated convex functions, respectively. The authors of [22,23,24,25,26] studied quantum Ostrowski inequalities for convex and coordinated convex functions. The notions of a ${}^{{\pi }_2}{D}_{p,q}$ difference operator and ${\left(p,q\right)}^{{\pi }_2}$-integral were employed by Latif and Chu in establishing their right estimates of Hermite–Hadamard-type inequalities [27]. These inequalities were used to prove Ostrowski’s inequalities. For further interesting advancements in this field, see [28,29,30,31,32,33]. However, in [34] Latif gave q-analogues of Hermite-Hadamard inequality of functions of two variables on finite rectangles in the plane. In [35] Vivas-Cortez, introduced new quantum estimates of Trapezium-Type Inequalities for Generalized $\varphi$-Convex Functions.

Motivated by the above-mentioned works [20,35], in this study, we develop two new identities to obtain fresh error bounds for Simpson’s one-third rule and Simpson’s three-eighths rule named the Newton’s rule, by coupling with the quantum integration convexity property of the functions over a rectangular domain. Our work presents a novel contribution by successfully integrating various existing and new findings from the literature, resulting in a significant unification of knowledge.

The rest of this study is structured as follows. Section 2 provides a concise overview of the history and applications of quantum calculus. Section 3 furnishes an abridged explanation of the terminologies and concepts underlying quantum calculus, as well as the relevant inequalities. Section 4 formulates the coordinated version of right quantum integral identities. Section 5 outlines the key findings, comprising coordinated versions of right quantum Simpson- and right quantum Newton-type inequalities, as well as related outcomes that were unearthed through appropriate fabrication. Section 6 presents the concluding remarks and highlights of this innovative research.

2. Quantum Calculus Preliminaries

In this section, we present some basic definitions and inequalities.

Definition 1

([35]). Let $\nu ,\mu >0$ and $\sigma =\left(\sigma \right(0),\dots ,\sigma ($ķ$),\dots )$ be a bounded sequence of positive real numbers. A nonempty set Δ to be generalized convex if

$$\begin{array}{ccc}{\pi }_1+t_1{\check{R}}_{\mu ,\nu ,\sigma }({\pi }_2-{\pi }_1)\in \Delta ,& \forall {\pi }_1,{\pi }_2\in \Delta ,& t\in \left[0,1\right].\end{array}$$

Here, ${\check{R}}_{\mu ,\nu ,\sigma }(.)$ is Raina’s function, which is as follows:

$${\check{R}}_{\mu ,\nu ,\sigma }\left(\check{z}\right)={\check{R}}_{\mu ,\nu }^{\sigma \left(0\right),\sigma \left(1\right),\dots }\left(\check{z}\right)=\sum _{k=0}^{\infty }\frac{\sigma \left(ķ\right)}{\Gamma \left(\mu +\nu \right)}{\check{z}}^{\mathsf{ķ}},$$

(1)

where $\left|\check{z}\right|<R$ and $\Gamma \left(.\right)$ is the well-known gamma function. For further details, see [36].

Definition 2

([35]). Let $\nu ,\mu >0$ and $\sigma =\left(\sigma \right(0),\dots ,\sigma ($ķ$),\dots )$ be a bounded sequence of positive real numbers. A function $F:\Delta \to {\mathbb{R}}^{+}$ is said to be generalized convex if

$$\begin{array}{ccc}F({\pi }_1+t{\check{R}}_{\mu ,\nu ,\sigma }({\pi }_2-{\pi }_1)\le (1-t)F\left({\pi }_1\right)+tF\left({\pi }_2\right),& \forall {\pi }_1,{\pi }_2\in \pounds ,& t\in [0,1].\end{array}$$

(2)

Definition 3

([37]). For μ,$\sigma >0$ and σ = $\left(\sigma \left(0\right),\sigma \left(1\right),\dots ,\sigma \left(\mathsf{ķ}\right),\dots \right)$, this is assumed to be a bounded sequence of ${\mathbb{R}}^{+}$. A nonempty set $\Delta$ is known to be a coordinated generalized convex set

$$\left({\pi }_1+t{\check{R}}_{\mu ,\nu ,\sigma }({\pi }_2-{\pi }_1),{\pi }_3+s{\check{R}}_{\mu ,\nu ,\sigma }({\pi }_4-{\pi }_3)\right)\in \Delta$$

(3)

which holds for all $t,s\in \left[0,1\right]$, while $\left({\pi }_1,{\pi }_3\right)\times \left({\pi }_2,{\pi }_4\right)\in$Δ and ${\check{R}}_{\mu ,\nu }^{\sigma }\left(.\right)$ denotes Raina’s function.

Definition 4

([37]). For μ,$\sigma >0$ and σ = $\left(\sigma \left(0\right),\sigma \left(1\right),\dots ,\sigma \left(\mathsf{ķ}\right),\dots \right)$, this is assumed to be a bounded sequence of ${\mathbb{R}}^{+}$. A mapping $F:\Delta \to {\mathbb{R}}^{+}$ is said to be coordinated generalized convex if

$$\begin{array}{ccc}& & F\left({\pi }_1+t{\check{R}}_{\mu ,\nu ,\sigma }({\pi }_2-{\pi }_1),{\pi }_3+s{\check{R}}_{\mu ,\nu ,\sigma }({\pi }_4-{\pi }_3)\right)\hfill \\ & \le & tsF\left({\pi }_2,{\pi }_4\right)+t\left(1-s\right)F\left({\pi }_2,{\pi }_3\right)+\left(1-t\right)sF\left({\pi }_1,{\pi }_3\right),\hfill \end{array}$$

(4)

holds for all $t,s\in \left[0,1\right]$ and $\left({\pi }_1,{\pi }_3\right)\times \left({\pi }_2,{\pi }_4\right)\in \Delta .$

Definition 5.

In [8], F. H. Jackson gave the q-Jackson integral from 0 to ${\pi }_2$ for $0<q<1$ as

$$\underset{0}{\overset{{\pi }_2}{\int }}\mathsf{Ϝ}\left(x\right)\begin{array}{c}{d}_q\left(x\right)\end{array}=\left(1-q\right){\pi }_2\sum _{n=0}^{\infty }{q}^n\mathsf{Ϝ}\left({\pi }_2{q}^n\right),$$

(5)

provided that the sum converges absolutely. Moreover, he gave the q-Jackson integral in an arbitrary interval $[{\pi }_1,{\pi }_2]$ as follows:

$$\underset{{\pi }_1}{\overset{{\pi }_2}{\int }}\mathsf{Ϝ}\left(x\right)\begin{array}{c}{d}_q\left(x\right)\end{array}=\underset{0}{\overset{{\pi }_2}{\int }}\mathsf{Ϝ}\left(x\right)\begin{array}{c}{d}_q\left(x\right)\end{array}-\underset{0}{\overset{{\pi }_1}{\int }}\mathsf{Ϝ}\left(x\right)\begin{array}{c}{d}_q\left(x\right)\end{array}.$$

Definition 6

([9]). For a continuous function $\mathsf{Ϝ}:\left[{\pi }_1,{\pi }_2\right]\to \mathbb{R},$ the left quantum derivative $\left(q_{{\pi }_1}-derivative\right)$ of ϝ at $x\in \left[{\pi }_1,{\pi }_2\right]$ is as characterized below:

$$\begin{array}{c}{}_{{\pi }_1}{D}_q\mathsf{Ϝ}\left(x\right)\end{array}=\frac{\mathsf{Ϝ}\left(x\right)-\mathsf{Ϝ}\left(qx+\left(1-q\right){\pi }_1\right)}{\left(1-q\right)\left(x-{\pi }_1\right)},x\ne {\pi }_1.$$

(6)

For $x={\pi }_1$, we state that ${}_{{\pi }_1}{D}_q\mathsf{Ϝ}\left({\pi }_1\right)={lim}_{x\to {\pi }_1}{}_{{\pi }_1}{D}_q\mathsf{Ϝ}\left(x\right)$ if it exists and is finite.

Definition 7

([10]). For a continuous function $\mathsf{Ϝ}:\left[{\pi }_1,{\pi }_2\right]\to \mathbb{R},$ the right quantum derivative $\left(q^{{\pi }_2}-derivative\right)$ of ϝ at $x\in \left[{\pi }_1,{\pi }_2\right]$ is characterized by the expression

$$\begin{array}{c}{}^{{\pi }_2}{D}_q\mathsf{Ϝ}\left(x\right)\end{array}=\frac{\mathsf{Ϝ}\left(qx+\left(1-q\right){\pi }_2\right)-\mathsf{Ϝ}\left(x\right)}{\left(1-q\right)\left({\pi }_2-x\right)},x\ne {\pi }_2.$$

(7)

For $x={\pi }_2$, we state that ${}^{{\pi }_2}{D}_q\mathsf{Ϝ}\left({\pi }_2\right)={lim}_{x\to {\pi }_2}{}^{{\pi }_2}{D}_q\mathsf{Ϝ}\left(x\right)$ if it exists and is finite.

Definition 8

([9]). Let $\mathsf{Ϝ}:\left[{\pi }_1,{\pi }_2\right]\to \mathbb{R}$ be a continuous function. The left definite quantum integral $\left(q_{{\pi }_1}-\mathit{definite}\mathit{integral}\right)$ on $\left[{\pi }_1,{\pi }_2\right]$ is defined as

$$\begin{array}{ccc}\hfill \underset{{\pi }_1}{\overset{{\pi }_2}{\int }}\mathsf{Ϝ}\left(x\right)\begin{array}{c}{}_{{\pi }_1}{d}_q\left(x\right)\end{array}& =& \left(1-q\right)\left({\pi }_2-{\pi }_1\right)\sum _{n=0}^{\infty }{q}^n\mathsf{Ϝ}\left(q^n{\pi }_2+\left(1-q^n\right){\pi }_1\right)\hfill \\ & =& \left({\pi }_2-{\pi }_1\right)\underset{0}{\overset{1}{\int }}\mathsf{Ϝ}\left(\left(1-\tau \right){\pi }_1+\tau {\pi }_2\right)\begin{array}{c}{d}_q\left(\tau \right)\end{array}.\hfill \end{array}$$

(8)

On the other hand, S. Bermudo et al. [10] gave the following new definition:

Definition 9.

Let $\mathsf{Ϝ}:\left[{\pi }_1,{\pi }_2\right]\to \mathbb{R}$ be a continuous function. The right definite quantum $\left(q^{{\pi }_2}-definiteintegral\right)$ on $\left[{\pi }_1,{\pi }_2\right]$ is defined as

$$\begin{array}{ccc}\hfill \underset{{\pi }_1}{\overset{{\pi }_2}{\int }}\mathsf{Ϝ}\left(x\right)\begin{array}{c}{}^{{\pi }_2}{d}_q\left(x\right)\end{array}& =& \left(1-q\right)\left({\pi }_2-{\pi }_1\right)\sum _{n=0}^{\infty }{q}^n\mathsf{Ϝ}\left(q^n{\pi }_1+\left(1-q^n\right){\pi }_2\right)\hfill \\ & =& \left({\pi }_2-{\pi }_1\right)\underset{0}{\overset{1}{\int }}\mathsf{Ϝ}\left(\tau {\pi }_1+\left(1-\tau \right){\pi }_2\right)\begin{array}{c}{d}_q\left(\tau \right)\end{array}.\hfill \end{array}$$

(9)

For more details about $q^{{\pi }_2}$ integrals and the corresponding inequalities, see [10].

Now, let us give the following notation, which will be used many times in the later sections (see [4]):

$${\left[n\right]}_q=\frac{q^n-1}{q-1}.$$

Moreover, for our main results, we give the lemma

Lemma 1

([9]). We have the equality

$$\underset{{\pi }_1}{\overset{{\pi }_2}{\int }}{\left(x-{\pi }_1\right)}^{\alpha }\begin{array}{c}{}_{{\pi }_1}{d}_q\left(x\right)\end{array}=\frac{{\left({\pi }_2-{\pi }_1\right)}^{\alpha +1}}{{\left[\alpha +1\right]}_q}$$

for $\alpha \in \mathbb{R}\setminus \left\{-1\right\}.$

In [26], H. Budak et al. proved the following variant of the quantum Ostrowski inequality using the $q_{{\pi }_1}$ and $q^{{\pi }_2}$ integrals:

Theorem 2

([26]). Let $\mathsf{Ϝ}:\left[{\pi }_1,{\pi }_2\right]\subset \mathbb{R}\to \mathbb{R}$ be a function and ${}^{{\pi }_2}{D}_q\mathsf{Ϝ}$ and ${}_{{\pi }_1}{D}_q\mathsf{Ϝ}$ be two continuous and integrable functions on $\left[{\pi }_1,{\pi }_2\right].$ If $\left|\begin{array}{c}{}^{{\pi }_2}{D}_q\end{array}\mathsf{Ϝ}\left(\tau \right)\right|\le M$ and $\left|\begin{array}{c}{}_{{\pi }_1}{D}_q\end{array}\mathsf{Ϝ}\left(\tau \right)\right|\le M$ for all $\tau \in \left[{\pi }_1,{\pi }_2\right],$ then we have the following quantum Ostrowski-type inequality for all $x\in \left[{\pi }_1,{\pi }_2\right]$:

$$\left|\mathsf{Ϝ}\left(x\right)-\frac{1}{{\pi }_2-{\pi }_1}\left[\underset{{\pi }_1}{\overset{x}{\int }}\mathsf{Ϝ}\left(\tau \right)\begin{array}{c}{}_{{\pi }_1}{d}_q\left(\tau \right)\end{array}+\underset{x}{\overset{{\pi }_2}{\int }}\mathsf{Ϝ}\left(\tau \right)\begin{array}{c}{}^{{\pi }_2}{d}_q\left(\tau \right)\end{array}\right]\right|\le \frac{qM}{\left({\pi }_2-{\pi }_1\right)}\left[\frac{{\left(x-{\pi }_1\right)}^2+{\left({\pi }_2-x\right)}^2}{{\left[2\right]}_q}\right]$$

(10)

where $0<q<1.$

In [38], H. Budak et al. gave the following definitions for the $q_{{\pi }_1{\pi }_3}$, $q_{{\pi }_1}^{{\pi }_4}$, $q_{{\pi }_2}^{{\pi }_3}$, and $q^{{\pi }_2{\pi }_4}$ integrals and the related inequalities of the quantum Hermite–Hadamard type:

Definition 10

([34,38]). Suppose that $\mathsf{Ϝ}:\left[{\pi }_1,{\pi }_2\right]$×$\left[{\pi }_3,{\pi }_4\right]\subset {\mathbb{R}}^2\to \mathbb{R}$ is a continuous function. Then, the following $q_{{\pi }_1{\pi }_3}$, $q_{{\pi }_1}^{{\pi }_4},$$q_{{\pi }_3}^{{\pi }_2}$, and $q^{{\pi }_2{\pi }_4}$ integrals on $\left[{\pi }_1,{\pi }_2\right]\times \left[{\pi }_3,{\pi }_4\right]$ are defined by

$$\begin{array}{ccc}& & \underset{{\pi }_1}{\overset{x}{\int }}\underset{{\pi }_3}{\overset{y}{\int }}\mathsf{Ϝ}\left(\tau ,s\right)\begin{array}{c}{}_{{\pi }_3}{d}_{q_2}\left(s\right)\end{array}\begin{array}{c}{}_{{\pi }_1}{d}_{q_1}\left(\tau \right)\end{array}\hfill \\ & =& \left(1-q_1\right)\left(1-q_2\right)\left(x-{\pi }_1\right)\left(y-{\pi }_3\right)\sum _{n=0}^{\infty }\sum _{m=0}^{\infty }{q}_1^n{q}_2^m\mathsf{Ϝ}\left(q_1^{n}x+\left(1-q_1^n\right){\pi }_1,q_2^{m}y+\left(1-q_2^m\right){\pi }_3\right);\hfill \end{array}$$

(11)

$$\begin{array}{ccc}& & \underset{{\pi }_1}{\overset{x}{\int }}\underset{y}{\overset{{\pi }_4}{\int }}\mathsf{Ϝ}\left(\tau ,s\right)\begin{array}{c}{d}_{q_2}\left(s\right)\end{array}\begin{array}{c}{}_{{\pi }_1}{d}_{q_1}\left(\tau \right)\end{array}\hfill \\ & =& \left(1-q_1\right)\left(1-q_2\right)\left(x-{\pi }_1\right)\left({\pi }_4-y\right)\sum _{n=0}^{\infty }\sum _{m=0}^{\infty }{q}_1^n{q}_2^m\mathsf{Ϝ}\left(q_1^{n}x+\left(1-q_1^n\right){\pi }_1,q_2^{m}y+\left(1-q_2^m\right){\pi }_4\right);\hfill \end{array}$$

(12)

$$\begin{array}{ccc}& & \underset{x}{\overset{{\pi }_2}{\int }}\underset{{\pi }_3}{\overset{y}{\int }}\mathsf{Ϝ}\left(\tau ,s\right)\begin{array}{c}{}_{{\pi }_3}{d}_{q_2}\left(s\right)\end{array}\begin{array}{c}{}^{{\pi }_2}{d}_{q_1}\left(\tau \right)\end{array}\hfill \\ & =& \left(1-q_1\right)\left(1-q_2\right)\left({\pi }_2-x\right)\left(y-{\pi }_3\right)\sum _{n=0}^{\infty }\sum _{m=0}^{\infty }{q}_1^n{q}_2^m\mathsf{Ϝ}\left(q_1^{n}x+\left(1-q_1^n\right){\pi }_2,q_2^{m}y+\left(1-q_2^m\right){\pi }_3\right);\hfill \end{array}$$

(13)

and

$$\begin{array}{ccc}& & \underset{x}{\overset{{\pi }_2}{\int }}\underset{y}{\overset{{\pi }_4}{\int }}\mathsf{Ϝ}\left(\tau ,s\right)\begin{array}{c}{d}_{q_2}\left(s\right)\end{array}\begin{array}{c}{}^{{\pi }_2}{d}_{q_1}\left(\tau \right)\end{array}\hfill \\ & =& \left(1-q_1\right)\left(1-q_2\right)\left({\pi }_2-x\right)\left({\pi }_4-y\right)\sum _{n=0}^{\infty }\sum _{m=0}^{\infty }{q}_1^n{q}_2^m\mathsf{Ϝ}\left(q_1^{n}x+\left(1-q_1^n\right){\pi }_2,q_2^{m}y+\left(1-q_2^m\right){\pi }_4\right).\hfill \end{array}$$

(14)

for $\left(x,y\right)\in \left[{\pi }_1,{\pi }_2\right]\times \left[{\pi }_3,{\pi }_4\right].$

Definition 11

([34]). Let $\mathsf{Ϝ}:\left[{\pi }_1,{\pi }_2\right]\times \left[{\pi }_3,{\pi }_4\right]\subseteq {\mathbb{R}}^2\to \mathbb{R}$ be a continuous function of two variables. The partial $q_1$ derivatives, $q_2$ derivatives, and $q_1{q}_2$ derivatives at $\left(x,y\right)\in \left[{\pi }_1,{\pi }_2\right]\times \left[{\pi }_3,{\pi }_4\right]$ can be defined as

$$\begin{array}{ccc}\hfill \frac{{}_{{\pi }_1}{\partial }_{q_1}\mathsf{Ϝ}\left(x,y\right)}{{}_{{\pi }_1}{\partial }_{q_1}x}& =& \frac{\mathsf{Ϝ}\left(q_{1}x+\left(1-q_1\right){\pi }_1,y\right)-\mathsf{Ϝ}\left(x,y\right)}{\left(1-q_1\right)\left(x-{\pi }_1\right)},x\ne {\pi }_1;\hfill \\ \hfill \frac{{}_{{\pi }_3}{\partial }_{q_2}\mathsf{Ϝ}\left(x,y\right)}{{}_{{\pi }_3}{\partial }_{q_2}y}& =& \frac{\mathsf{Ϝ}\left(x,q_{2}y+\left(1-q_2\right){\pi }_3\right)-\mathsf{Ϝ}\left(x,y\right)}{\left(1-q_2\right)\left(y-{\pi }_3\right)},y\ne {\pi }_3;\hfill \\ \hfill \frac{{}_{{\pi }_1,{\pi }_3}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}\left(x,y\right)}{{}_{{\pi }_1}{\partial }_{q_1}x{}_{{\pi }_3}{\partial }_{q_2}y}& =& \frac{1}{\left(x-{\pi }_1\right)\left(y-{\pi }_3\right)\left(1-q_1\right)\left(1-q_2\right)}\left[\mathsf{Ϝ}\left(q_{1}x+\left(1-q_1\right){\pi }_1,q_{2}y+\left(1-q_2\right){\pi }_3\right)\right.\hfill \\ & & \left.-\mathsf{Ϝ}\left(q_{1}x+\left(1-q_1\right){\pi }_1,y\right)-\mathsf{Ϝ}\left(x,q_{2}y+\left(1-q_2\right){\pi }_3\right)+\mathsf{Ϝ}\left(x,y\right)\right],x\ne {\pi }_1,y\ne {\pi }_3;\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \frac{{}^{{\pi }_2}{\partial }_{q_1}\mathsf{Ϝ}\left(x,y\right)}{{}^{{\pi }_2}{\partial }_{q_1}x}& =& \frac{\mathsf{Ϝ}\left(q_{1}x+\left(1-q_1\right){\pi }_2,y\right)-\mathsf{Ϝ}\left(x,y\right)}{\left(1-q_1\right)\left({\pi }_2-x\right)},x\ne {\pi }_2;\hfill \\ \hfill \frac{{}^{{\pi }_4}{\partial }_{q_2}\mathsf{Ϝ}\left(x,y\right)}{{}^{{\pi }_4}{\partial }_{q_2}y}& =& \frac{\mathsf{Ϝ}\left(x,q_{2}y+\left(1-q_2\right){\pi }_4\right)-\mathsf{Ϝ}\left(x,y\right)}{\left(1-q_2\right)\left({\pi }_4-y\right)},y\ne {\pi }_4;\hfill \\ \hfill \frac{{}_{{\pi }_1}^{{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}\left(x,y\right)}{{}_{{\pi }_1}{\partial }_{q_1}x{}^{{\pi }_4}{\partial }_{q_2}y}& =& \frac{1}{\left(x-{\pi }_1\right)\left({\pi }_4-y\right)\left(1-q_1\right)\left(1-q_2\right)}\left[\mathsf{Ϝ}\left(q_{1}x+\left(1-q_1\right){\pi }_1,q_{2}y+\left(1-q_2\right){\pi }_4\right)\right.\hfill \\ & & \left.-\mathsf{Ϝ}\left(q_{1}x+\left(1-q_1\right){\pi }_1,y\right)-\mathsf{Ϝ}\left(x,q_{2}y+\left(1-q_2\right){\pi }_4\right)+\mathsf{Ϝ}\left(x,y\right)\right],x\ne {\pi }_1,y\ne {\pi }_4;\hfill \\ \hfill \frac{{}_{{\pi }_3}^{{\pi }_2}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}\left(x,y\right)}{{}^{{\pi }_2}{\partial }_{q_1}{x}_{{\pi }_3}{\partial }_{q_2}y}& =& \frac{1}{\left({\pi }_2-x\right)\left(y-{\pi }_3\right)\left(1-q_1\right)\left(1-q_2\right)}\left[\mathsf{Ϝ}\left(q_{1}x+\left(1-q_1\right){\pi }_2,q_{2}y+\left(1-q_2\right){\pi }_3\right)\right.\hfill \\ & & \left.-\mathsf{Ϝ}\left(q_{1}x+\left(1-q_1\right){\pi }_2,y\right)-\mathsf{Ϝ}\left(x,q_{2}y+\left(1-q_2\right){\pi }_3\right)+\mathsf{Ϝ}\left(x,y\right)\right],x\ne {\pi }_2,y\ne {\pi }_3;\hfill \\ \hfill \frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}\left(x,y\right)}{{}^{{\pi }_2}{\partial }_{q_1}x{}_{}^{{\pi }_4}{\partial }_{q_2}y}& =& \frac{1}{\left({\pi }_2-x\right)\left({\pi }_4-y\right)\left(1-q_1\right)\left(1-q_2\right)}\left[\mathsf{Ϝ}\left(q_{1}x+\left(1-q_1\right){\pi }_2,q_{2}y+\left(1-q_2\right){\pi }_4\right)\right.\hfill \\ & & \left.-\mathsf{Ϝ}\left(q_{1}x+\left(1-q_1\right){\pi }_2,y\right)-\mathsf{Ϝ}\left(x,q_{2}y+\left(1-q_2\right){\pi }_4\right)+\mathsf{Ϝ}\left(x,y\right)\right],x\ne {\pi }_2,y\ne {\pi }_4.\hfill \end{array}$$

For clarity, we adopted the following notations to facilitate comprehension of this paper:

$$\begin{array}{ccc}{\check{R}}_{{\mu }_1,{\nu }_1}^{{\sigma }_1}({\pi }_1-{\pi }_2)& =& {\acute{O}}_1\\ {\check{R}}_{{\mu }_2,{\nu }_2}^{{\sigma }_2}({\pi }_3-{\pi }_4)& =& {\acute{O}}_2\\ {\check{R}}_{{\mu }_1,{\nu }_1}^{{\sigma }_1}({\pi }_2-{\pi }_1)& =& {\stackrel{`}{O}}_1\\ {\check{R}}_{{\mu }_2,{\nu }_2}^{{\sigma }_2}({\pi }_4-{\pi }_3)& =& {\stackrel{`}{O}}_2\end{array}$$

(15)

3. Identities

In this section, we prove two identities with the right quantum integral which are useful for the formulation of quantum Simpson- and quantum Newton-type inequalities.

Lemma 2.

Let $\mathsf{Ϝ}:\left[{\pi }_2+{\acute{O}}_1,{\pi }_2\right]\times \left[{\pi }_4+{\acute{O}}_2,{\pi }_4\right]\subseteq \Delta \to \mathbb{R}$ be a twice partially $q_1{q}_2$-differentiable function on $\left({\pi }_2+{\acute{O}}_1,{\pi }_2\right)\times \left({\pi }_4+{\acute{O}}_2,{\pi }_4\right)$ such that ${\displaystyle \frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}\left(t,s\right)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}}$ is continuous and integrable. Then, we have the following identity:

$$\begin{array}{ccc}& & q_1{q}_2\stackrel{`}{O}{\stackrel{`}{O}}_2{\int }_0^1{\int }_0^1{\Lambda }_{q_1}t{\Lambda }_{q_2}s\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2+t{\acute{O}}_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}t{}^{{\pi }_4}{\partial }_{q_2}s}{}^{{\pi }_2}{d}_{q_1}t{}^{{\pi }_4}{d}_{q_2}s\hfill \\ & =& \frac{1}{{\left[6\right]}_{q_1}{\left[6\right]}_{q_2}}\left[q_1{q}_2^2{\left[4\right]}_{q_2}\mathsf{Ϝ}\left({\pi }_2,{\pi }_4+\frac{{\acute{O}}_2}{{\left[2\right]}_{q_2}}\right)+q_1^2{q}_2^2{\left[4\right]}_{q_1}{\left[4\right]}_{q_2}\mathsf{Ϝ}\left({\pi }_2+\frac{{\acute{O}}_1}{{\left[2\right]}_{q_1}},{\pi }_4+\frac{{\acute{O}}_2}{{\left[2\right]}_{q2}}\right)\right.\hfill \\ & & \left.+q_1^2{\left[4\right]}_{q_1}\mathsf{Ϝ}\left({\pi }_2+\frac{{\acute{O}}_1}{{\left[2\right]}_{q_1}},{\pi }_4+{\acute{O}}_2\right)+q_1^2{q}_2{\left[4\right]}_{q_1}\mathsf{Ϝ}\left({\pi }_2+\frac{{\acute{O}}_1}{{\left[2\right]}_{q_1}},{\pi }_4\right)+q_2^2{\left[4\right]}_{q_2}\mathsf{Ϝ}\left({\pi }_2+{\acute{O}}_1,{\pi }_4+\frac{{\acute{O}}_2}{{\left[2\right]}_{q_2}}\right)\right]\hfill \\ & & +\frac{\mathsf{Ϝ}\left({\pi }_2+{\acute{O}}_1,{\pi }_4+{\acute{O}}_2\right)+q_1{q}_2\mathsf{Ϝ}({\pi }_2,{\pi }_4)+q_1\mathsf{Ϝ}\left({\pi }_2,{\pi }_4+{\acute{O}}_2\right)+q_2\mathsf{Ϝ}\left({\pi }_2+{\acute{O}}_1,{\pi }_4\right)}{{\left[6\right]}_{q_1}{\left[6\right]}_{q_2}}\hfill \\ & & -\frac{1}{{\stackrel{`}{O}}_1{\left[6\right]}_{q_2}}{\int }_{{\pi }_2+{\acute{O}}_1}^{{\pi }_2}\left[q_2\mathsf{Ϝ}\left(x,{\pi }_4\right)+q_2^2{\left[4\right]}_{q_2}\mathsf{Ϝ}\left(x,{\pi }_4+\frac{{\acute{O}}_2}{{\left[2\right]}_{q_2}}\right)+\mathsf{Ϝ}\left(x,{\pi }_4+{\acute{O}}_2\right)\right]{}^{{\pi }_2}{d}_{q_1}\left(x\right)\hfill \\ & & -\frac{1}{{\stackrel{`}{O}}_2{\left[6\right]}_{q_1}}{\int }_{{\pi }_4+{\acute{O}}_2}^{{\pi }_4}\left[q_1\mathsf{Ϝ}\left({\pi }_2,y\right)+q_1^2{\left[4\right]}_{q_1}\mathsf{Ϝ}\left({\pi }_2+\frac{{\acute{O}}_1}{{\left[2\right]}_{q_1}},y\right)+\mathsf{Ϝ}({\pi }_2+{\acute{O}}_1,y)\right]{}^{{\pi }_4}{d}_{q_2}\left(y\right)\hfill \\ & & +\frac{1}{{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2}{\int }_{{\pi }_2+{\acute{O}}_1}^{{\pi }_2}{\int }_{{\pi }_4+{\acute{O}}_2}^{{\pi }_4}\mathsf{Ϝ}(x,y){}^{{\pi }_2}{d}_{q_1}x{}^{{\pi }_4}{d}_{q_2}\left(y\right),\hfill \end{array}$$

(16)

where

$${\Lambda }_{q_1}\left(t\right)=\left\{\begin{array}{cc}t-\frac{1}{{\left[6\right]}_{q_1}},& t\in \left[0,\frac{1}{{\left[2\right]}_{q_1}}\right),\\ t-\frac{{\left[5\right]}_{q_{{}_1}}}{{\left[6\right]}_{q_1}},& t\in \left[\frac{1}{{\left[2\right]}_{q_1}},1\right];\end{array}\right.and,{\Lambda }_{q_2}\left(s\right)=\left\{\begin{array}{cc}s-\frac{1}{{\left[6\right]}_{q_2}},& s\in \left[0,\frac{1}{{\left[2\right]}_{q_2}}\right),\\ s-\frac{{\left[5\right]}_{q_{{}_2}}}{{\left[6\right]}_{q_2}},& s\in \left[\frac{1}{{\left[2\right]}_{q_2}},1\right].\end{array}\right.$$

Proof. 

By using the basic properties of the q integral and definitions ${\Lambda }_{q_1}\left(t\right)$ and ${\Lambda }_{q_2}\left(s\right)$, we have

$$\begin{array}{cc}\hfill & {\int }_0^1{\int }_0^1{\Lambda }_{q_1}\left(t\right){\Lambda }_{q_2}\left(s\right)\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1{q}_2}^2\mathsf{Ϝ}({\pi }_2+t{\acute{O}}_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}{d}_{q_1}\left(t\right)d_{q_2}\left(s\right)\hfill \\ \hfill & =\left(\frac{{\left[5\right]}_{q_2}-1}{{\left[6\right]}_{q_2}}\right)\left(\frac{{\left[5\right]}_{q_1}-1}{{\left[6\right]}_{q_1}}\right){\int }_0^{\frac{1}{{\left[2\right]}_{q_1}}}{\int }_0^{\frac{1}{{\left[2\right]}_{q_2}}}\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2+t{\acute{O}}_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}{d}_{q_2}\left(s\right)d_{q_1}\left(t\right)\hfill \\ \hfill & +\left(\frac{{\left[5\right]}_{q_2}-1}{{\left[6\right]}_{q_2}}\right){\int }_0^{\frac{1}{{\left[2\right]}_{q_2}}}{\int }_0^1\left(t-\frac{{\left[5\right]}_{q_1}}{{\left[6\right]}_{q_1}}\right)\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2+t{\acute{O}}_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}{d}_{q_2}\left(s\right)d_{q_1}\left(t\right)\hfill \\ \hfill & +\left(\frac{{\left[5\right]}_{q_1}-1}{{\left[6\right]}_{q_1}}\right){\int }_0^{\frac{1}{{\left[2\right]}_{q_1}}}{\int }_0^1\left(s-\frac{{\left[5\right]}_{q_2}}{{\left[6\right]}_{q_2}}\right)\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2+t{\acute{O}}_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}{d}_{q_2}\left(s\right)d_{q_1}\left(t\right)\hfill \\ \hfill & +{\int }_0^1{\int }_0^1\left(s-\frac{{\left[5\right]}_{q_2}}{{\left[6\right]}_{q_2}}\right)\left(t-\frac{{\left[5\right]}_{q_1}}{{\left[6\right]}_{q_1}}\right)\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2+t{\acute{O}}_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}{d}_{q_2}\left(s\right)d_{q_1}\left(t\right)\hfill \end{array}$$

$${\int }_0^1{\int }_0^1{\Lambda }_{q_1}\left(t\right){\Lambda }_{q_2}\left(s\right)\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1{q}_2}^2\mathsf{Ϝ}({\pi }_2+t{\acute{O}}_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}{d}_{q_1}\left(t\right)d_{q_2}\left(s\right)=I_1+I_2+I_3+I_4$$

(17)

By using Definition 9 for the right quantum integrals, the quantum integral $I_1$ becomes

$$\begin{array}{ccc}\hfill I_1& =& \frac{\left({\left[5\right]}_{q_1}-1\right)\left({\left[5\right]}_{q_2}-1\right)}{{\left[6\right]}_{q_1}{\left[6\right]}_{q_2}{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2}\left[\mathsf{Ϝ}\left({\pi }_2+\frac{{\acute{O}}_1}{{\left[2\right]}_{q_1}},{\pi }_4+\frac{{\acute{O}}_2}{{\left[2\right]}_{q_2}}\right)+\mathsf{Ϝ}({\pi }_2,{\pi }_4)\right.\hfill \\ & & \left.-\mathsf{Ϝ}\left({\pi }_2,{\pi }_4+\frac{{\acute{O}}_2}{{\left[2\right]}_{q_2}}\right)-\mathsf{Ϝ}\left({\pi }_2+\frac{{\acute{O}}_1}{{\left[2\right]}_{q_1}},{\pi }_4\right)\right].\hfill \end{array}$$

Now, solving the quantum integrals involved in $I_2$ by using Definition 7 yields

$$\begin{array}{ccc}& & {\int }_0^{\frac{1}{{\left[2\right]}_{q_2}}}{\int }_0^{1}t\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^{2}F({\pi }_2+t{\acute{O}}_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q1}\left(t\right){}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}{d}_{q_1}s{d}_{q_2}t\hfill \\ & =& \frac{1}{{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2}\left[\frac{1}{{\stackrel{`}{O}}_1{q}_1}{\int }_{{\pi }_2+{\acute{O}}_1}^{{\pi }_2}\mathsf{Ϝ}\left(x,{\pi }_4\right){}^{{\pi }_2}{d}_{q_1}\left(x\right)+\frac{1}{q_{{}_1}}F\left({\pi }_2+{\acute{O}}_1,{\pi }_4+\frac{{\acute{O}}_2}{{\left[2\right]}_{q_{{}_2}}}\right)\right.\hfill \\ & & \left.-\frac{1}{{\stackrel{`}{O}}_1{q}_1}{\int }_{{\pi }_2+{\acute{O}}_1}^{{\pi }_2}F\left(x,{\pi }_4+\frac{{\acute{O}}_2}{{\left[2\right]}_{q_2}}\right){}^{{\pi }_2}{d}_{q_1}\left(x\right)-\frac{1}{q_{{}_1}}F\left({\pi }_2+{\acute{O}}_1,{\pi }_4\right)\right].\hfill \end{array}$$

(18)

By using similar arguments, we have

$$\begin{array}{ccc}& & {\int }_0^{\frac{1}{{\left[2\right]}_{q_2}}}{\int }_0^1\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2+t{\acute{O}}_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q1}\left(t\right){}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}{d}_{q_2}\left(s\right)d_{q_1}\left(t\right)=\frac{1}{{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2}\left[\mathsf{Ϝ}({\pi }_2,{\pi }_4)-\mathsf{Ϝ}\left({\pi }_2,{\pi }_4+\frac{{\acute{O}}_2}{{\left[2\right]}_{q_2}}\right)\right.\hfill \\ & & \left.-\mathsf{Ϝ}({\pi }_2+{\acute{O}}_1,{\pi }_4)+\mathsf{Ϝ}\left({\pi }_2+{\acute{O}}_1,{\pi }_4+\frac{{\acute{O}}_2}{{\left[2\right]}_{q_2}}\right)\right].\hfill \end{array}$$

(19)

By putting the values from Equations (18) and (19) into $I_2$, we have

$$\begin{array}{cc}\hfill I_2& =\frac{{\left[5\right]}_{q_2}-1}{{\stackrel{`}{O}}_{12}{R}_{u_2,v_2}^{{\sigma }_2}({\pi }_4,{\pi }_2){\left[6\right]}_{q_2}}\left[\frac{1}{{\stackrel{`}{O}}_1{q}_1}{\int }_{{\pi }_2+{\acute{O}}_1}^{{\pi }_2}\mathsf{Ϝ}\left(x,{\pi }_4\right){}^{{\pi }_2}{d}_{q_1}\left(x\right)-\frac{1}{q_{{}_1}}\mathsf{Ϝ}\left({\pi }_2+{\acute{O}}_1,{\pi }_4\right)\right.\hfill \\ \hfill & \left.-\frac{1}{{\stackrel{`}{O}}_1{q}_1}{\int }_{{\pi }_2+{\acute{O}}_1}^{{\pi }_2}\mathsf{Ϝ}{\left(x,{\pi }_4+\frac{{\acute{O}}_2}{{\left[2\right]}_{q_2}}\right)}^{{\pi }_2}{d}_{q_1}x+\frac{1}{q_{{}_1}}\mathsf{Ϝ}\left({\pi }_2+{\acute{O}}_1,{\pi }_4+\frac{{\acute{O}}_2}{{\left[2\right]}_{q_{{}_2}}}\right)\right]-\frac{{\left[5\right]}_{q_1}\left({\left[5\right]}_{q_2}-1\right)}{{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2{\left[6\right]}_{q_1}{\left[6\right]}_{q_2}}\hfill \\ \hfill & \times \left[\mathsf{Ϝ}({\pi }_2,{\pi }_4)-\mathsf{Ϝ}\left({\pi }_2,{\pi }_4+\frac{{\acute{O}}_2}{{\left[2\right]}_{q_2}}\right)-\mathsf{Ϝ}({\pi }_2+{\acute{O}}_1,{\pi }_4)+\mathsf{Ϝ}\left({\pi }_2+{\acute{O}}_1,{\pi }_4+\frac{{\acute{O}}_2}{{\left[2\right]}_{q_2}}\right)\right].\hfill \end{array}$$

Similarly, $I_3$ is evaluated as

$$\begin{array}{cc}\hfill I_3& =\frac{{\left[5\right]}_{q_1}-1}{{\stackrel{`}{O}}_{12}{R}_{u_2,v_2}^{{\sigma }_2}({\pi }_4,{\pi }_2){\left[6\right]}_{q_1}}\left[\frac{1}{{\stackrel{`}{O}}_2{q}_2}{\int }_{{\pi }_4+{\acute{O}}_2}^{{\pi }_4}\mathsf{Ϝ}\left({\pi }_2,y\right){}^{{\pi }_4}{d}_{q_2}\left(y\right)\right.\hfill \\ \hfill & -\frac{\mathsf{Ϝ}\left({\pi }_2,{\pi }_4+{\acute{O}}_2\right)}{q_{{}_2}}-\frac{1}{{\stackrel{`}{O}}_2{q}_2}{\int }_{{\pi }_4+{\acute{O}}_2}^{{\pi }_4}\mathsf{Ϝ}\left({\pi }_2+\frac{{\acute{O}}_1}{{\left[2\right]}_{q_1}},y\right){}^{{\pi }_4}{d}_{q_2}\left(y\right)\hfill \\ \hfill & \left.+\frac{1}{q_{{}_2}}\mathsf{Ϝ}\left({\pi }_2+\frac{{\acute{O}}_1}{{\left[2\right]}_{q_1}},{\pi }_4+{\acute{O}}_2\right)\right]-\frac{{\left[5\right]}_{q_2}\left({\left[5\right]}_{q_1}-1\right)}{{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2{\left[6\right]}_{q_1}{\left[6\right]}_{q_2}}\left[\mathsf{Ϝ}({\pi }_2,{\pi }_4)\right.\hfill \\ \hfill & \left.-\mathsf{Ϝ}\left({\pi }_2+\frac{{\acute{O}}_1}{{\left[2\right]}_{q_1}},{\pi }_4\right)-\mathsf{Ϝ}({\pi }_2,{\pi }_4+{\acute{O}}_2)+\mathsf{Ϝ}\left({\pi }_2+\frac{{\acute{O}}_1}{{\left[2\right]}_{q_1}},{\pi }_4+{\acute{O}}_2\right)\right].\hfill \end{array}$$

Solving the quantum integrals in $I_4$ yields

$${\int }_0^1{\int }_0^1\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2+t{\acute{O}}_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q1}\left(t\right){}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}{d}_{q_2}\left(s\right)d_{q_1}\left(t\right)=\frac{1}{{\stackrel{`}{O}}_1{\stackrel{`}{O}}_{2}ts}$$

(20)

$$\begin{array}{ccc}& & \left[\mathsf{Ϝ}({\pi }_2+{\acute{O}}_1,{\pi }_4+{\acute{O}}_2)+\mathsf{Ϝ}({\pi }_2,{\pi }_4)-\mathsf{Ϝ}({\pi }_2+{\acute{O}}_1,{\pi }_4)\right.\hfill \\ & & \left.-\mathsf{Ϝ}({\pi }_2,{\pi }_4+{\acute{O}}_2)\right]\hfill \end{array}$$

$${\int }_0^1{\int }_0^{1}t\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2+t{\acute{O}}_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q1}\left(t\right){}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}{d}_{q_2}\left(s\right)d_{q_1}\left(t\right)=\frac{1}{{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2}$$

$$\begin{array}{cc}\hfill & \times \left\{\frac{1}{{\stackrel{`}{O}}_1{q}_1}{\int }_{{\pi }_2+{\acute{O}}_1}^{{\pi }_2}\mathsf{Ϝ}{\left(x,{\pi }_4\right)}^{{\pi }_2}{d}_{q_1}\left(x\right)-\frac{\mathsf{Ϝ}\left({\pi }_2+{\acute{O}}_1,{\pi }_4\right)}{q_{{}_1}}-\frac{1}{{\stackrel{`}{O}}_1{q}_1}\right.\hfill \\ \hfill & \left.\times {\int }_{{\pi }_2+{\acute{O}}_1}^{{\pi }_2}\mathsf{Ϝ}{\left(x,{\pi }_4+{\acute{O}}_2\right)}^{{\pi }_2}{d}_{q_1}\left(x\right)+\frac{\mathsf{Ϝ}\left({\pi }_2+{\acute{O}}_1,{\pi }_4+{\acute{O}}_2\right)}{q_{{}_1}}\right\}\hfill \end{array}$$

$${\int }_0^1{\int }_0^{1}s\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2+t{\acute{O}}_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q1}\left(t\right){}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}{d}_{q_2}\left(s\right)d_{q_1}\left(t\right)=\frac{1}{{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2}$$

(21)

$$\begin{array}{cc}\hfill & \times \left\{\frac{1}{{\stackrel{`}{O}}_2{q}_2}{\int }_{{\pi }_4+{\acute{O}}_2}^{{\pi }_4}\mathsf{Ϝ}{\left({\pi }_2,y\right)}^{{\pi }_2}{d}_{q_2}\left(y\right)-\frac{1}{q_{{}_2}}\mathsf{Ϝ}\left({\pi }_2,{\pi }_4+{\acute{O}}_2\right)-\frac{1}{{\stackrel{`}{O}}_2{q}_2}\right.\hfill \\ \hfill & \left.\times {\int }_{{\pi }_4+{\acute{O}}_2}^{{\pi }_4}\mathsf{Ϝ}{\left({\pi }_2+{\acute{O}}_1,y\right)}^{{\pi }_2}{d}_{q_2}\left(y\right)+\frac{1}{q_2}\mathsf{Ϝ}\left({\pi }_2+{\acute{O}}_1,{\pi }_4+{\acute{O}}_2\right)\right\}.\hfill \end{array}$$

In addition, we have

$${\int }_0^1{\int }_0^{1}ts\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2+t{\acute{O}}_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q1}\left(t\right){}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}{d}_{q_2}\left(s\right)d_{q_1}\left(t\right)=\frac{1}{{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2}$$

(22)

$$\begin{array}{cc}\hfill & \times \left[\frac{1}{{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2{q}_1{q}_2}{\int }_{{\pi }_2+{\acute{O}}_1}^{{\pi }_2}{\int }_{{\pi }_4+{\acute{O}}_2}^{{\pi }_4}\mathsf{Ϝ}(x,y){}^{{\pi }_2}{d}_{q_1}\left(x\right){}^{{\pi }_4}{d}_{q_2}\left(y\right)-\frac{1}{q_1{q}_2{\stackrel{`}{O}}_1}{\int }_{{\pi }_2+{\acute{O}}_1}^{{\pi }_2}\mathsf{Ϝ}{(x,{\pi }_4+{\acute{O}}_2)}^{{\pi }_2}{d}_{q_1}\left(x\right)\right.\hfill \\ \hfill & \left.+\frac{1}{q_1{q}_2}\mathsf{Ϝ}({\pi }_2+{\acute{O}}_1,{\pi }_4+{\acute{O}}_2)-\frac{1}{q_1{q}_2{\stackrel{`}{O}}_2}{\int }_{{\pi }_4+{\acute{O}}_2}^{{\pi }_4}\mathsf{Ϝ}({\pi }_2+{\acute{O}}_1,x){}^{{\pi }_4}{d}_{q_2}\left(y\right)\right].\hfill \end{array}$$

Using (20)–(22) we obtain

$$\begin{array}{cc}\hfill I_4& =\frac{1}{{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2}\left[\left\{\frac{1}{q_1{q}_2{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2}{\int }_{{\pi }_2+{\acute{O}}_1}^{{\pi }_2}{\int }_{{\pi }_4+{\acute{O}}_2}^{{\pi }_4}\mathsf{Ϝ}{(x,y)}^{{\pi }_2}{d}_{q_1}\left(x\right){}^{{\pi }_4}{d}_{q_2}\left(y\right)-\frac{1}{q_1{q}_2{\stackrel{`}{O}}_1}{\int }_{{\pi }_2+{\acute{O}}_1}^{{\pi }_2}\right.\right.\hfill \\ \hfill & \left.\times \mathsf{Ϝ}{(x,{\pi }_4+{\acute{O}}_2)}^{{\pi }_2}{d}_{q_1}\left(x\right)+\frac{\mathsf{Ϝ}({\pi }_2+{\acute{O}}_1,{\pi }_4+{\acute{O}}_2)}{q_1{q}_2}-\frac{1}{{\stackrel{`}{O}}_2}{\int }_{{\pi }_4+{\acute{O}}_2}^{{\pi }_4}\mathsf{Ϝ}{({\pi }_2+{\acute{O}}_1,y)}^{{\pi }_2}{d}_{q_2}\left(y\right)\right\}\hfill \\ \hfill & -\frac{{\left[5\right]}_{q_2}}{{\left[6\right]}_{q_2}}\left\{\frac{1}{{\stackrel{`}{O}}_1{q}_1}{\int }_{{\pi }_4+{\acute{O}}_2}^{{\pi }_4}\mathsf{Ϝ}\left({\pi }_2,y\right){}^{{\pi }_4}{d}_{q_2}\left(y\right)-\frac{1}{{\stackrel{`}{O}}_2{q}_2}{\int }_{{\pi }_4+{\acute{O}}_2}^{{\pi }_4}\mathsf{Ϝ}{\left({\pi }_2+{\acute{O}}_1,y\right)}^{{\pi }_4}{d}_{q_2}\left(y\right)\right.\hfill \\ \hfill & \left.+\frac{1}{q_2}\mathsf{Ϝ}\left({\pi }_2+{\acute{O}}_1,{\pi }_4+{\acute{O}}_2\right)-\frac{1}{q_{{}_2}}\mathsf{Ϝ}\left({\pi }_2,{\pi }_4+{\acute{O}}_2\right)\right\}-\frac{{\left[5\right]}_{q_2}}{{\left[6\right]}_{q_2}}\left\{\frac{1}{{\acute{O}}_1{q}_1}{\int }_{{\pi }_2+{\acute{O}}_1}^{{\pi }_2}\mathsf{Ϝ}{\left(x,{\pi }_4\right)}^{{\pi }_2}{d}_{q_1}\left(x\right)\right.\hfill \\ \hfill & \left.-\frac{1}{q_{{}_1}}\mathsf{Ϝ}\left({\pi }_2+{\acute{O}}_1,{\pi }_4\right)+\frac{1}{q_{{}_1}}\mathsf{Ϝ}\left({\pi }_2+{\acute{O}}_1,{\pi }_4+{\acute{O}}_2\right)-\frac{1}{{\acute{O}}_1{q}_1}{\int }_{{\pi }_2+{\acute{O}}_1}^{{\pi }_2}\mathsf{Ϝ}{\left(x,{\pi }_4+{\acute{O}}_2\right)}^{{\pi }_2}{d}_{q_1}\left(x\right)\right\}\hfill \\ \hfill & \left.+\frac{{\left[5\right]}_{q_1}{\left[5\right]}_{q_2}}{{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2{\left[6\right]}_{q_1}{\left[6\right]}_{q_2}}\mathsf{Ϝ}({\pi }_2,{\pi }_4)-\mathsf{Ϝ}({\pi }_2+{\acute{O}}_1,{\pi }_4)-\mathsf{Ϝ}({\pi }_2,{\pi }_4+{\acute{O}}_2)+\mathsf{Ϝ}({\pi }_2+{\acute{O}}_1,{\pi }_4+{\acute{O}}_2)\right].\hfill \end{array}$$

By using integrals $(I_1-I_4)$ in Equation (17) and multiplying ${\stackrel{`}{O}}_1{\stackrel{`}{O}}_2$ on both sides, and after some simplification, we obtain the required result. □

Remark 1.

If we set ${\stackrel{`}{O}}_1={\pi }_2-{\pi }_1,{\acute{O}}_1={\pi }_1-{\pi }_2$, ${\acute{O}}_2=$(${\pi }_3-{\pi }_4),$ and ${\stackrel{`}{O}}_2={\pi }_4-{\pi }_3$ in Lemma 2, then we have an identity:

$$\begin{array}{cc}\hfill & q_1{q}_2\left({\pi }_2-{\pi }_1\right)({\pi }_4-{\pi }_3){\int }_0^1{\int }_0^1{\Lambda }_{q_1}\left(t\right){\Lambda }_{q_2}\left(s\right)\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2+t\left({\pi }_1-{\pi }_2\right),{\pi }_4+s\left({\pi }_3-{\pi }_4\right))}{{}^{{\pi }_2}{\partial }_{q1}\left(t\right){}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}{d}_{q_1}\left(t\right)d_{q_2}\left(s\right)=\frac{1}{{\left[6\right]}_{q_1}{\left[6\right]}_{q_2}}\hfill \\ \hfill & \left[q_1^2{q}_2{\left[4\right]}_{q_1}\mathsf{Ϝ}\left({\pi }_2+\frac{{\pi }_1-{\pi }_2}{{\left[2\right]}_{q_1}},{\pi }_4\right)+q_1{q}_2^2{\left[4\right]}_{q_2}\mathsf{Ϝ}\left({\pi }_2,{\pi }_4+\frac{{\pi }_3-{\pi }_4}{{\left[2\right]}_{q_2}}\right)+q_2^2{\left[4\right]}_{q_2}\mathsf{Ϝ}\left({\pi }_1,{\pi }_4+\frac{{\pi }_3-{\pi }_4}{{\left[2\right]}_{q_2}}\right)\right.\hfill \\ \hfill & \left.+q_1^2{q}_2^2{\left[4\right]}_{q_1}{\left[4\right]}_{q_2}\mathsf{Ϝ}\left({\pi }_2+\frac{{\pi }_1-{\pi }_2}{{\left[2\right]}_{q_1}},{\pi }_4+\frac{{\pi }_3-{\pi }_4}{{\left[2\right]}_{q2}}\right)+q_1^2{\left[4\right]}_{q_1}\mathsf{Ϝ}\left({\pi }_2+\frac{{\pi }_1-{\pi }_2}{{\left[2\right]}_{q_1}},{\pi }_3\right)\right]\hfill \\ \hfill & -\frac{1}{{\left[6\right]}_{q_2}({\pi }_2-{\pi }_1)}{\int }_{{\pi }_1}^{{\pi }_2}{\left[q_2\mathsf{Ϝ}\left(x,{\pi }_4\right)+q_2^2{\left[4\right]}_{q_2}\mathsf{Ϝ}\left(x,{\pi }_4+\frac{{\pi }_3-{\pi }_4}{{\left[2\right]}_{q_2}}\right)+\mathsf{Ϝ}\left(x,{\pi }_3\right)\right]}^{{\pi }_2}{d}_{q_1}\left(x\right)\hfill \\ \hfill & -\frac{1}{{\left[6\right]}_{q_1}({\pi }_2-{\pi }_1)}{\int }_{{\pi }_3}^{{\pi }_4}{\left[q_1\mathsf{Ϝ}\left({\pi }_2,y\right)+q_1^2{\left[4\right]}_{q_1}\mathsf{Ϝ}\left({\pi }_2+\frac{{\pi }_1-{\pi }_2}{{\left[2\right]}_{q1}},y\right)+\mathsf{Ϝ}({\pi }_1,y)\right]}^{{\pi }_2}{d}_{q_2}\left(y\right)\hfill \\ \hfill & +\frac{q_1\mathsf{Ϝ}\left({\pi }_2,{\pi }_3\right)+\mathsf{Ϝ}\left({\pi }_1,{\pi }_3)\right)+q_2\mathsf{Ϝ}\left({\pi }_1,{\pi }_4\right)+q_1{q}_2\mathsf{Ϝ}\left({\pi }_2,{\pi }_4\right)}{{\left[6\right]}_{q_1}{\left[6\right]}_{q_2}}+\frac{{\int }_{{\pi }_1}^{{\pi }_2}{\int }_{{\pi }_3}^{{\pi }_4}\mathsf{Ϝ}{(x,y)}^{{\pi }_2}{d}_{q_1}{\left(x\right)}^{{\pi }_4}{d}_{q_2}\left(y\right)}{\left({\pi }_2-{\pi }_1\right)({\pi }_4-{\pi }_3)},\hfill \end{array}$$

where

$${\Lambda }_{q_1}\left(t\right)=\left\{\begin{array}{cc}t-\frac{1}{{\left[6\right]}_{q_1}},& t\in \left[0,\frac{1}{{\left[2\right]}_{q_1}}\right),\\ t-\frac{{\left[5\right]}_{q_{{}_1}}}{{\left[6\right]}_{q_1}},& t\in \left[\frac{1}{{\left[2\right]}_{q_1}},1\right];\end{array}\right.\mathit{and},{\Lambda }_{q_2}\left(s\right)=\left\{\begin{array}{cc}s-\frac{1}{{\left[6\right]}_{q_2}},& s\in \left[0,\frac{1}{{\left[2\right]}_{q_2}}\right),\\ s-\frac{{\left[5\right]}_{q_{{}_2}}}{{\left[6\right]}_{q_2}},& s\in \left[\frac{1}{{\left[2\right]}_{q_2}},1\right].\end{array}\right.$$

Lemma 3.

Let $\mathsf{Ϝ}:\left[{\pi }_2+{\acute{O}}_1,{\pi }_2\right]\times \left[{\pi }_4+{\acute{O}}_2,{\pi }_4\right]\subseteq \Delta \to \mathbb{R}$ be a twice partially $q_1{q}_2$-differentiable function on $\left({\pi }_2+{\acute{O}}_1,{\pi }_2\right)$×$\left({\pi }_4+{\acute{O}}_2,{\pi }_4\right)$ such that the partial $q_1{q}_2$ derivative

${\displaystyle \frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}\left(t,s\right)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}}$ is continuous and integrable on $\left[{\pi }_2+{\acute{O}}_1,{\pi }_2\right]$×$\left[{\pi }_4+{\acute{O}}_2,{\pi }_4\right],$ where $0\le q_1<1$ and $0\le q_1<1.$ Then, the following identity holds:

$$\begin{array}{cc}\hfill & q_1{q}_2{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2{\int }_0^1{\int }_0^1{\Lambda }_{q_1}\left(t\right){\Lambda }_{q_2}\left(s\right)\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1{q}_2}^2\mathsf{Ϝ}({\pi }_2+t{\acute{O}}_1,{\pi }_4+s{\acute{O}}_2)}{{\partial }_{q_1}\left(t\right){\partial }_{q_2}\left(s\right)}{}^{{\pi }_1}{d}_{q_1}\left(t\right){}^{{\pi }_2}{d}_{q_2}\left(s\right)\hfill \\ \hfill & =\frac{1}{{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2}{\int }_{{\pi }_2+{\eta }_1}^{{\pi }_2}{\int }_{{\pi }_4+{\eta }_2}^{{\pi }_4}\mathsf{Ϝ}\left(x,y\right){}^{{\pi }_2}{d}_{q_1}\left(x\right){}^{{\pi }_4}{d}_{q_2}\left(y\right)-\frac{1}{{\stackrel{`}{O}}_1{\left[8\right]}_{q_2}}{\int }_{{\pi }_2+{\eta }_1}^{{\pi }_2}\left[\mathsf{Ϝ}\left(x,{\pi }_4+{\acute{O}}_2\right)\right.\hfill \\ \hfill & \left.+\frac{q_2^2{\left[6\right]}_{q_2}}{{\left[2\right]}_{q_2}}\mathsf{Ϝ}\left(x,{\pi }_4+\frac{{\left[2\right]}_{q_2}{\acute{O}}_2}{{\left[3\right]}_{q_2}}\right)+q_{{}_2}\mathsf{Ϝ}\left(x,{\pi }_4\right)+\frac{q_2^3{\left[6\right]}_{q_2}}{{\left[2\right]}_{q_2}}\mathsf{Ϝ}\left(x,{\pi }_4+\frac{{\acute{O}}_2}{{\left[3\right]}_{q_2}}\right)\right]{}^{{\pi }_2}{d}_{q_1}\left(x\right)\hfill \\ \hfill & -\frac{1}{{\stackrel{`}{O}}_2{\left[8\right]}_{q_1}}{\int }_{{\pi }_4+{\eta }_2}^{{\pi }_4}\left[\frac{q_1^3{\left[6\right]}_{q_1}}{{\left[2\right]}_{q_1}}\mathsf{Ϝ}\left({\pi }_2+\frac{{\acute{O}}_1}{{\left[3\right]}_{q_1}},y\right)+\mathsf{Ϝ}\left({\pi }_2+{\acute{O}}_2,y\right)+q_1\mathsf{Ϝ}\left({\pi }_2,y\right)+\frac{q_{{}_1}^2{\left[6\right]}_{q_1}}{{\left[2\right]}_{q_1}}\right.\hfill \\ \hfill & \left.\times \mathsf{Ϝ}\left({\pi }_2+\frac{{\left[2\right]}_{q_1}{\acute{O}}_1}{{\left[3\right]}_{q_1}},y\right)d_{q_2}\left(y\right)\right]{}^{{\pi }_4}{d}_{q_2}\left(y\right)+\frac{1}{{\left[8\right]}_{q_2}{\left[8\right]}_{q_1}}\left[\frac{q_{{}_2}^3{\left[6\right]}_{q_2}}{{\left[2\right]}_{q_2}}\mathsf{Ϝ}\left({\pi }_2,{\pi }_4+\frac{{\acute{O}}_2}{{\left[3\right]}_{q_2}}\right)+q_2\mathsf{Ϝ}\left({\pi }_2,{\pi }_4\right)\right.\hfill \\ \hfill & \left.+\mathsf{Ϝ}\left({\pi }_2,{\pi }_4+{\acute{O}}_2\right)+\frac{q_2^2{\left[6\right]}_{q_2}}{{\left[2\right]}_{q_2}}\mathsf{Ϝ}\left({\pi }_2,{\pi }_4+\frac{{\left[2\right]}_{q_2}{\acute{O}}_2}{{\left[3\right]}_{q_2}}\right)\right]+\frac{1}{{\left[8\right]}_{q_1}{\left[8\right]}_{q_2}}\left[\mathsf{Ϝ}\left({\pi }_2+{\acute{O}}_1,{\pi }_4\right)\right.\hfill \\ \hfill & \left.+\frac{q_{{}_1}^3{\left[6\right]}_{q_1}}{{\left[2\right]}_{q_1}}\mathsf{Ϝ}\left({\pi }_2+\frac{{\acute{O}}_1}{{\left[3\right]}_{q_1}},{\pi }_4\right)+\frac{q_1^2{\left[6\right]}_{q1}}{{\left[2\right]}_{q_1}}\mathsf{Ϝ}\left({\pi }_2+\frac{{\left[2\right]}_{q_1}{\acute{O}}_1}{{\left[3\right]}_{q_1}},{\pi }_4\right)+\mathsf{Ϝ}\left({\pi }_2+{\acute{O}}_1,{\pi }_4+{\acute{O}}_2\right)\right],\hfill \end{array}$$

(23)

where

$${\Lambda }_{q_1}\left(t\right)=\left\{\begin{array}{ccc}t-\frac{1}{{\left[8\right]}_{q_1}},& if& t\in \left[0,\frac{1}{{\left[3\right]}_{q_1}}\right),\\ t-\frac{1}{{\left[2\right]}_{q_1}},& if& t\in \left[\frac{1}{{\left[3\right]}_{q_1}},\frac{{\left[2\right]}_{q_1}}{{\left[3\right]}_{q_1}}\right),\\ t-\frac{{\left[7\right]}_{q_1}}{[8_{q_1}},& if& t\in \left[\frac{{\left[2\right]}_{q_1}}{{\left[3\right]}_{q_1}},1\right];\end{array}\right.and,{\Lambda }_{q_2}\left(s\right)=\left\{\begin{array}{ccc}t-\frac{1}{{\left[8\right]}_{q_2}},& if& s\in \left[0,\frac{1}{{\left[3\right]}_{q_2}}\right),\\ t-\frac{1}{{\left[2\right]}_{q_2}},& if& s\in \left[\frac{1}{{\left[3\right]}_{q_2}},\frac{{\left[2\right]}_{q_2}}{{\left[3\right]}_{q_2}}\right),\\ t-\frac{{\left[7\right]}_{q_2}}{[8_{q_2}},& if& s\in \left[\frac{{\left[2\right]}_{q_2}}{{\left[3\right]}_{q_2}},1\right].\end{array}\right.$$

Proof. 

Through the fundamental properties of q integrals and ${\Lambda }_{q_2}\left(s\right),{\Lambda }_{q_1}\left(t\right)$, we obtain that

$$\begin{array}{cc}\hfill & {\int }_0^1{\int }_0^1{\Lambda }_{q_1}\left(t\right){\Lambda }_{q_2}\left(s\right)\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1{q}_2}^2\mathsf{Ϝ}({\pi }_2+t{\acute{O}}_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}{d}_{q_1}\left(t\right)d_{q_2}\left(s\right)\hfill \\ \hfill & =\frac{{\left[8\right]}_{q_2}-{\left[2\right]}_{q_2}}{{\left[8\right]}_{q_2}{\left[2\right]}_{q_2}}{\int }_0^{\frac{1}{{\left[3\right]}_{q_2}}}\left[\frac{{\left[8\right]}_{q_1}-{\left[2\right]}_{q_1}}{{\left[8\right]}_{q_1}{\left[2\right]}_{q_1}}{\int }_0^{\frac{1}{{\left[3\right]}_{q_1}}}\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2+t{\acute{O}}_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}{d}_{q_1}\left(t\right)\right.\hfill \\ \hfill & +\frac{{\left[7\right]}_{q_1}{\left[2\right]}_{q_1}-{\left[8\right]}_{q_1}}{{\left[8\right]}_{q_1}{\left[2\right]}_{q_1}}{\int }_0^{\frac{{\left[2\right]}_{q_1}}{{\left[3\right]}_{q_1}}}\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2+t{\acute{O}}_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}{d}_{q_1}\left(t\right)\hfill \\ \hfill & \left.+{\int }_0^1\left(t-\frac{{\left[7\right]}_{q_1}}{{\left[8\right]}_{q_1}}\right)\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2+t{\acute{O}}_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}{d}_{q_1}\left(t\right)\right]d_{q_2}\left(s\right)\hfill \\ \hfill & +\frac{{\left[7\right]}_{q_2}{\left[2\right]}_{q_2}-{\left[8\right]}_{q_2}}{{\left[8\right]}_{q_2}{\left[2\right]}_{q_2}}{\int }_0^{\frac{{\left[2\right]}_{q_2}}{{\left[3\right]}_{q_2}}}\left[\frac{{\left[8\right]}_{q_1}-{\left[2\right]}_{q_1}}{{\left[8\right]}_{q_1}{\left[2\right]}_{q_1}}{\int }_0^{\frac{1}{{\left[3\right]}_{q_1}}}\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2+t{\acute{O}}_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}{d}_{q_1}\left(t\right)\right.\hfill \\ \hfill & +\frac{{\left[7\right]}_{q_1}{\left[2\right]}_{q_1}-{\left[8\right]}_{q_1}}{{\left[8\right]}_{q_1}{\left[2\right]}_{q_1}}{\int }_0^{\frac{{\left[2\right]}_{q_1}}{{\left[3\right]}_{q_1}}}\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2+t{\acute{O}}_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}{d}_{q_1}\left(t\right)\hfill \\ \hfill & \left.+{\int }_0^1\left(t-\frac{{\left[7\right]}_{q_1}}{{\left[8\right]}_{q_1}}\right)\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2+t{\acute{O}}_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}{d}_{q_1}\left(t\right)\right]d_{q_2}\left(s\right)\hfill \\ \hfill & +{\int }_0^1\left(s-\frac{{\left[7\right]}_{q_2}}{{\left[8\right]}_{q_2}}\right)\left[\frac{{\left[8\right]}_{q_1}-{\left[2\right]}_{q_1}}{{\left[8\right]}_{q_1}{\left[2\right]}_{q_1}}{\int }_0^{\frac{1}{{\left[3\right]}_{q_1}}}\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2+t{\acute{O}}_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}{d}_{q_1}\left(t\right)\right.\hfill \\ \hfill & +\frac{{\left[7\right]}_{q_1}{\left[2\right]}_{q_1}-{\left[8\right]}_{q_1}}{{\left[8\right]}_{q_1}{\left[2\right]}_{q_1}}{\int }_0^{\frac{{\left[2\right]}_{q_1}}{{\left[3\right]}_{q_1}}}\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2+t{\acute{O}}_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}{d}_{q_1}\left(t\right)\hfill \\ \hfill & \left.+{\int }_0^1\left(t-\frac{{\left[7\right]}_{q_1}}{{\left[8\right]}_{q_1}}\right)\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2+t{\acute{O}}_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}{d}_{q_1}\left(t\right)\right]d_{q_2}\left(s\right).\hfill \end{array}$$

(24)

By following the arguments similar to those which were used in the proof of Lemma 2 and multiplying with ${\stackrel{`}{O}}_1{\stackrel{`}{O}}_2$ on both sides, we have the required identity. □

Remark 2.

If we replace ${\stackrel{`}{O}}_1={\pi }_2-{\pi }_1,{\acute{O}}_1={\pi }_1-{\pi }_2$, ${\acute{O}}_2={\pi }_3-{\pi }_4,$ and ${\stackrel{`}{O}}_2={\pi }_4-{\pi }_3$ in Lemma 3, then we have

$$\begin{array}{cc}\hfill & q_1{q}_2\left({\pi }_2-{\pi }_1\right)({\pi }_4-{\pi }_3){\int }_0^1{\int }_0^1{\Lambda }_{q_1}\left(t\right){\Lambda }_{q_2}\left(s\right)\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1{q}_2}^2\mathsf{Ϝ}({\pi }_2+t{\acute{O}}_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}{d}_{q_1}\left(t\right)d_{q_2}\left(s\right)\hfill \\ \hfill & =\frac{1}{\left({\pi }_2-{\pi }_1\right)({\pi }_4-{\pi }_3)}{\int }_{{\pi }_1}^{{\pi }_2}{\int }_{{\pi }_3}^{{\pi }_4}\mathsf{Ϝ}{\left(x,y\right)}^{{\pi }_2}{d}_{q_1}{\left(x\right)}^{{\pi }_4}{d}_{q_2}\left(y\right)-\frac{1}{\left({\pi }_2-{\pi }_1\right){\left[8\right]}_{q_2}}{\int }_{{\pi }_1}^{{\pi }_2}\left[\frac{q_2^3{\left[6\right]}_{q_2}}{{\left[2\right]}_{q_2}}\mathsf{Ϝ}\left(x,{\pi }_4+\frac{({\pi }_3-{\pi }_4)}{{\left[3\right]}_{q_2}}\right)\right.\hfill \\ \hfill & \left.+q_{{}_2}\mathsf{Ϝ}\left(x,{\pi }_4\right)+\frac{q_2^2{\left[6\right]}_{q_2}}{{\left[2\right]}_{q_2}}\mathsf{Ϝ}\left(x,{\pi }_4+\frac{{\left[2\right]}_{q_2}({\pi }_3-{\pi }_4)}{{\left[3\right]}_{q_2}}\right)d_{q_1}\left(x\right)+\mathsf{Ϝ}\left(x,{\pi }_3\right)\right]{}^{{\pi }_2}{d}_{q_1}\left(x\right)-\frac{1}{({\pi }_4-{\pi }_3){\left[8\right]}_{q_1}}{\int }_{{\pi }_4+{\eta }_2}^{{\pi }_4}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \times \left[\mathsf{Ϝ}\left({\pi }_3,y\right)+\frac{q_1^3{\left[6\right]}_{q_1}}{{\left[2\right]}_{q_1}}\mathsf{Ϝ}\left({\pi }_2+\frac{({\pi }_1-{\pi }_2)}{{\left[3\right]}_{q_1}},y\right)+\frac{q_{{}_1}^2{\left[6\right]}_{q_1}}{{\left[2\right]}_{q_1}}\mathsf{Ϝ}\left({\pi }_2+\frac{{\left[2\right]}_{q_1}({\pi }_3-{\pi }_4)}{{\left[3\right]}_{q_1}},y\right)d_{q_2}\left(y\right)+q_1\mathsf{Ϝ}\left({\pi }_2,y\right)\right]{}^{{\pi }_4}{d}_{q_2}\left(y\right)\hfill \\ \hfill & +\frac{1}{{\left[8\right]}_{q_2}{\left[8\right]}_{q_1}}\left[\mathsf{Ϝ}\left({\pi }_2,{\pi }_3\right)+\frac{q_{{}_2}^3{\left[6\right]}_{q_2}}{{\left[2\right]}_{q_2}}\mathsf{Ϝ}\left({\pi }_2,{\pi }_4+\frac{({\pi }_3-{\pi }_4)}{{\left[3\right]}_{q_2}}\right)+\frac{q_2^2{\left[6\right]}_{q_2}}{{\left[2\right]}_{q_2}}\mathsf{Ϝ}\left({\pi }_2,{\pi }_4+\frac{{\left[2\right]}_{q_2}({\pi }_3-{\pi }_4)}{{\left[3\right]}_{q_2}}\right)+q_2\mathsf{Ϝ}\left({\pi }_2,{\pi }_4\right)\right]\hfill \\ \hfill & +\frac{1}{{\left[8\right]}_{q_1}{\left[8\right]}_{q_2}}\left[\mathsf{Ϝ}\left({\pi }_1,{\pi }_4\right)+\frac{q_{{}_1}^3{\left[6\right]}_{q_1}}{{\left[2\right]}_{q_1}}\mathsf{Ϝ}\left({\pi }_2+\frac{({\pi }_1-{\pi }_2)}{{\left[3\right]}_{q_1}},{\pi }_4\right)+\frac{q_1^2{\left[6\right]}_{q1}}{{\left[2\right]}_{q_1}}\mathsf{Ϝ}\left({\pi }_2+\frac{{\left[2\right]}_{q_1}({\pi }_1-{\pi }_2)}{{\left[3\right]}_{q_1}},{\pi }_4\right)+\mathsf{Ϝ}\left({\pi }_1,{\pi }_3\right)\right]\hfill \end{array}$$

4. Main Results

Within this section, we present fresh inequalities of the Simpson and Newton types for generalized coordinated convex functions by utilizing Lemmas 2 and 3, respectively. We shall commence this segment by introducing certain concepts that will be useful in verifying our outcomes:

$$\begin{array}{cc}\hfill A_1\left(q\right)& =\frac{2q^2{\left[2\right]}_q^2+{\left[6\right]}_q^2\left({\left[6\right]}_q-{\left[3\right]}_q\right)}{{\left[2\right]}_q^3{\left[3\right]}_q{\left[6\right]}_q^3},\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill B_1\left(q\right)& =2\frac{q{\left[3\right]}_q{\left[6\right]}_q-q^2}{{\left[2\right]}_q{\left[3\right]}_q{\left[6\right]}_q^3}+\frac{1}{{\left[2\right]}_q^3}\left(\frac{q+q^2}{{\left[3\right]}_q}-\frac{q^2+2q}{{\left[6\right]}_q}\right),\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill A_2\left(q\right)& =\frac{2q^2{\left[5\right]}_q^3}{{\left[2\right]}_q{\left[3\right]}_q{\left[6\right]}_q^3}+\frac{{\left[6\right]}_q\left(1+{\left[2\right]}_q^3\right)-{\left[3\right]}_q{\left[5\right]}_q\left(1+{\left[2\right]}_q^2\right)}{{\left[2\right]}_q^3{\left[3\right]}_q{\left[6\right]}_q},\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill B_2\left(q\right)& =2\frac{q{\left[5\right]}_q^2{\left[6\right]}_q{\left[3\right]}_q-q^2{\left[5\right]}_q^3}{{\left[2\right]}_q{\left[3\right]}_q{\left[6\right]}_q^3}+\frac{q^2}{{\left[2\right]}_q{\left[3\right]}_q}-\frac{q{\left[5\right]}_q}{{\left[2\right]}_q{\left[6\right]}_q}-\frac{1}{{\left[2\right]}_q^3}\left[\frac{{\left[5\right]}_q(q^2+2q)}{{\left[6\right]}_q}-\frac{q+q^2}{{\left[3\right]}_q}\right],\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill A_3\left(q\right)& =\frac{2q^2{\left[3\right]}_q^3+{\left[8\right]}_q^2\left({\left[8\right]}_q{\left[2\right]}_q-{\left[3\right]}_q^2\right)}{{\left[2\right]}_q{\left[3\right]}_q^4{\left[8\right]}_q^3},\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill B_3\left(q\right)& =2\frac{q{\left[8\right]}_q{\left[3\right]}_q-q^2}{{\left[2\right]}_q{\left[3\right]}_q{\left[8\right]}_q^3}+\frac{{\left[3\right]}_q^2-{\left[2\right]}_q}{{\left[2\right]}_q{\left[3\right]}_q^4}+\frac{1-{\left[2\right]}_q{\left[3\right]}_q}{{\left[2\right]}_q{\left[3\right]}_q^2{\left[2\right]}_q},\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill A_4\left(q\right)& =\frac{2q^2}{{\left[2\right]}_q^4{\left[3\right]}_q}+\frac{{\left[2\right]}_q^2(1+{\left[2\right]}_q^3)-{\left[3\right]}_q^2(1+{\left[2\right]}_q^2)}{{\left[2\right]}_q^2{\left[3\right]}_q^4},\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill B_4\left(q\right)& =\frac{2q}{{\left[2\right]}_q^3}-\frac{q}{{\left[3\right]}_q^2}-\frac{q^2}{{\left[3\right]}_q^2}-A_4\left(q\right),\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill A_5\left(q\right)& =\frac{2q^2{\left[7\right]}_q^3}{{\left[2\right]}_q{\left[3\right]}_q{\left[8\right]}_q^3}+\frac{{\left[8\right]}_q{\left[2\right]}_q\left({\left[3\right]}_q^3+{\left[2\right]}_q^3\right)-{\left[7\right]}_q{\left[3\right]}_q^2\left({\left[2\right]}_q^2+{\left[3\right]}_q^2\right)}{{\left[2\right]}_q{\left[3\right]}_q^4{\left[8\right]}_q},\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill B_5\left(q\right)& =2\frac{q{\left[8\right]}_q{\left[7\right]}_q^2{\left[3\right]}_q-q^2{\left[7\right]}_q^3}{{\left[2\right]}_q{\left[3\right]}_q{\left[8\right]}_q^3}+\frac{q^2}{{\left[2\right]}_q{\left[3\right]}_q}-\frac{q{\left[7\right]}_q}{{\left[2\right]}_q{\left[8\right]}_q}+\frac{{\left[2\right]}_q({\left[3\right]}_q^2-{\left[2\right]}_q^2)}{{\left[3\right]}_q^4}-\frac{(q^2+q){\left[2\right]}_q{\left[7\right]}_q}{{\left[8\right]}_q{\left[3\right]}_q^2}.\hfill \end{array}$$

(34)

4.1. Simpson-Type Inequalities

Theorem 3.

We assume that the conditions of Lemma 2 hold. If $\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1{q}_2}^2\mathsf{Ϝ}({\pi }_2+t{\acute{O}}_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|$ is a $generalizedconvexfunction$ and integrable on Δ, then the following inequality holds for the right quantum integrals:

$$\begin{array}{cc}\hfill & \left|\frac{1}{{\left[6\right]}_{q_1}{\left[6\right]}_{q_2}}\left[q_1^2{\left[4\right]}_{q_1}\mathsf{Ϝ}\left({\pi }_2+\frac{{\acute{O}}_1}{{\left[2\right]}_{q_1}},{\pi }_4+{\acute{O}}_2\right)+q_1^2{q}_2{\left[4\right]}_{q_1}\mathsf{Ϝ}\left({\pi }_2+\frac{{\acute{O}}_1}{{\left[2\right]}_{q_1}},{\pi }_4\right)\right.\right.\hfill \\ \hfill & +q_1^2{q}_2^2{\left[4\right]}_{q_1}{\left[4\right]}_{q_2}\mathsf{Ϝ}\left({\pi }_2+\frac{{\acute{O}}_1}{{\left[2\right]}_{q_1}},{\pi }_4+\frac{{\acute{O}}_2}{{\left[2\right]}_{q2}}\right)+q_1{q}_2^2{\left[4\right]}_{q_2}\mathsf{Ϝ}\left({\pi }_2,{\pi }_4+\frac{{\acute{O}}_2}{{\left[2\right]}_{q_2}}\right)+q_2^2{\left[4\right]}_{q_2}\hfill \\ \hfill & \left.\times \mathsf{Ϝ}\left({\pi }_2+{\acute{O}}_1,{\pi }_4+\frac{{\acute{O}}_2}{{\left[2\right]}_{q2}}\right)\right]+\frac{1}{{\left[6\right]}_{q_1}{\left[6\right]}_{q_2}}\left[q_1\mathsf{Ϝ}\left({\pi }_2,{\pi }_4+{\acute{O}}_2\right)+q_2\mathsf{Ϝ}\left({\pi }_2+{\acute{O}}_1,{\pi }_4\right)\right.\hfill \\ \hfill & \left.+\mathsf{Ϝ}\left({\pi }_2+{\acute{O}}_1,{\pi }_4+{\acute{O}}_2\right)+q_1{q}_2\mathsf{Ϝ}({\pi }_2,{\pi }_4)\right]-\frac{1}{{\left[6\right]}_{q_2}{\acute{O}}_1}{\int }_{{\pi }_2+{\acute{O}}_1}^{{\pi }_2}\left[q_2\mathsf{Ϝ}\left(x,{\pi }_4\right)\right.\hfill \\ \hfill & \left.+\mathsf{Ϝ}\left(x,{\pi }_4+{\acute{O}}_2\right)+q_2^2{\left[4\right]}_{q_2}\mathsf{Ϝ}\left(x,{\pi }_4+\frac{{\acute{O}}_2}{{\left[2\right]}_{q_2}}\right)\right]{}^{{\pi }_2}{d}_{q_1}\left(x\right)-\frac{1}{{\left[6\right]}_{q_1}{\stackrel{`}{O}}_2}{\int }_{{\pi }_4+{\acute{O}}_2}^{{\pi }_4}\hfill \\ \hfill & \times \left[q_1\mathsf{Ϝ}\left({\pi }_2,y\right)+q_1^2{\left[4\right]}_{q_1}\mathsf{Ϝ}\left({\pi }_2+\frac{{\acute{O}}_1}{{\left[2\right]}_{q1}},y\right)+\mathsf{Ϝ}({\pi }_2+{\acute{O}}_1,y)\right]{}^{{\pi }_4}{d}_{q_2}\left(y\right)\hfill \\ \hfill & \left.+\frac{1}{{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2}{\int }_{{\pi }_2+{\acute{O}}_1}^{{\pi }_2}{\int }_{{\pi }_4+{\acute{O}}_2}^{{\pi }_4}\mathsf{Ϝ}{(x,y)}^{{\pi }_2}{d}_{q_1}{\left(x\right)}^{{\pi }_4}{d}_{q_2}\left(y\right)\right|\hfill \end{array}$$

$$\begin{array}{cc}\hfill & ⩽q_1{q}_2{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2\left[\left(A_1\left(q_1\right)+A_2\left(q_1\right)\right)\left(A_1\left(q_2\right)+A_2\left(q_2\right)\right)\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|\right.\hfill \\ \hfill & +\left(B_1\left(q_2\right)+B_2\left(q_2\right)\right)\left(A_1\left(q_1\right)+A_2\left(q_1\right)\right)\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|+\left(B_1\left(q_1\right)+B_2\left(q_1\right)\right)\left(A_1\left(q_2\right)+A_2\left(q_2\right)\right)\hfill \\ \hfill & \times \left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|\left.+\left(B_1\left(q_1\right)+B_2\left(q_1\right)\right)\left(B_1\left(q_2\right)+B_2\left(q_2\right)\right)\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|\right].\hfill \end{array}$$

Proof. 

By taking the modulus on the right side of the inequality in Lemma 2 and using the properties of the modulus along with the definition of pre-invex, we have the inequality

$$\begin{array}{cc}\hfill & \left|q_1{q}_2{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}{\Lambda }_{q_1}\left(t\right){\Lambda }_{q_2}\left(s\right)\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2+t{\acute{O}}_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}{d}_{q_1}\left(t\right)d_{q_2}\left(s\right)\right|\hfill \\ \hfill & ⩽q_1{q}_2{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2\underset{0}{\overset{1}{\int }}{\Lambda }_{q_2}\left(s\right)\left[\underset{0}{\overset{1}{\int }}{\Lambda }_{q_1}\left(t\right)t\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|\right.\hfill \\ \hfill & \left.+\left(1-t\right)\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|d_{q_1}\left(t\right)\right]d_{q_2}\left(s\right).\hfill \end{array}$$

(35)

Using the definitions of ${\Lambda }_{q_1}\left(t\right)$ and ${\Lambda }_{q_2}\left(s\right)$, the integral on the right side of Equation (35) yields

$$\begin{array}{c}\underset{0}{\overset{1}{\int }}{\Lambda }_{q_2}\left(s\right)\left[\left\{A_1\left(q_1\right)+A_2\left(q_1\right)\right\}\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|+\left\{B_1\left(q_1\right)+B_2\left(q_1\right)\right\}\right.\hfill \\ \left.\times \left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|\right]d_{q_2}\left(s\right)⩽A_1\left(q_2\right)\left[\left\{A_1\left(q_1\right)+A_2\left(q_1\right)\right\}\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|\right.\\ \left.+\left\{B_1\left(q_1\right)+B_2\left(q_1\right)\right\}\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|\right]+B_1\left(q_2\right)\left[\left\{\left(A_1\left(q_1\right)+A_2\left(q_1\right)\right)\right\}\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|\right.\\ \left.+\left\{B_1\left(q_1\right)+B_2\left(q_1\right)\right\}\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|\right]+A_2\left(q_2\right)\left[\left\{A_1\left(q_1\right)+A_2\left(q_1\right)\right\}\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|\right.\\ \left.+\left\{B_1\left(q_1\right)+B_2\left(q_1\right)\right\}\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|\right]+B_2\left(q_2\right)\left[\left\{A_1\left(q_1\right)+A_2\left(q_1\right)\right\}\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|\right.\\ \hfill \left.+\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|\left\{B_1\left(q_1\right)+B_2\left(q_1\right)\right\}\right],\end{array}$$

Multiplying $q_1{q}_2{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2$ on both sides and some simple shuffling gives the required result. □

Corollary 1.

If we have $q\to 1^{-},$ then Theorem 3 gives us the following quantum Simpson-type inequality:

$$\begin{array}{c}\left|\frac{1}{9}\left[\mathsf{Ϝ}\left({\pi }_2+\frac{{\acute{O}}_1}{2},{\pi }_4+{\acute{O}}_2\right)+\mathsf{Ϝ}\left({\pi }_2+{\acute{O}}_1,{\pi }_4+\frac{{\acute{O}}_2}{2}\right)+4\mathsf{Ϝ}\left({\pi }_2+\frac{{\acute{O}}_1}{2},{\pi }_4+\frac{{\acute{O}}_2}{2}\right)\right.\right.\hfill \\ \left.+\mathsf{Ϝ}\left({\pi }_2,{\pi }_4+\frac{{\acute{O}}_2}{2}\right)+\mathsf{Ϝ}\left({\pi }_2+\frac{{\acute{O}}_1}{2},{\pi }_4\right)\right]+\frac{1}{36}\left[\mathsf{Ϝ}\left({\pi }_2,{\pi }_4+{\acute{O}}_2\right)+\mathsf{Ϝ}\left({\pi }_2+{\acute{O}}_1,{\pi }_4+{\acute{O}}_2\right)\right.\\ \left.+\mathsf{Ϝ}\left({\pi }_2+{\acute{O}}_1,{\pi }_4\right)+\mathsf{Ϝ}({\pi }_2,{\pi }_4)\right]-\frac{1}{6{\eta }_1({\pi }_1,{\pi }_2)}\underset{{\pi }_2+{\acute{O}}_1}{\overset{{\pi }_2}{\int }}\left[\mathsf{Ϝ}\left(x,{\pi }_4\right)+4\mathsf{Ϝ}\left(x,{\pi }_4+\frac{{\acute{O}}_2}{2}\right)\right.\\ \left.+\mathsf{Ϝ}\left(x,{\pi }_4+{\acute{O}}_2\right)\right]{}^{{\pi }_2}{d}_{q_1}\left(x\right)-\frac{1}{6{\eta }_2({\pi }_1,{\pi }_2)}\underset{{\pi }_4+{\acute{O}}_2}{\overset{{\pi }_4}{\int }}\left[+4\mathsf{Ϝ}\left({\pi }_2+\frac{{\acute{O}}_1}{2},y\right)+\mathsf{Ϝ}({\pi }_2+{\acute{O}}_1,y)\right.\\ \left.\left.+\mathsf{Ϝ}\left({\pi }_2,y\right)\right]{}^{{\pi }_4}{d}_{q_2}\left(y\right)+\frac{1}{{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2}{\int }_{{\pi }_2+{\acute{O}}_1}^{{\pi }_2}{\int }_{{\pi }_4+{\acute{O}}_2}^{{\pi }_4}\mathsf{Ϝ}{(x,y)}^{{\pi }_2}{d}_{q_1}{\left(x\right)}^{{\pi }_4}{d}_{q_2}\left(y\right)\right|⩽{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2\\ \hfill \times \frac{25}{5184}\left[\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3))}{{}^{{\pi }_2}\partial {\left(t\right)}^{{\pi }_4}\partial \left(s\right)}\right|+\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4))}{{}^{{\pi }_2}\partial {\left(t\right)}^{{\pi }_4}\partial \left(s\right)}\right|+\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_3))}{{}^{{\pi }_2}\partial {\left(t\right)}^{{\pi }_4}\partial \left(s\right)}\right|+\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_4))}{{}^{{\pi }_2}\partial {\left(t\right)}^{{\pi }_4}\partial \left(s\right)}\right|\right]\end{array}$$

Remark 3.

If we replace ${\stackrel{`}{O}}_1={\pi }_2-{\pi }_1,{\acute{O}}_1={\pi }_1-{\pi }_2$, ${\acute{O}}_2={\pi }_3-{\pi }_4,$ and ${\stackrel{`}{O}}_2={\pi }_4-{\pi }_3$ in Theorem 3, then we have the identity

$$\begin{array}{cc}\hfill & \left|\frac{1}{{\left[6\right]}_{q_1}{\left[6\right]}_{q_2}}\left[q_1^2{\left[4\right]}_{q_1}\mathsf{Ϝ}\left(\frac{q_1{\pi }_2+{\pi }_1}{{\left[2\right]}_{q1}},{\pi }_3\right)+q_1^2{q}_2^2{\left[4\right]}_{q_1}{\left[4\right]}_{q_2}\mathsf{Ϝ}\left(\frac{q_1{\pi }_2+{\pi }_1}{{\left[2\right]}_{q1}},\frac{q_2{\pi }_4+{\pi }_3}{{\left[2\right]}_{q_2}}\right)+q_1{q}_2^2{\left[4\right]}_{q_2}\mathsf{Ϝ}\left({\pi }_2,\frac{q_2{\pi }_4+{\pi }_3}{{\left[2\right]}_{q_2}}\right)\right.\right.\hfill \\ \hfill & \left.+q_1^2{q}_2{\left[4\right]}_{q_1}\mathsf{Ϝ}\left(\frac{q_1{\pi }_2+{\pi }_1}{{\left[2\right]}_{q1}},{\pi }_4\right)+q_2^2{\left[4\right]}_{q_2}\mathsf{Ϝ}\left({\pi }_1,\frac{q_2{\pi }_4+{\pi }_3}{{\left[2\right]}_{q_2}}\right)\right]-\frac{1}{{\left[6\right]}_{q_2}({\pi }_3-{\pi }_4)}\underset{{\pi }_1}{\overset{{\pi }_2}{\int }}\left[q_2\mathsf{Ϝ}\left(x,{\pi }_4\right)+q_2^2{\left[4\right]}_{q_2}\mathsf{Ϝ}\left(x,\frac{q_2{\pi }_4+{\pi }_3}{{\left[2\right]}_{q_2}}\right)\right.\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \left.+\mathsf{Ϝ}\left(x,{\pi }_3\right)\right]{}^{{\pi }_2}{d}_{q_1}\left(x\right)+\frac{q_1\mathsf{Ϝ}\left({\pi }_2,{\pi }_3\right)+\mathsf{Ϝ}\left({\pi }_1,{\pi }_3\right)+q_2\mathsf{Ϝ}\left({\pi }_1,{\pi }_4\right)+q_1{q}_2\mathsf{Ϝ}({\pi }_2,{\pi }_4)}{{\left[6\right]}_{q_1}{\left[6\right]}_{q_2}}-\frac{1}{{\left[6\right]}_{q_1}({\pi }_2-{\pi }_1)}\underset{{\pi }_3}{\overset{{\pi }_4}{\int }}\left[q_1\mathsf{Ϝ}\left({\pi }_2,y\right)\right.\hfill \\ \hfill & \left.\left.+q_1^2{\left[4\right]}_{q_1}\mathsf{Ϝ}\left(\frac{q_1{\pi }_2+{\pi }_1}{{\left[2\right]}_{q1}},y\right)+\mathsf{Ϝ}({\pi }_1,y)\right]{}^{{\pi }_4}{d}_{q_2}\left(y\right)+\frac{1}{\left({\pi }_2-{\pi }_1\right)({\pi }_4-{\pi }_3)}\underset{{\pi }_1}{\overset{{\pi }_2}{\int }}\underset{{\pi }_3}{\overset{{\pi }_4}{\int }}\mathsf{Ϝ}{(x,y)}^{{\pi }_2}{d}_{q_1}{\left(x\right)}^{{\pi }_4}{d}_{q_2}\left(y\right)\right|\hfill \end{array}$$

$$\begin{array}{cc}\hfill & ⩽({\pi }_2-{\pi }_1)({\pi }_4-{\pi }_3)\left[\left(A_1\left(q_1\right)+A_2\left(q_1\right)\right)\left(A_1\left(q_2\right)+A_2\left(q_2\right)\right)\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3))}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|\right.\hfill \\ \hfill & +\left(B_1\left(q_2\right)+B_2\left(q_2\right)\right)\left(A_1\left(q_1\right)+A_2\left(q_1\right)\right)\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4))}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|+\left(B_1\left(q_1\right)+B_2\left(q_1\right)\right)\left(A_1\left(q_2\right)+A_2\left(q_2\right)\right)\hfill \\ \hfill & \left.\times \left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_3))}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|+\left(B_1\left(q_1\right)+B_2\left(q_1\right)\right)\left(B_1\left(q_2\right)+B_2\left(q_2\right)\right)\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_4))}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|\right].\hfill \end{array}$$

Remark 4.

If we replace ${\stackrel{`}{O}}_1={\pi }_2-{\pi }_1,{\acute{O}}_1={\pi }_1-{\pi }_2$, ${\acute{O}}_2={\pi }_3-{\pi }_4,$ and ${\stackrel{`}{O}}_2={\pi }_4-{\pi }_3$ in Theorem 3 along with $q_1,q_2\to 1^{-},$ then we have the identity

$$\begin{array}{cc}\hfill & \left|\frac{1}{9}\left[\mathsf{Ϝ}\left(\frac{{\pi }_1-{\pi }_2}{2},{\pi }_3\right)+\mathsf{Ϝ}\left(\frac{{\pi }_1-{\pi }_2}{2},{\pi }_4\right)+4\mathsf{Ϝ}\left(\frac{{\pi }_1-{\pi }_2}{2},\frac{{\pi }_3-{\pi }_4}{2}\right)+\mathsf{Ϝ}\left({\pi }_2,\frac{{\pi }_3-{\pi }_4}{2}\right)+\mathsf{Ϝ}\left({\pi }_1,\frac{{\pi }_3-{\pi }_4}{2}\right)\right]\right.\hfill \\ \hfill & +\frac{1}{36}\left[\mathsf{Ϝ}\left({\pi }_2,{\pi }_3\right)+\mathsf{Ϝ}\left({\pi }_1,{\pi }_3\right)+\mathsf{Ϝ}\left({\pi }_1,{\pi }_4\right)+\mathsf{Ϝ}({\pi }_2,{\pi }_4)\right]-\frac{1}{6\left({\pi }_2-{\pi }_1\right)}{\int }_{{\pi }_1}^{{\pi }_2}\left[4\mathsf{Ϝ}\left(x,\frac{{\pi }_3+{\pi }_4}{2}\right)\right.\hfill \\ \hfill & \left.+\mathsf{Ϝ}\left(x,{\pi }_4\right)+\mathsf{Ϝ}\left(x,{\pi }_3\right)\right]{}^{{\pi }_2}d\left(x\right)-\frac{1}{6\left({\pi }_2-{\pi }_1\right)}{\int }_{{\pi }_3}^{{\pi }_4}\left[\mathsf{Ϝ}\left({\pi }_2,y\right)+4\mathsf{Ϝ}\left(\frac{{\pi }_1+{\pi }_2}{2},y\right)+\mathsf{Ϝ}({\pi }_1,y)\right]{}^{{\pi }_4}d\left(y\right)\hfill \\ \hfill & \left.+\frac{1}{\left({\pi }_2-{\pi }_1\right)({\pi }_4-{\pi }_3)}{\int }_{{\pi }_1}^{{\pi }_2}{\int }_{{\pi }_3}^{{\pi }_4}\mathsf{Ϝ}(x,y){}^{{\pi }_2}d{\left(x\right)}^{{\pi }_4}d\left(y\right)\right|⩽\frac{25}{5184}({\pi }_2-{\pi }_1)({\pi }_4-{\pi }_3)\left[\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}\partial {\left(t\right)}^{{\pi }_4}\partial \left(s\right)}\right|\right.\hfill \\ \hfill & \left.+\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4)}{{}^{{\pi }_2}\partial {\left(t\right)}^{{\pi }_4}\partial \left(s\right)}\right|+\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_3)}{{}^{{\pi }_2}\partial {\left(t\right)}^{{\pi }_4}\partial \left(s\right)}\right|+\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_4)}{{}^{{\pi }_2}\partial {\left(t\right)}^{{\pi }_4}\partial \left(s\right)}\right|\right].\hfill \end{array}$$

Theorem 4.

We assume that the conditions of Lemma 2 hold. If ${\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1{q}_2}^2\mathsf{Ϝ}({\pi }_2+t{\acute{O}}_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}$ is a $generalizedconvexfunction$ and integrable on Δ, where $p_1>1$ and ${\displaystyle \frac{1}{r_1}}+{\displaystyle \frac{1}{p_1}}=1,$ then we have the following inequality:

$$\left|\frac{1}{{\left[6\right]}_{q_1}{\left[6\right]}_{q_2}}\left[q_1^2{\left[4\right]}_{q_1}\mathsf{Ϝ}\left({\pi }_2+\frac{{\acute{O}}_1}{{\left[2\right]}_{q_1}},{\pi }_4+{\acute{O}}_2\right)+q_2^2{\left[4\right]}_{q_2}\mathsf{Ϝ}\left({\pi }_2+{\acute{O}}_1,{\pi }_4+\frac{{\acute{O}}_2}{{\left[2\right]}_{q2}}\right)+q_1^2{q}_2^2{\left[4\right]}_{q_1}{\left[4\right]}_{q_2}\right.\right.$$

(36)

$$\begin{array}{c}\left.\times \mathsf{Ϝ}\left({\pi }_2+\frac{{\acute{O}}_1}{{\left[2\right]}_{q_1}},{\pi }_4+\frac{{\acute{O}}_2}{{\left[2\right]}_{q2}}\right)+q_1^2{q}_2{\left[4\right]}_{q_1}\mathsf{Ϝ}\left({\pi }_2+\frac{{\acute{O}}_1}{{\left[2\right]}_{q_1}},{\pi }_4\right)+q_1{q}_2^2{\left[4\right]}_{q_2}\mathsf{Ϝ}\left({\pi }_2,{\pi }_4+\frac{{\acute{O}}_2}{{\left[2\right]}_{q_2}}\right)\right]\hfill \\ +\frac{q_1\mathsf{Ϝ}\left({\pi }_2,{\pi }_4+{\acute{O}}_2\right)+q_1{q}_2\mathsf{Ϝ}({\pi }_2,{\pi }_4)+\mathsf{Ϝ}\left({\pi }_2+{\acute{O}}_1,{\pi }_4+{\acute{O}}_2\right)+q_2\mathsf{Ϝ}\left({\pi }_2+{\acute{O}}_1,{\pi }_4\right)}{{\left[6\right]}_{q_1}{\left[6\right]}_{q_2}}\\ -\frac{1}{{\left[6\right]}_{q_2}{\stackrel{`}{O}}_1}\underset{{\pi }_2+{\acute{O}}_1}{\overset{{\pi }_2}{\int }}{\left[q_2\mathsf{Ϝ}\left(x,{\pi }_4\right)+q_2^2{\left[4\right]}_{q_2}\mathsf{Ϝ}\left(x,{\pi }_4+\frac{{\acute{O}}_2}{{\left[2\right]}_{q_2}}\right)+\mathsf{Ϝ}\left(x,{\pi }_4+{\acute{O}}_2\right)\right]}^{{\pi }_2}{d}_{q_1}\left(x\right)\\ -\frac{1}{{\left[6\right]}_{q_1}{\eta }_2({\pi }_2,{\pi }_1)}\underset{{\pi }_4+{\acute{O}}_2}{\overset{{\pi }_4}{\int }}{\left[q_1\mathsf{Ϝ}\left({\pi }_2,y\right)+q_1^2{\left[4\right]}_{q_1}\mathsf{Ϝ}\left({\pi }_2+\frac{{\acute{O}}_1}{{\left[2\right]}_{q1}},y\right)+\mathsf{Ϝ}({\pi }_2+{\acute{O}}_1,y)\right]}^{{\pi }_4}{d}_{q_2}\left(y\right)\end{array}$$

$$\begin{array}{c}\left.+\frac{1}{{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2}\underset{{\pi }_2+{\acute{O}}_1}{\overset{{\pi }_2}{\int }}\underset{{\pi }_4+{\acute{O}}_2}{\overset{{\pi }_4}{\int }}\mathsf{Ϝ}{(x,y)}^{{\pi }_2}{d}_{q_1}{\left(x\right)}^{{\pi }_4}{d}_{q_2}\left(y\right)\right|\le q_1{q}_2{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2\hfill \\ \times {\left(\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}{\left|{\Lambda }_{q_2}\left(s\right){\Lambda }_{q_1}\left(t\right)\right|}^{r_1}{}^{{\pi }_4}{d}_{q_2}{\left(s\right)}^{{\pi }_2}{d}_{q_1}\left(t\right)\right)}^{\frac{1}{r_1}}\left[\frac{1}{{\left[2\right]}_{q_1}{\left[2\right]}_{q_2}}{\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+\frac{q_2}{{\left[2\right]}_{q_1}{\left[2\right]}_{q_2}}\right.\\ \hfill {\left.\times {\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+\frac{q_1}{{\left[2\right]}_{q_1}{\left[2\right]}_{q_2}}{\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+\frac{q_1{q}_2}{{\left[2\right]}_{q_1}{\left[2\right]}_{q_2}}{\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right]}^{\frac{1}{p_1}},\end{array}$$

where $0<q_1<1$ and $0<q_2<1$.

Proof. 

From the integrals on the right side of Equation (35), and by applying the well-known quantum Hölders integral inequality, it is found that

$$\begin{array}{cc}\hfill & \left|q_1{q}_2{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}{\Lambda }_{q_1}\left(t\right){\Lambda }_{q_2}\left(s\right)\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2+t{\acute{O}}_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}{d}_{q_1}\left(t\right)d_{q_2}\left(s\right)\right|\hfill \\ \hfill & \le q_1{q}_2{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2{\left(\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}{\left|{\Lambda }_{q_2}\left(s\right){\Lambda }_{q_1}\left(t\right)\right|}^{r_1}{d}_{q_2}\left(s\right)d_{q_1}\left(t\right)\right)}^{\frac{1}{r_1}}\left[\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}\left(ts{\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right.\right.\hfill \\ \hfill & +\left(1-t\right)s{\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+\left(1-t\right)\left(1-s\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\hfill \\ \hfill & {\left.\left.+t(1-s){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)d_{q_1}\left(t\right)d_{q_2}\left(s\right)\right]}^{\frac{1}{p_1}}.\hfill \end{array}$$

(37)

By applying the technique of Lemma 1 when $\alpha =0$ to the above integrals, we have the following results:

$${\int }_0^1{\int }_0^{1}ts{d}_{q_1}\left(t\right)d_{q_2}\left(s\right)=\left({\int }_0^{1}t{d}_{q_1}\left(t\right)\right)\left({\int }_0^{1}s{d}_{q_2}\left(s\right)\right)=\frac{1}{{\left[2\right]}_{q_1}{\left[2\right]}_{q_2}},$$

(38)

$${\int }_0^1{\int }_0^{1}t\left(1-s\right)d_{q_1}\left(t\right)d_{q_2}\left(s\right)=\frac{{\left[2\right]}_{q_2}-1}{{\left[2\right]}_{q_1}{\left[2\right]}_{q_2}},$$

(39)

$${\int }_0^1{\int }_0^1\left(1-t\right)s{d}_{q_1}\left(t\right)d_{q_2}\left(s\right)=\frac{{\left[2\right]}_{q_1}-1}{{\left[2\right]}_{q_1}{\left[2\right]}_{q_2}},$$

(40)

$${\int }_0^1{\int }_0^1\left(1-t\right)\left(1-s\right)d_{q_1}\left(t\right)d_{q_2}\left(s\right)=\frac{\left({\left[2\right]}_{q_1}-1\right)\left({\left[2\right]}_{q_2}-1\right)}{{\left[2\right]}_{q_1}{\left[2\right]}_{q_2}}.$$

(41)

By using all these values in Equation (37), we find the required result. □

Corollary 2.

By replacing ${\acute{O}}_1={\pi }_1-{\pi }_2,{\stackrel{`}{O}}_1={\pi }_2-{\pi }_1,{\acute{O}}_2={\pi }_3-{\pi }_4$, and ${\stackrel{`}{O}}_2={\pi }_4-{\pi }_3$ in Theorem 4, we obtain the following inequality:

$$\begin{array}{c}\left|\frac{1}{{\left[6\right]}_{q_1}{\left[6\right]}_{q_2}}\left[q_1^2{\left[4\right]}_{q_1}\mathsf{Ϝ}\left(\frac{q_1{\pi }_2+{\pi }_1}{{\left[2\right]}_{q_1}},{\pi }_3\right)+q_2^2{\left[4\right]}_{q_2}\mathsf{Ϝ}\left({\pi }_1,\frac{q_2{\pi }_4+{\pi }_3}{{\left[2\right]}_{q2}}\right)+q_1^2{q}_2^2{\left[4\right]}_{q_1}{\left[4\right]}_{q_2}\right.\right.\hfill \\ \times \mathsf{Ϝ}\left(\frac{q_1{\pi }_2+{\pi }_1}{{\left[2\right]}_{q_1}},\frac{q_2{\pi }_4+{\pi }_3}{{\left[2\right]}_{q2}}\right)+q_1^2{q}_2{\left[4\right]}_{q_1}\mathsf{Ϝ}\left(\frac{q_1{\pi }_2+{\pi }_1}{{\left[2\right]}_{q_1}},{\pi }_4\right)+q_1{q}_2^2{\left[4\right]}_{q_2}\\ \left.\times \mathsf{Ϝ}\left({\pi }_2,\frac{q_2{\pi }_4+{\pi }_3}{{\left[2\right]}_{q2}}\right)\right]+\frac{q_1\mathsf{Ϝ}\left({\pi }_2,{\pi }_4\right)+\mathsf{Ϝ}\left({\pi }_1,{\pi }_4\right)+q_2\mathsf{Ϝ}\left({\pi }_1,{\pi }_4\right)+q_1{q}_2\mathsf{Ϝ}({\pi }_2,{\pi }_4)}{{\left[6\right]}_{q_1}{\left[6\right]}_{q_2}}-\frac{1}{{\left[6\right]}_{q_2}({\pi }_2-{\pi }_1)}\\ \times \underset{{\pi }_1}{\overset{{\pi }_2}{\int }}\left[q_2^2{\left[4\right]}_{q_2}\mathsf{Ϝ}\left(x,\frac{q_2{\pi }_4+{\pi }_3}{{\left[2\right]}_{q2}}\right)+\mathsf{Ϝ}\left(x,{\pi }_3\right)+q_2\mathsf{Ϝ}\left(x,{\pi }_4\right)\right]{}^{{\pi }_2}{d}_{q_1}\left(x\right)+\underset{{\pi }_1}{\overset{{\pi }_2}{\int }}\underset{{\pi }_3}{\overset{{\pi }_4}{\int }}\frac{\mathsf{Ϝ}{(x,y)}^{{\pi }_2}{d}_{q_1}{\left(x\right)}^{{\pi }_4}{d}_{q_2}\left(y\right)}{\left({\pi }_2-{\pi }_1\right)({\pi }_4-{\pi }_3)}\\ \left.-\frac{1}{{\left[6\right]}_{q_1}({\pi }_2-{\pi }_1)}\underset{{\pi }_3}{\overset{{\pi }_4}{\int }}\left[q_1\mathsf{Ϝ}\left({\pi }_2,y\right)+q_1^2{\left[4\right]}_{q_1}\mathsf{Ϝ}\left(\frac{q_1{\pi }_2+{\pi }_1}{{\left[2\right]}_{q_1}},y\right)+\mathsf{Ϝ}\left({\pi }_1,y\right)\right]{}^{{\pi }_4}{d}_{q_2}\left(y\right)\right|\\ \le q_1{q}_2\left({\pi }_2-{\pi }_1\right)({\pi }_4-{\pi }_3){\left(\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}{\left|{\Lambda }_{q_2}\left(s\right){\Lambda }_{q_1}\left(t\right)\right|}^{r_1}{d}_{q_2}\left(s\right)d_{q_1}\left(t\right)\right)}^{\frac{1}{r_1}}\left[\frac{1}{{\left[2\right]}_{q_1}{\left[2\right]}_{q_2}}{\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+\frac{q_2}{{\left[2\right]}_{q_1}{\left[2\right]}_{q_2}}\right.\\ \hfill {\left.\times {\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+\frac{q_1}{{\left[2\right]}_{q_1}{\left[2\right]}_{q_2}}{\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+\frac{q_1{q}_2}{{\left[2\right]}_{q_1}{\left[2\right]}_{q_2}}{\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right]}^{\frac{1}{p_1}}.\end{array}$$

Theorem 5.

We assume that the conditions of Lemma 2 hold. If ${\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2+t{\acute{O}}_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}$ is generalized convex and an integrable function on Δ, where $p_1>1$, then

$$\begin{array}{cc}\hfill & \left|\frac{1}{{\left[6\right]}_{q_1}{\left[6\right]}_{q_2}}\left[q_1^2{\left[4\right]}_{q_1}\mathsf{Ϝ}\left({\pi }_2+\frac{{\acute{O}}_1}{{\left[2\right]}_{q_1}},{\pi }_4+{\acute{O}}_2\right)+q_2^2{\left[4\right]}_{q_2}\mathsf{Ϝ}\left({\pi }_2+{\acute{O}}_1,{\pi }_4+\frac{{\acute{O}}_2}{{\left[2\right]}_{q2}}\right)+q_1^2{q}_2^2{\left[4\right]}_{q_1}{\left[4\right]}_{q_2}\right.\right.\hfill \\ \hfill & \times \mathsf{Ϝ}\left({\pi }_2+\frac{{\acute{O}}_1}{{\left[2\right]}_{q_1}},{\pi }_4+\frac{{\acute{O}}_2}{{\left[2\right]}_{q2}}\right)+q_1{q}_2^2{\left[4\right]}_{q_2}\mathsf{Ϝ}\left({\pi }_2,{\pi }_4+\frac{{\acute{O}}_2}{{\left[2\right]}_{q_2}}\right)\left.+q_1^2{q}_2{\left[4\right]}_{q_1}\mathsf{Ϝ}\left({\pi }_2+\frac{{\acute{O}}_1}{{\left[2\right]}_{q_1}},{\pi }_4\right)\right]\hfill \\ \hfill & +\frac{q_1\mathsf{Ϝ}\left({\pi }_2,{\pi }_4+{\acute{O}}_2\right)+q_2\mathsf{Ϝ}\left({\pi }_2+{\acute{O}}_1,{\pi }_4\right)+\mathsf{Ϝ}\left({\pi }_2+{\acute{O}}_1,{\pi }_4+{\acute{O}}_2\right)+q_1{q}_2\mathsf{Ϝ}({\pi }_2,{\pi }_4)}{{\left[6\right]}_{q_1}{\left[6\right]}_{q_2}}\hfill \\ \hfill & -\frac{1}{{\left[6\right]}_{q_2}{\stackrel{`}{O}}_1}{\int }_{{\pi }_2+{\acute{O}}_1}^{{\pi }_2}\left[q_2\mathsf{Ϝ}\left(x,{\pi }_4\right)+\mathsf{Ϝ}\left(x,{\pi }_4+{\acute{O}}_2\right)+q_2^2{\left[4\right]}_{q_2}\mathsf{Ϝ}\left(x,{\pi }_4+\frac{{\acute{O}}_2}{{\left[2\right]}_{q_2}}\right)\right]{}^{{\pi }_2}{d}_{q_1}\left(x\right)\hfill \\ \hfill & -\frac{1}{{\left[6\right]}_{q_1}{\stackrel{`}{O}}_2}{\int }_{{\pi }_4+{\acute{O}}_2}^{{\pi }_4}\left[q_1\mathsf{Ϝ}\left({\pi }_2,y\right)+q_1^2{\left[4\right]}_{q_1}\mathsf{Ϝ}\left({\pi }_2+\frac{{\acute{O}}_1}{{\left[2\right]}_{q1}},y\right)+\mathsf{Ϝ}({\pi }_2+{\acute{O}}_1,y)\right]{}^{{\pi }_4}{d}_{q_2}\left(y\right)\hfill \end{array}$$

$$\begin{array}{c}\left.+\frac{1}{{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2}{\int }_{{\pi }_2+{\acute{O}}_1}^{{\pi }_2}{\int }_{{\pi }_4+{\acute{O}}_2}^{{\pi }_4}\mathsf{Ϝ}{(x,y)}^{{\pi }_2}{d}_{q_1}{\left(x\right)}^{{\pi }_4}{d}_{q_2}\left(y\right)\right|\le q_1{q}_2{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2\hfill \\ \times {\left[\left(\frac{2q_1}{{\left[2\right]}_{q_1}{\left[6\right]}_{q_1}^2}+\frac{q_1^3{\left[3\right]}_{q_1}-q_1}{{\left[6\right]}_{q_1}{\left[2\right]}_{q_1}^3}\right)\left(\frac{2q_2}{{\left[2\right]}_{q_2}{\left[6\right]}_{q_2}^2}+\frac{q_2^3{\left[3\right]}_{q_2}-q_2}{{\left[6\right]}_{q_2}{\left[2\right]}_{q_2}^3}\right)\right]}^{1-\frac{1}{p_1}}\left[A_1{(q}_1)\left(A_1{(q}_2){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+B_1{(q}_2)\right.\right.\\ {\left.\left.\times {\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4))}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)+B_1\left(q_1\right)\left(A_1\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+B_1\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)\right]}^{\frac{1}{p_1}}\\ +q_1{q}_2{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2{\left[\left(\frac{2q_1}{{\left[2\right]}_{q_1}{\left[6\right]}_{q_1}^2}+\frac{q_1^3{\left[3\right]}_{q_1}-q_1}{{\left[6\right]}_{q_1}{\left[2\right]}_{q_1}^3}\right)\left(\frac{2q_2{\left[5\right]}_{q_2}^2}{{\left[2\right]}_{q_2}{\left[6\right]}_{q_2}^2}+\frac{1}{{\left[2\right]}_{q_2}}-\frac{{\left[5\right]}_{q_2}}{{\left[6\right]}_{q_2}}-\frac{{\left[2\right]}_{q_2}^2{\left[5\right]}_{q_2}-{\left[6\right]}_{q_2}}{{\left[6\right]}_{q_2}{\left[2\right]}_{q_2}^3}\right)\right]}^{1-\frac{1}{p_1}}\end{array}$$

$$\begin{array}{c}\times \left[A_1\left(q_1\right)\left(A_2\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+B_2\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)+B_1\left(q_1\right)\right.\hfill \\ {\left.\times \left(A_2\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+B_2\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)\right]}^{\frac{1}{p_1}}+q_1{q}_2{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2\\ \times {\left[\left(\frac{2q_1{\left[5\right]}_{q_1}^2}{{\left[2\right]}_{q_1}{\left[6\right]}_{q_1}^2}+\frac{1}{{\left[2\right]}_{q_1}}-\frac{{\left[5\right]}_{q_1}}{{\left[6\right]}_{q_1}}-\frac{{\left[2\right]}_{q_1}^2{\left[5\right]}_{q_1}-{\left[6\right]}_{q_1}}{{\left[6\right]}_{q_1}{\left[2\right]}_{q_1}^3}\right)\left(\frac{2q_2}{{\left[2\right]}_{q_2}{\left[6\right]}_{q_2}^2}+\frac{q_1^3{\left[3\right]}_{q_2}-q_2}{{\left[6\right]}_{q_2}{\left[2\right]}_{q_2}^3}\right)\right]}^{1-\frac{1}{p_1}}\\ \times \left[A_2\left(q_1\right)\left(A_1\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+B_1\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)\right.\\ {\left.+B_2\left(q_1\right)\left(A_1\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+B_1\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)\right]}^{\frac{1}{p_1}}+q_1{q}_2{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2\\ \hfill \times {\left[\left(\frac{2q_1{\left[5\right]}_{q_1}^2}{{\left[2\right]}_{q_1}{\left[6\right]}_{q_1}^2}+\frac{1}{{\left[2\right]}_{q_1}}-\frac{{\left[5\right]}_{q_1}}{{\left[6\right]}_{q_1}}-\frac{{\left[2\right]}_{q_1}^2{\left[5\right]}_{q_1}-{\left[6\right]}_{q_1}}{{\left[6\right]}_{q_1}{\left[2\right]}_{q_1}^3}\right)\left(\frac{2q_2{\left[5\right]}_{q_2}^2}{{\left[2\right]}_{q_2}{\left[6\right]}_{q_2}^2}+\frac{1}{{\left[2\right]}_{q_2}}-\frac{{\left[5\right]}_{q_2}}{{\left[6\right]}_{q_2}}-\frac{{\left[2\right]}_{q_2}^2{\left[5\right]}_{q_2}-{\left[6\right]}_{q_2}}{{\left[6\right]}_{q_2}{\left[2\right]}_{q_2}^3}\right)\right]}^{1-\frac{1}{p_1}}\end{array}$$

$$\begin{array}{cc}\hfill & \times \left[A_2\left(q_1\right)\left(A_2\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+B_2\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)\right.\hfill \\ \hfill & {\left.+B_2\left(q_1\right)\left(A_2\left(q_2\right){\left|\frac{{}^{{\pi }_2,d}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+B_2\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)\right]}^{\frac{1}{p_1}}\hfill \end{array}$$

where $0<q_1<1$, $0<q_2<1$, and $A_1\left(q_1\right),A_2\left(q_1\right),A_3\left(q_1\right),A_4\left(q_1\right),B_1\left(q_2\right),B_2\left(q_2\right),B_3\left(q_2\right)$, and $B_4\left(q_2\right)$ are given in Equations (25)–(30).

Proof. 

By applying the power mean inequality for integrals on the right side of Equation (35) (used in the proof of Theorem 3) along with definition of preinvexity of ${\left|{\displaystyle \frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2+t{\acute{O}}_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}}\right|}^{p_1},$ we have

$$\left|q_1{q}_2{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2{\int }_0^1{\int }_0^1{\Lambda }_{q_1}\left(t\right){\Lambda }_{q_2}\left(s\right)\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2+t{\acute{O}}_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}{}^{{\pi }_1}{d}_{q_1}\left(t\right){}^{{\pi }_2}{d}_{q_2}\left(s\right)\right|$$

(42)

$$\begin{array}{cc}\hfill & \le q_1{q}_2{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2{\left\{\left({\int }_0^{\frac{1}{{\left[2\right]}_{q_1}}}\left|t-\frac{1}{{\left[6\right]}_{q_1}}\right|d_{q_1}\left(t\right)\right)\left({\int }_0^{\frac{1}{{\left[2\right]}_{q_2}}}\left|s-\frac{1}{{\left[6\right]}_{q_2}}\right|d_{q_2}\left(s\right)\right)\right\}}^{1-\frac{1}{p_1}}\left[{\int }_0^{\frac{1}{{\left[2\right]}_{q_2}}}\left|s-\frac{1}{{\left[6\right]}_{q_2}}\right|\right.\hfill \\ \hfill & \times \left\{{\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\left({\int }_0^{\frac{1}{{\left[2\right]}_{q_1}}}t\left|t-\frac{1}{{\left[6\right]}_{q_1}}\right|d_{q_1}\left(t\right)\right)+{\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right.\hfill \\ \hfill & {\left.\left.\times \left({\int }_0^{\frac{1}{{\left[2\right]}_{q_1}}}\left(1-t\right)\left|t-\frac{1}{{\left[6\right]}_{q_1}}\right|d_{q_1}\left(t\right)\right)\right\}{d}_{q_2}\left(s\right)\right]}^{\frac{1}{p_1}}+q_1{q}_2{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2\left\{\left({\int }_0^{\frac{1}{{\left[2\right]}_{q_1}}}\left|t-\frac{1}{{\left[6\right]}_{q_1}}\right|d_{q_1}\left(t\right)\right)\right.\hfill \\ \hfill & {\left.\times \left({\int }_{\frac{1}{{\left[2\right]}_{q_2}}}^1\left|s-\frac{{\left[5\right]}_{q_2}}{{\left[6\right]}_{q_2}}\right|d_{q_2}\left(s\right)\right)\right\}}^{1-\frac{1}{p_1}}\left[{\int }_{\frac{1}{{\left[2\right]}_{q_2}}}^1\left|s-\frac{{\left[5\right]}_{q_2}}{{\left[6\right]}_{q_2}}\right|\left\{{\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right.\right.\hfill \end{array}$$

$$\begin{array}{cc}\hfill & {\left.\left.\times \left({\int }_0^{\frac{1}{{\left[2\right]}_{q_1}}}t\left|t-\frac{1}{{\left[6\right]}_{q_1}}\right|d_{q_1}\left(t\right)\right)+{\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\left({\int }_0^{\frac{1}{{\left[2\right]}_{q_1}}}\left(1-t\right)\left|t-\frac{1}{{\left[6\right]}_{q_1}}\right|d_{q_1}\left(t\right)\right)\right\}{d}_{q_2}\left(s\right)\right]}^{\frac{1}{p_1}}\hfill \\ \hfill & +q_1{q}_2{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2{\left\{\left({\int }_{\frac{1}{{\left[2\right]}_{q_1}}}^1\left|t-\frac{{\left[5\right]}_{q_1}}{{\left[6\right]}_{q_1}}\right|d_{q_1}\left(t\right)\right)\left({\int }_0^{\frac{1}{{\left[2\right]}_{q_2}}}\left|s-\frac{1}{{\left[6\right]}_{q_2}}\right|d_{q_2}\left(s\right)\right)\right\}}^{1-\frac{1}{p_1}}\left[{\int }_0^{\frac{1}{{\left[2\right]}_{q_2}}}\left|s-\frac{1}{{\left[6\right]}_{q_2}}\right|\right.\hfill \\ \hfill & \times \left\{{\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\left({\int }_{\frac{1}{{\left[2\right]}_{q_1}}}^{1}t\left|t-\frac{{\left[5\right]}_{q_1}}{{\left[6\right]}_{q_1}}\right|d_{q_1}\left(t\right)\right)+{\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right.\hfill \\ \hfill & {\left.\left.\times \left({\int }_{\frac{1}{{\left[2\right]}_{q_1}}}^1\left(1-t\right)\left|t-\frac{{\left[5\right]}_{q_1}}{{\left[6\right]}_{q_1}}\right|d_{q_1}\left(t\right)\right)\right\}{d}_{q_2}\left(s\right)\right]}^{\frac{1}{p_1}}+q_1{q}_2{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2\left\{\left({\int }_{\frac{1}{{\left[2\right]}_{q_1}}}^1\left|t-\frac{{\left[5\right]}_{q_1}}{{\left[6\right]}_{q_1}}\right|d_{q_1}\left(t\right)\right)\right.\hfill \\ \hfill & {\left.\times \left({\int }_{\frac{1}{{\left[2\right]}_{q_2}}}^1\left|s-\frac{{\left[5\right]}_{q_2}}{{\left[6\right]}_{q_2}}\right|d_{q_2}\left(s\right)\right)\right\}}^{1-\frac{1}{p_1}}\left[{\int }_{\frac{1}{{\left[2\right]}_{q_2}}}^1\left|s-\frac{{\left[5\right]}_{q_2}}{{\left[6\right]}_{q_2}}\right|\left\{{\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right.\right.\hfill \\ \hfill & {\left.\left.\times \left({\int }_{\frac{1}{{\left[2\right]}_{q_1}}}^{1}t\left|t-\frac{{\left[5\right]}_{q_1}}{{\left[6\right]}_{q_1}}\right|d_{q_1}\left(t\right)\right)+{\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\left({\int }_{\frac{1}{{\left[2\right]}_{q_1}}}^1\left(1-t\right)\left|t-\frac{{\left[5\right]}_{q_1}}{{\left[6\right]}_{q_1}}\right|d_{q_1}\left(t\right)\right)\right\}{d}_{q_2}\left(s\right)\right]}^{\frac{1}{p_1}}\hfill \\ \hfill & \le q_1{q}_2{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2{\left\{\left(\frac{2q_1}{{\left[2\right]}_{q_1}{\left[6\right]}_{q_1}^2}+\frac{q_1^3{\left[3\right]}_{q_1}-q_1}{{\left[6\right]}_{q_1}{\left[2\right]}_{q_1}^3}\right)\left(\frac{2q_2}{{\left[2\right]}_{q_2}{\left[6\right]}_{q_2}^2}+\frac{q_2^3{\left[3\right]}_{q_2}-q_2}{{\left[6\right]}_{q_2}{\left[2\right]}_{q_2}^3}\right)\right\}}^{1-\frac{1}{p_1}}\hfill \\ \hfill & \times \left[A_1\left(q_1\right)\left({\int }_0^{\frac{1}{{\left[2\right]}_{q_2}}}s\left|s-\frac{1}{{\left[6\right]}_{q_2}}\right|d_{q_2}\left(s\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+{\int }_0^{\frac{1}{{\left[2\right]}_{q_2}}}\left(1-s\right)\left|s-\frac{1}{{\left[6\right]}_{q_2}}\right|d_{q_2}\left(s\right)\right.\right.\hfill \\ \hfill & \left.\times {\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)+B_1\left(q_1\right)\left({\int }_0^{\frac{1}{{\left[2\right]}_{q_2}}}s\left|s-\frac{1}{{\left[6\right]}_{q_2}}\right|{\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}{d}_{q_2}\left(s\right)\right.\hfill \\ \hfill & {\left.\left.+{\int }_0^{\frac{1}{{\left[2\right]}_{q_2}}}\left(1-s\right)\left|s-\frac{1}{{\left[6\right]}_{q_2}}\right|d_{q_2}\left(s\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)\right]}^{\frac{1}{p_1}}+q_1{q}_2{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2\hfill \\ \hfill & \times {\left\{\left(\frac{2q_1}{{\left[2\right]}_{q_1}{\left[6\right]}_{q_1}^2}+\frac{q_1^3{\left[3\right]}_{q_1}-q_1}{{\left[6\right]}_{q_1}{\left[2\right]}_{q_1}^3}\right)\left(\frac{2q_{{}_2}{\left[5\right]}_{q_2}^2}{{\left[2\right]}_{q_2}{\left[6\right]}_{q_2}^2}+\frac{1}{{\left[2\right]}_{q_2}}-\frac{{\left[5\right]}_{q_2}}{{\left[6\right]}_{q_2}}-\frac{{\left[2\right]}_{q_2}^2{\left[5\right]}_{q_2}-{\left[6\right]}_{q_2}}{{\left[6\right]}_{q_2}{\left[2\right]}_{q_2}^3}\right)\right\}}^{1-\frac{1}{p_1}}\left[A_2\left(q_1\right){\int }_{\frac{1}{{\left[2\right]}_{q_2}}}^1\right.\hfill \\ \hfill & \times \left(s\left|s-\frac{{\left[5\right]}_{q_2}}{{\left[6\right]}_{q_2}}\right|{\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}{d}_{q_2}\left(s\right)+{\int }_{\frac{1}{{\left[2\right]}_{q_2}}}^1\left(1-s\right)\left|s-\frac{{\left[5\right]}_{q_2}}{{\left[6\right]}_{q_2}}\right|d_{q_2}\left(s\right)\right.\hfill \\ \hfill & \left.\times {\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)+B_1\left(q_1\right)\left({\int }_{\frac{1}{{\left[2\right]}_{q_2}}}^{1}s\left|s-\frac{1}{{\left[6\right]}_{q_2}}\right|{\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}{d}_{q_2}\left(s\right)\right.\hfill \\ \hfill & {\left.\left.+{\int }_{\frac{1}{{\left[2\right]}_{q_2}}}^1\left(1-s\right)\left|s-\frac{1}{{\left[6\right]}_{q_2}}\right|d_{q_2}\left(s\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)\right]}^{\frac{1}{p_1}}+q_1{q}_2{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2\hfill \\ \hfill & \times {\left\{\left(\frac{2q_1{\left[5\right]}_{q_1}^2}{{\left[2\right]}_{q_1}{\left[6\right]}_{q_1}^2}+\frac{1}{{\left[2\right]}_{q_1}}-\frac{{\left[5\right]}_{q_1}}{{\left[6\right]}_{q_1}}-\frac{{\left[2\right]}_{q_1}^2{\left[5\right]}_{q_1}-{\left[6\right]}_{q_1}}{{\left[6\right]}_{q_1}{\left[2\right]}_{q_1}^3}\right)\left(\frac{2q_2}{{\left[2\right]}_{q_2}{\left[6\right]}_{q_2}^2}+\frac{q_1^3{\left[3\right]}_{q_2}-q_2}{{\left[6\right]}_{q_2}{\left[2\right]}_{q_2}^3}\right)\right\}}^{1-\frac{1}{p_1}}\left[A_2\left(q_1\right){\int }_{\frac{1}{{\left[2\right]}_{q_2}}}^1\right.\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \times \left(s\left|s-\frac{1}{{\left[6\right]}_{q_2}}\right|d_{q_2}\left(s\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+{\int }_{\frac{1}{{\left[2\right]}_{q_2}}}^1\left(1-s\right)\left|s-\frac{1}{{\left[6\right]}_{q_2}}\right|d_{q_2}\left(s\right)\right.\hfill \\ \hfill & \left.\times {\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)+B_2\left(q_1\right)\left({\int }_{\frac{1}{{\left[2\right]}_{q_2}}}^{1}s\left|s-{\displaystyle \frac{1}{{\left[6\right]}_{q_2}}}\right|d_{q_2}\left(s\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right.\hfill \\ \hfill & {\left.+\left.{\int }_{\frac{1}{{\left[2\right]}_{q_2}}}^1\left(1-s\right)\left|s-{\displaystyle \frac{1}{{\left[6\right]}_{q_2}}}\right|d_{q_2}\left(s\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)\right]}^{\frac{1}{p_1}}+q_1{q}_2{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2\hfill \\ \hfill & \times {\left\{\left(\frac{2q_1{\left[5\right]}_{q_1}^2}{{\left[2\right]}_{q_1}{\left[6\right]}_{q_1}^2}+\frac{1}{{\left[2\right]}_{q_1}}-\frac{{\left[5\right]}_{q_1}}{{\left[6\right]}_{q_1}}-\frac{{\left[2\right]}_{q_1}^2{\left[5\right]}_{q_1}-{\left[6\right]}_{q_1}}{{\left[6\right]}_{q_1}{\left[2\right]}_{q_1}^3}\right)\left(\frac{2q_2{\left[5\right]}_{q_2}^2}{{\left[2\right]}_{q_2}{\left[6\right]}_{q_2}^2}+\frac{1}{{\left[2\right]}_{q_2}}-\frac{{\left[5\right]}_{q_2}}{{\left[6\right]}_{q_2}}-\frac{{\left[2\right]}_{q_2}^2{\left[5\right]}_{q_2}-{\left[6\right]}_{q_2}}{{\left[6\right]}_{q_2}{\left[2\right]}_{q_2}^3}\right)\right\}}^{1-\frac{1}{p_1}}\hfill \\ \hfill & \times \left[A_2\left(q_1\right)\left({\int }_{\frac{1}{{\left[2\right]}_{q_2}}}^{1}s\left|s-\frac{{\left[5\right]}_{q_2}}{{\left[6\right]}_{q_2}}\right|{\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}{d}_{q_2}\left(s\right)+{\int }_{\frac{1}{{\left[2\right]}_{q_2}}}^1\left(1-s\right)\left|s-{\displaystyle \frac{{\left[5\right]}_{q_2}}{{\left[6\right]}_{q_2}}}\right|d_{q_2}\left(s\right)\right.\right.\hfill \\ \hfill & \left.\times {\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)+B_2\left(q_1\right)\left({\int }_{\frac{1}{{\left[2\right]}_{q_2}}}^{1}s\left|s-\frac{{\left[5\right]}_{q_2}}{{\left[6\right]}_{q_2}}\right|d_{q_2}\left(s\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right.\hfill \\ \hfill & {\left.\left.+{\int }_{\frac{1}{{\left[2\right]}_{q_2}}}^1\left(1-s\right)\left|s-\frac{{\left[5\right]}_{q_2}}{{\left[6\right]}_{q_2}}\right|d_{q_2}\left(s\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)\right]}^{\frac{1}{p_1}}.\hfill \end{array}$$

□

Corollary 3.

If we take the limit $q_1,q_2\to 1^{-}$ in Theorem 5, then we have the following inequality:

$$\begin{array}{cc}\hfill & \left|\frac{1}{9}\left[\mathsf{Ϝ}\left({\pi }_2+\frac{{\acute{O}}_1}{2},{\pi }_4+{\acute{O}}_2\right)+\mathsf{Ϝ}\left({\pi }_2+\frac{{\acute{O}}_1}{2},{\pi }_4\right)+4\mathsf{Ϝ}\left({\pi }_2+\frac{{\acute{O}}_1}{2},{\pi }_4+\frac{{\acute{O}}_2}{2}\right)\right.\right.\hfill \\ \hfill & \left.+\mathsf{Ϝ}\left({\pi }_2+{\acute{O}}_1,{\pi }_4+\frac{{\acute{O}}_2}{2}\right)+\mathsf{Ϝ}\left({\pi }_2,{\pi }_4+\frac{{\acute{O}}_2}{2}\right)\right]+{\displaystyle \frac{1}{{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2}}{\int }_{{\pi }_2+{\acute{O}}_1}^{{\pi }_2}{\int }_{{\pi }_4+{\acute{O}}_2}^{{\pi }_4}\hfill \\ \hfill & \times \mathsf{Ϝ}{(x,y)}^{{\pi }_2}d{\left(x\right)}^{{\pi }_4}d\left(y\right)-\frac{1}{6{\eta }_2({\pi }_1,{\pi }_2)}{\int }_{{\pi }_4+{\acute{O}}_2}^{{\pi }_4}\left[4\mathsf{Ϝ}\left({\pi }_2+\frac{{\acute{O}}_1}{2},y\right)+\mathsf{Ϝ}({\pi }_2+{\acute{O}}_1,y)\right.\hfill \\ \hfill & \left.+\mathsf{Ϝ}\left({\pi }_2,y\right)\right]{}^{{\pi }_4}d\left(y\right)-\frac{1}{6{\eta }_1({\pi }_1,{\pi }_2)}{\int }_{{\pi }_2+{\acute{O}}_1}^{{\pi }_2}\left[\mathsf{Ϝ}\left(x,{\pi }_4+{\acute{O}}_2\right)+\mathsf{Ϝ}\left(x,{\pi }_4\right)+4\mathsf{Ϝ}\left(x,{\pi }_4+\frac{{\acute{O}}_2}{2}\right)\right]{}^{{\pi }_2}d\left(x\right)\hfill \\ \hfill & \left.+\frac{\mathsf{Ϝ}\left({\pi }_2,{\pi }_4+{\acute{O}}_2\right)+\mathsf{Ϝ}\left({\pi }_2+{\acute{O}}_1,{\pi }_4\right)+\mathsf{Ϝ}\left({\pi }_2+{\acute{O}}_1,{\pi }_4+{\acute{O}}_2\right)+\mathsf{Ϝ}({\pi }_2,{\pi }_4)}{36}\right|\hfill \\ \hfill & \le {\stackrel{`}{O}}_1{\stackrel{`}{O}}_2{\left[\frac{25}{5184}\right]}^{1-\frac{1}{p_1}}\left[\frac{29}{1296}\left(\frac{29}{1296}{\left|\frac{{\partial }^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{\partial \left(t\right)\partial \left(s\right)}\right|}^{p_1}+\frac{61}{1296}{\left|\frac{{\partial }^2\mathsf{Ϝ}({\pi }_1,{\pi }_4))}{\partial \left(t\right)\partial \left(s\right)}\right|}^{p_1}\right)+\frac{61}{1296}\right.\hfill \\ \hfill & {\left.\times \left(\frac{29}{1296}{\left|\frac{{\partial }^2\mathsf{Ϝ}({\pi }_2,{\pi }_3)}{\partial \left(t\right)\partial \left(s\right)}\right|}^{p_1}+\frac{61}{1296}{\left|\frac{{\partial }^2\mathsf{Ϝ}({\pi }_2,{\pi }_4)}{\partial \left(t\right)\partial \left(s\right)}\right|}^{p_1}\right)\right]}^{\frac{1}{p_1}}+\left[\frac{29}{1296}\left(\frac{61}{1296}{\left|\frac{{\partial }^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{\partial \left(t\right)\partial \left(s\right)}\right|}^{p_1}+\frac{29}{1296}{\left|\frac{{\partial }^2\mathsf{Ϝ}({\pi }_1,{\pi }_4)}{\partial \left(t\right)\partial \left(s\right)}\right|}^{p_1}\right.\right.\hfill \\ \hfill & {\left.+\frac{61}{1296}\left(\frac{61}{1296}{\left|\frac{{\partial }^2\mathsf{Ϝ}({\pi }_2,{\pi }_3)}{\partial \left(t\right)\partial \left(s\right)}\right|}^{p_1}+\frac{29}{1296}{\left|\frac{{\partial }^2\mathsf{Ϝ}({\pi }_2,{\pi }_4)}{\partial \left(t\right)\partial \left(s\right)}\right|}^{p_1}\right)\right]}^{\frac{1}{p_1}}+\left[\frac{61}{1296}\left(\frac{29}{1296}{\left|\frac{{\partial }^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}\partial {\left(t\right)}^{{\pi }_4}\partial \left(s\right)}\right|}^{p_1}+\frac{61}{1296}\right.\right.\hfill \\ \hfill & {\left.\left.\times {\left|\frac{{\partial }^2\mathsf{Ϝ}({\pi }_1,{\pi }_4)}{\partial \left(t\right)\partial \left(s\right)}\right|}^{p_1}\right)+\frac{29}{1296}\left(\frac{29}{1296}{\left|\frac{{\partial }^2\mathsf{Ϝ}({\pi }_2,{\pi }_3)}{\partial \left(t\right)\partial \left(s\right)}\right|}^{p_1}+\frac{61}{1296}{\left|\frac{{\partial }^2\mathsf{Ϝ}({\pi }_2,{\pi }_4)}{\partial \left(t\right)\partial \left(s\right)}\right|}^{p_1}\right)\right]}^{\frac{1}{p_1}}+\left[\frac{61}{1296}\left(\frac{61}{1296}\right.\right.\hfill \\ \hfill & {\left.\left.\times {\left|\frac{{\partial }^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{\partial \left(t\right)\partial \left(s\right)}\right|}^{p_1}+\frac{29}{1296}{\left|\frac{{\partial }^2\mathsf{Ϝ}({\pi }_1,{\pi }_4)}{\partial \left(t\right)\partial \left(s\right)}\right|}^{p_1}\right)+\frac{29}{1296}\left(\frac{61}{1296}{\left|\frac{{\partial }^2\mathsf{Ϝ}({\pi }_2,{\pi }_3)}{\partial \left(t\right)\partial \left(s\right)}\right|}^{p_1}+\frac{29}{1296}{\left|\frac{{\partial }^2\mathsf{Ϝ}({\pi }_2,{\pi }_4)}{\partial \left(t\right)\partial \left(s\right)}\right|}^{p_1}\right)\right]}^{\frac{1}{p_1}}\hfill \end{array}$$

Remark 5.

By replacing ${\acute{O}}_1={\pi }_1-{\pi }_2,{\stackrel{`}{O}}_1={\pi }_2-{\pi }_1,{\acute{O}}_2=$(${\pi }_3-{\pi }_4)$, and ${\stackrel{`}{O}}_2={\pi }_4-{\pi }_3$ in Theorem 5, we have the following inequality:

$$\begin{array}{cc}\hfill & \left|\frac{1}{{\left[6\right]}_{q_1}{\left[6\right]}_{q_2}}\left[q_1^2{\left[4\right]}_{q_1}\mathsf{Ϝ}\left({\pi }_2+\frac{{\pi }_1-{\pi }_2}{{\left[2\right]}_{q_1}},{\pi }_3\right)+q_1^2{q}_2^2{\left[4\right]}_{q_1}{\left[4\right]}_{q_2}\mathsf{Ϝ}\left({\pi }_2+\frac{({\pi }_1-{\pi }_2}{{\left[2\right]}_{q_1}}),{\pi }_4+\frac{{\pi }_3-{\pi }_4}{{\left[2\right]}_{q2}}\right)\right.\right.\hfill \\ \hfill & \left.+q_1^2{q}_2{\left[4\right]}_{q_1}\mathsf{Ϝ}\left({\pi }_2+\frac{{\pi }_1-{\pi }_2}{{\left[2\right]}_{q_1}},{\pi }_4\right)+q_1{q}_2^2{\left[4\right]}_{q_2}\mathsf{Ϝ}\left({\pi }_2,{\pi }_4+\frac{{\pi }_3-{\pi }_4}{{\left[2\right]}_{q_2}}\right)+q_2^2{\left[4\right]}_{q_2}\mathsf{Ϝ}\left({\pi }_1,{\pi }_4+\frac{{\pi }_3-{\pi }_4}{{\left[2\right]}_{q2}}\right)\right]\hfill \\ \hfill & +\frac{q_1\mathsf{Ϝ}\left({\pi }_2,{\pi }_3\right)+\mathsf{Ϝ}\left({\pi }_1,{\pi }_3\right)+q_2\mathsf{Ϝ}\left({\pi }_1,{\pi }_4\right)+q_1{q}_2\mathsf{Ϝ}({\pi }_2,{\pi }_4)}{{\left[6\right]}_{q_1}{\left[6\right]}_{q_2}}+{\int }_{{\pi }_1}^{{\pi }_2}{\int }_{{\pi }_3}^{{\pi }_4}\frac{\mathsf{Ϝ}(x,y){}^{{\pi }_2}{d}_{q_1}\left(x\right){}^{{\pi }_4}{d}_{q_2}\left(y\right)}{\left({\pi }_2-{\pi }_1\right)({\pi }_4-{\pi }_3)}\hfill \\ \hfill & -\frac{1}{{\left[6\right]}_{q_2}\left({\pi }_2-{\pi }_1\right)}{\int }_{{\pi }_1}^{{\pi }_2}\left[\mathsf{Ϝ}\left(x,{\pi }_3\right)+q_2\mathsf{Ϝ}\left(x,{\pi }_4\right)+q_2^2{\left[4\right]}_{q_2}\mathsf{Ϝ}\left(x,{\pi }_4+\frac{{\pi }_3-{\pi }_4}{{\left[2\right]}_{q_2}}\right)\right]{}^{{\pi }_2}{d}_{q_1}\left(x\right)\hfill \\ \hfill & \left.-\frac{1}{{\left[6\right]}_{q_1}\left({\pi }_4-{\pi }_3\right)}{\int }_{{\pi }_3}^{{\pi }_4}\left[q_1\mathsf{Ϝ}\left({\pi }_2,y\right)+q_1^2{\left[4\right]}_{q_1}\mathsf{Ϝ}\left({\pi }_2+\frac{({\pi }_1-{\pi }_2)}{{\left[2\right]}_{q1}},y\right)+\mathsf{Ϝ}({\pi }_1,y)\right]{}^{{\pi }_4}{d}_{q_2}\left(y\right)\right|\hfill \\ \hfill & \le q_1{q}_2\left({\pi }_2-{\pi }_1\right)({\pi }_4-{\pi }_3)\left[{\left\{\left(\frac{2q_1}{{\left[2\right]}_{q_1}{\left[6\right]}_{q_1}^2}+\frac{q_1^3{\left[3\right]}_{q_1}-q_1}{{\left[6\right]}_{q_1}{\left[2\right]}_{q_1}^3}\right)\left(\frac{2q_2}{{\left[2\right]}_{q_2}{\left[6\right]}_{q_2}^2}+\frac{q_2^3{\left[3\right]}_{q_2}-q_2}{{\left[6\right]}_{q_2}{\left[2\right]}_{q_2}^3}\right)\right\}}^{1-\frac{1}{p_1}}\left\{A_1\left(q_1\right)\left(A_1\left(q_2\right)\right.\right.\right.\hfill \\ \hfill & \times \left.{\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+B_1\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4))}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)+B_1\left(q_1\right)\left(A_1\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right.\hfill \\ \hfill & {\left.\left.+B_1\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)\right\}}^{{}^{\frac{1}{p_1}}}+\left\{\left(\frac{2q_2{\left[5\right]}_{q_2}^2}{{\left[2\right]}_{q_2}{\left[6\right]}_{q_2}^2}+\frac{1}{{\left[2\right]}_{q_2}}-\frac{{\left[5\right]}_{q_2}}{{\left[6\right]}_{q_2}}-\frac{{\left[2\right]}_{q_2}^2{\left[5\right]}_{q_2}-{\left[6\right]}_{q_2}}{{\left[6\right]}_{q_2}{\left[2\right]}_{q_2}^3}\right)\right.\hfill \\ \hfill & {\left.\times \left(\frac{2q_1}{{\left[2\right]}_{q_1}{\left[6\right]}_{q_1}^2}+\frac{q_1^3{\left[3\right]}_{q_1}-q_1}{{\left[6\right]}_{q_1}{\left[2\right]}_{q_1}^3}\right)\right\}}^{1-\frac{1}{p_1}}\left\{A_1\left(q_1\right)\left(A_2\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+B_2\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)\right.\hfill \end{array}$$

$$\begin{array}{ccc}& & {\left.+B_1\left(q_1\right)\left(A_2\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+B_2\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)\right\}}^{{}^{\frac{1}{p_1}}}\hfill \\ & & +{\left\{\left(\frac{2q_1{\left[5\right]}_{q_1}^2}{{\left[2\right]}_{q_1}{\left[6\right]}_{q_1}^2}+\frac{1}{{\left[2\right]}_{q_1}}-\frac{{\left[5\right]}_{q_1}}{{\left[6\right]}_{q_1}}-\frac{{\left[2\right]}_{q_1}^2{\left[5\right]}_{q_1}-{\left[6\right]}_{q_1}}{{\left[6\right]}_{q_1}{\left[2\right]}_{q_1}^3}\right)\left(\frac{2q_2}{{\left[2\right]}_{q_2}{\left[6\right]}_{q_2}^2}+\frac{q_1^3{\left[3\right]}_{q_2}-q_2}{{\left[6\right]}_{q_2}{\left[2\right]}_{q_2}^3}\right)\right\}}^{1-\frac{1}{p_1}}\hfill \\ & & \times \left\{A_2\left(q_1\right)\left(A_1\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+B_1\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)\right.\hfill \\ & & {\left.+B_2\left(q_1\right)\left(A_1\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+B_1\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)\right\}}^{\frac{1}{p_1}}\hfill \\ & & +\left\{\left(2q_1\frac{{\left[5\right]}_{q_1}^2}{{\left[2\right]}_{q_1}{\left[6\right]}_{q_1}^2}+\frac{1}{{\left[2\right]}_{q_1}}-\frac{{\left[5\right]}_{q_1}}{{\left[6\right]}_{q_1}}-\frac{{\left[2\right]}_{q_1}^2{\left[5\right]}_{q_1}-{\left[6\right]}_{q_1}}{{\left[6\right]}_{q_1}{\left[2\right]}_{q_1}^3}\right)\right.\hfill \\ & & \times {\left.\left(2q_2\frac{{\left[5\right]}_{q_2}^2}{{\left[2\right]}_{q_2}{\left[6\right]}_{q_2}^2}+\frac{1}{{\left[2\right]}_{q_2}}-\frac{{\left[5\right]}_{q_2}}{{\left[6\right]}_{q_2}}-\frac{{\left[2\right]}_{q_2}^2{\left[5\right]}_{q_2}-{\left[6\right]}_{q_2}}{{\left[6\right]}_{q_2}{\left[2\right]}_{q_2}^3}\right)\right\}}^{1-\frac{1}{p_1}}\hfill \\ & & \times \left\{A_2\left(q_1\right)\left(A_2\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+B_2\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)\right.\hfill \\ & & {\left.\left.+B_2\left(q_1\right)\left(A_2\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+B_2\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)\right\}\right]}^{\frac{1}{p_1}}.\hfill \end{array}$$

4.2. Newton-Type Inequalities

Theorem 6.

It is assumed that the assumptions of Lemma 3 hold. If $\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2+t{\acute{O}}_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|$ is a $generalizedconvexfunction$ and integrable on Δ, then the following inequality holds for the right quantum integrals:

$$\begin{array}{cc}\hfill & \left|\frac{1}{{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2}{\int }_{{\pi }_2+{\acute{O}}_1}^{{\pi }_2}{\int }_{{\pi }_4+{\acute{O}}_2}^{{\pi }_4}\mathsf{Ϝ}{\left(x,y\right)}^{{\pi }_2}{d}_{q_1}{\left(x\right)}^{{\pi }_4}{d}_{q_2}\left(y\right)-\frac{1}{{\stackrel{`}{O}}_1{\left[8\right]}_{q_2}}{\int }_{{\pi }_2+{\acute{O}}_1}^{{\pi }_2}\left[q_{{}_2}\mathsf{Ϝ}\left(x,{\pi }_4\right)+\frac{q_2^3{\left[6\right]}_{q_2}}{{\left[2\right]}_{q_2}}\mathsf{Ϝ}\left(x,{\pi }_4+\frac{{\acute{O}}_2}{{\left[3\right]}_{q_2}}\right)\right.\right.\hfill \\ \hfill & \left.+\mathsf{Ϝ}\left(x,{\pi }_4+{\acute{O}}_2\right)+\frac{q_2^2{\left[6\right]}_{q_2}}{{\left[2\right]}_{q_2}}\mathsf{Ϝ}\left(x,{\pi }_4+\frac{{\left[2\right]}_{q_2}{\acute{O}}_2}{{\left[3\right]}_{q_2}}\right)\right]{}^{{\pi }_2}{d}_{q_1}\left(x\right)-\frac{1}{{\stackrel{`}{O}}_2{\left[8\right]}_{q_1}}{\int }_{{\pi }_4+{\acute{O}}_2}^{{\pi }_4}\left[\frac{q_1^3{\left[6\right]}_{q_1}}{{\left[2\right]}_{q_1}}\mathsf{Ϝ}\left({\pi }_2+\frac{{\acute{O}}_1}{{\left[3\right]}_{q_1}},y\right)\right.\hfill \\ \hfill & \left.+\mathsf{Ϝ}\left({\pi }_2+{\acute{O}}_2,y\right)+\frac{q_{{}_1}^2{\left[6\right]}_{q_1}}{{\left[2\right]}_{q_1}}\mathsf{Ϝ}\left({\pi }_2+\frac{{\left[2\right]}_{q_1}{\acute{O}}_1}{{\left[3\right]}_{q_1}},y\right)+q_1\mathsf{Ϝ}\left({\pi }_2,y\right)\right]{}^{{\pi }_4}{d}_{q_2}\left(y\right)+\frac{1}{{\left[8\right]}_{q_2}{\left[8\right]}_{q_1}}\hfill \\ \hfill & \times \left[\mathsf{Ϝ}\left({\pi }_2,{\pi }_4+{\acute{O}}_2\right)+\frac{q_2^2{\left[6\right]}_{q_2}}{{\left[2\right]}_{q_2}}\mathsf{Ϝ}\left({\pi }_2,{\pi }_4+\frac{{\left[2\right]}_{q_2}{\acute{O}}_2}{{\left[3\right]}_{q_2}}\right)+q_2\mathsf{Ϝ}\left({\pi }_2,{\pi }_4\right)+\frac{q_{{}_2}^3{\left[6\right]}_{q_2}}{{\left[2\right]}_{q_2}}\right.\hfill \\ \hfill & \left.\times \mathsf{Ϝ}\left({\pi }_2,{\pi }_4+\frac{{\acute{O}}_2}{{\left[3\right]}_{q_2}}\right)\right]+\frac{1}{{\left[8\right]}_{q_1}{\left[8\right]}_{q_2}}\left[\mathsf{Ϝ}\left({\pi }_2+{\acute{O}}_1,{\pi }_4\right)+\frac{q_{{}_1}^3{\left[6\right]}_{q_1}}{{\left[2\right]}_{q_1}}\mathsf{Ϝ}\left({\pi }_2+\frac{{\acute{O}}_1}{{\left[3\right]}_{q_1}},{\pi }_4\right)\right.\hfill \\ \hfill & \left.\left.+\frac{q_1^2{\left[6\right]}_{q1}}{{\left[2\right]}_{q_1}}\mathsf{Ϝ}\left({\pi }_2+\frac{{\left[2\right]}_{q_1}{\acute{O}}_1}{{\left[3\right]}_{q_1}},{\pi }_4\right)+\mathsf{Ϝ}\left({\pi }_2+{\acute{O}}_1,{\pi }_4+{\acute{O}}_2\right)\right]\right|\hfill \\ \hfill & \le q_1{q}_2{\acute{O}}_1{\acute{O}}_2\left(A_3\left(q_1\right)+A_4\left(q_1\right)+A_5\left(q_1\right)\right)\left[\left(A_3\left(q_2\right)+A_4\left(q_2\right)+A_5\left(q_2\right)\right)\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|\right.\hfill \\ \hfill & \left.+\left(B_3\left(q_2\right)+B_4\left(q_2\right)+B_5\left(q_2\right)\right)\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4))}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|\right]+q_1{q}_2{\acute{O}}_1{\acute{O}}_2\left(B_3\left(q_1\right)+B_4\left(q_1\right)+B_5\left(q_1\right)\right)\hfill \\ \hfill & \times \left[\left(A_3\left(q_2\right)+A_4\left(q_2\right)+A_5\left(q_2\right)\right)\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|+\left(B_3\left(q_2\right)+B_4\left(q_2\right)+B_5\left(q_2\right)\right)\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_4))}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|\right],\hfill \end{array}$$

(43)

where $0<q_1<1$, $0<q_2<1$, and $A_3\left(q_2\right),A_4\left(q_2\right),A_5\left(q_2\right),A_3\left(q_1\right),A_4\left(q_1\right),A_5\left(q_1\right),B_3\left(q_1\right),B_4\left(q_1\right)$, $B_5\left(q_1\right),B_3\left(q_2\right),B_4\left(q_2\right),$ and $B_5\left(q_2\right)$ are from Equations (25)–(30).

Proof. 

Following the same arguments used in the proof of Theorem 3, and by using Lemma 3, we have the required inequality in Equation (43). □

Remark 6.

If we replace ${\stackrel{`}{O}}_1={\pi }_2-{\pi }_1,{\stackrel{`}{O}}_2={\pi }_4-{\pi }_3,{\acute{O}}_1={\pi }_1-{\pi }_2,$ and ${\acute{O}}_2={\pi }_3-{\pi }_4$ in Theorem 6, then we have the following inequality:

$$\begin{array}{cc}\hfill & \left|{\int }_{{\pi }_1}^{{\pi }_2}{\int }_{{\pi }_3}^{{\pi }_4}\frac{\mathsf{Ϝ}\left(x,y\right){}^{{\pi }_2}{d}_{q_1}\left(x\right){}^{{\pi }_4}{d}_{q_2}\left(y\right)}{\left({\pi }_2-{\pi }_1\right)({\pi }_4-{\pi }_3)}-\frac{1}{\left({\pi }_2-{\pi }_1\right){\left[8\right]}_{q_2}}{\int }_{{\pi }_1}^{{\pi }_2}\left[\frac{q_2^3{\left[6\right]}_{q_2}}{{\left[2\right]}_{q_2}}\mathsf{Ϝ}\left(x,\frac{{\left[3\right]}_{q_2}{\pi }_4+({\pi }_3-{\pi }_4)}{{\left[3\right]}_{q_2}}\right)\right.\right.\hfill \\ \hfill & \left.+\frac{q_2^2{\left[6\right]}_{q_2}}{{\left[2\right]}_{q_2}}\mathsf{Ϝ}\left(x,\frac{{\left[3\right]}_{q_2}{\pi }_4+{\left[2\right]}_{q_2}({\pi }_3-{\pi }_4)}{{\left[3\right]}_{q_2}}\right)+q_{{}_2}\mathsf{Ϝ}\left(x,{\pi }_4\right)+\mathsf{Ϝ}\left(x,{\pi }_3\right)\right]{}^{{\pi }_2}{d}_{q_1}\left(x\right)-\frac{1}{({\pi }_4-{\pi }_3){\left[8\right]}_{q_1}}\hfill \\ \hfill & \times {\int }_{{\pi }_3}^{{\pi }_4}\left[\frac{q_1^3{\left[6\right]}_{q_1}}{{\left[2\right]}_{q_1}}\mathsf{Ϝ}\left(\frac{{\left[3\right]}_{q_1}{\pi }_2+({\pi }_1-{\pi }_2)}{{\left[3\right]}_{q_1}},y\right)+\frac{q_{{}_1}^2{\left[6\right]}_{q_1}}{{\left[2\right]}_{q_1}}\mathsf{Ϝ}\left(\frac{{\left[3\right]}_{q_1}{\pi }_2+{\left[2\right]}_{q_1}({\pi }_3-{\pi }_4)}{{\left[3\right]}_{q_1}},y\right)\right.\hfill \\ \hfill & \left.+\mathsf{Ϝ}\left({\pi }_3,y\right)+q_1\mathsf{Ϝ}\left({\pi }_2,y\right)\right]{}^{{\pi }_4}{d}_{q_2}\left(y\right)+\frac{1}{{\left[8\right]}_{q_2}{\left[8\right]}_{q_1}}\left[\mathsf{Ϝ}\left({\pi }_2,{\pi }_3\right)+\frac{q_{{}_2}^3{\left[6\right]}_{q_2}}{{\left[2\right]}_{q_2}}\mathsf{Ϝ}\left({\pi }_2,\frac{{\left[3\right]}_{q_2}{\pi }_4+({\pi }_3-{\pi }_4)}{{\left[3\right]}_{q_2}}\right)\right.\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \left.+\frac{q_2^2{\left[6\right]}_{q_2}}{{\left[2\right]}_{q_2}}\mathsf{Ϝ}\left({\pi }_2,\frac{{\left[3\right]}_{q_2}{\pi }_4+{\left[2\right]}_{q_2}({\pi }_3-{\pi }_4)}{{\left[3\right]}_{q_2}}\right)+q_2\mathsf{Ϝ}\left({\pi }_2,{\pi }_4\right)\right]+\frac{1}{{\left[8\right]}_{q_1}{\left[8\right]}_{q_2}}\left[\mathsf{Ϝ}\left({\pi }_1,{\pi }_4\right)+\mathsf{Ϝ}\left({\pi }_1,{\pi }_3\right)+\frac{q_{{}_1}^3{\left[6\right]}_{q_1}}{{\left[2\right]}_{q_1}}\right.\hfill \\ \hfill & \left.\left.\times \mathsf{Ϝ}\left(\frac{{\left[3\right]}_{q_1}{\pi }_2+({\pi }_1-{\pi }_2)}{{\left[3\right]}_{q_1}},{\pi }_4\right)+\frac{q_1^2{\left[6\right]}_{q1}}{{\left[2\right]}_{q_1}}\mathsf{Ϝ}\left(\frac{{\left[3\right]}_{q_1}{\pi }_2+{\left[2\right]}_{q_1}({\pi }_1-{\pi }_2)}{{\left[3\right]}_{q_1}},{\pi }_4\right)\right]\right|\hfill \\ \hfill & \le q_1{q}_2\left({\pi }_2-{\pi }_1\right)({\pi }_4-{\pi }_3)\left[\left(A_3\left(q_1\right)+A_4\left(q_1\right)+A_5\left(q_1\right)\right)\left\{\left(A_3\left(q_2\right)+A_4\left(q_2\right)+A_5\left(q_2\right)\right)\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|\right.\right.\hfill \\ \hfill & \left.+\left(B_3\left(q_2\right)+B_4\left(q_2\right)+B_5\left(q_2\right)\right)\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4))}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|\right\}+\left\{\left(B_3\left(q_1\right)+B_4\left(q_1\right)+B_5\left(q_1\right)\right)\right.\hfill \\ \hfill & \left.\left.\left(A_3\left(q_2\right)+A_4\left(q_2\right)+A_5\left(q_2\right)\right)\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|+\left(B_3\left(q_2\right)+B_4\left(q_2\right)+B_5\left(q_2\right)\right)\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_4))}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|\right\}\right].\hfill \end{array}$$

Corollary 4.

If we replace ${\stackrel{`}{O}}_1={\pi }_2-{\pi }_1,{\stackrel{`}{O}}_2={\pi }_4-{\pi }_3,{\acute{O}}_1={\pi }_1-{\pi }_2,$ and ${\acute{O}}_2=$(${\pi }_3-{\pi }_4)$ along with the limits $q_1\to 1^{-}and{q}_2\to 1^{-}$ in Theorem 6, then we have the following inequality:

$$\begin{array}{cc}\hfill & \left|\frac{1}{({\pi }_2-{\pi }_1)({\pi }_4-{\pi }_3)}{\int }_{{\pi }_1}^{{\pi }_2}{\int }_{{\pi }_3}^{{\pi }_4}\mathsf{Ϝ}\left(x,y\right){}^{{\pi }_2}d\left(x\right){}^{{\pi }_2}d\left(y\right)-\frac{1}{8({\pi }_2-{\pi }_1)}{\int }_{{\pi }_1}^{{\pi }_2}\left[\mathsf{Ϝ}\left(x,{\pi }_4\right)+\frac{6}{2}\mathsf{Ϝ}\left(x,\frac{{\pi }_3+2{\pi }_4}{3}\right)\right.\right.\hfill \\ \hfill & \left.+\mathsf{Ϝ}\left(x,{\pi }_3\right)+\frac{6}{2}\mathsf{Ϝ}\left(x,\frac{2{\pi }_3+{\pi }_4}{3}\right)\right]{}^{{\pi }_2}d\left(x\right)-\frac{1}{8({\pi }_4-{\pi }_3)}{\int }_{{\pi }_3}^{{\pi }_4}\left[\mathsf{Ϝ}\left({\pi }_3,y\right)+\frac{6}{2}\mathsf{Ϝ}\left(\frac{{\pi }_1+2{\pi }_2}{3},y\right)\right.+\mathsf{Ϝ}\left({\pi }_2,y\right)\hfill \\ \hfill & \left.+\frac{6}{2}\mathsf{Ϝ}\left(\frac{2{\pi }_1+{\pi }_2}{3},y\right)\right]{}^{{\pi }_4}d\left(y\right)+\frac{1}{64}\left[\mathsf{Ϝ}\left({\pi }_2,{\pi }_3\right)+\frac{6}{2}\mathsf{Ϝ}\left({\pi }_2,\frac{{\pi }_3+2{\pi }_4}{3}\right)+\mathsf{Ϝ}\left({\pi }_2,{\pi }_4\right)+\frac{6}{2}\mathsf{Ϝ}\left({\pi }_2,\frac{2{\pi }_3+{\pi }_4}{3}\right)\right]\hfill \\ \hfill & +\frac{1}{64}\left[\mathsf{Ϝ}\left({\pi }_1,{\pi }_4\right)+\frac{6}{2}\mathsf{Ϝ}\left(\frac{{\pi }_1+2{\pi }_2}{3},{\pi }_4\right)+\frac{6}{2}\mathsf{Ϝ}\left(\frac{2{\pi }_1+{\pi }_2}{3},{\pi }_4\right)+\mathsf{Ϝ}\left({\pi }_1,{\pi }_3\right)\right]\le {\stackrel{`}{O}}_1{\stackrel{`}{O}}_2{\left(\frac{25}{576}\right)}^2\hfill \\ \hfill & \left.\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}\partial {\left(t\right)}^{{\pi }_4}\partial \left(s\right)}\right|+\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }^2\mathsf{Ϝ}({\pi }_1,{\pi }_4))}{{}^{{\pi }_2}\partial {\left(t\right)}^{{\pi }_4}\partial \left(s\right)}\right|+\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }^2\mathsf{Ϝ}({\pi }_2,{\pi }_3)}{{}^{{\pi }_2}\partial {\left(t\right)}^{{\pi }_4}\partial \left(s\right)}\right|+\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }^2\mathsf{Ϝ}({\pi }_2,{\pi }_4))}{{}^{{\pi }_2}\partial {\left(t\right)}^{{\pi }_4}\partial \left(s\right)}\right|\right].\hfill \end{array}$$

Theorem 7.

We assume that the conditions of Lemma 3 hold. If ${\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2+t{\acute{O}}_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}$ is a $generalizedconvexfunction$ and integrable on Δ, where $p_1>1$ and ${\displaystyle \frac{1}{r_1}}+{\displaystyle \frac{1}{p_1}}=1$, then we have the following inequality:

$$\begin{array}{cc}\hfill & \left|\frac{1}{{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2}{\int }_{{\pi }_2+{\acute{O}}_1}^{{\pi }_2}{\int }_{{\pi }_4+{\acute{O}}_2}^{{\pi }_4}\mathsf{Ϝ}\left(x,y\right){}^{{\pi }_2}{d}_{q_1}\left(x\right){}^{{\pi }_4}{d}_{q_2}\left(y\right)-\frac{1}{{\stackrel{`}{O}}_1{\left[8\right]}_{q_2}}\right.\hfill \\ \hfill & {\int }_{{\pi }_2+{\acute{O}}_1}^{{\pi }_2}\left[q_{{}_2}\mathsf{Ϝ}\left(x,{\pi }_4\right)+\frac{q_2^3{\left[6\right]}_{q_2}}{{\left[2\right]}_{q_2}}\mathsf{Ϝ}\left(x,{\pi }_4+\frac{{\acute{O}}_2}{{\left[3\right]}_{q_2}}\right)+\mathsf{Ϝ}\left(x,{\pi }_4+{\acute{O}}_2\right)+\frac{q_2^2{\left[6\right]}_{q_2}}{{\left[2\right]}_{q_2}}\right.\hfill \\ \hfill & \left.\mathsf{Ϝ}\left(x,{\pi }_4+\frac{{\left[2\right]}_{q_2}{\acute{O}}_2}{{\left[3\right]}_{q_2}}\right)\right]{}^{{\pi }_2}{d}_{q_1}\left(x\right)-\frac{1}{{\stackrel{`}{O}}_2{\left[8\right]}_{q_1}}{\int }_{{\pi }_4+{\acute{O}}_2}^{{\pi }_4}\left[\mathsf{Ϝ}\left({\pi }_2+{\acute{O}}_2,y\right)\right.\hfill \\ \hfill & \left.+\frac{q_1^3{\left[6\right]}_{q_1}}{{\left[2\right]}_{q_1}}\mathsf{Ϝ}\left({\pi }_2+\frac{{\acute{O}}_1}{{\left[3\right]}_{q_1}},y\right)+\frac{q_{{}_1}^2{\left[6\right]}_{q_1}}{{\left[2\right]}_{q_1}}\mathsf{Ϝ}\left({\pi }_2+\frac{{\left[2\right]}_{q_1}{\acute{O}}_1}{{\left[3\right]}_{q_1}},y\right)+q_1\mathsf{Ϝ}\left({\pi }_2,y\right)\right]{}^{{\pi }_4}{d}_{q_2}\left(y\right)\hfill \\ \hfill & +\frac{1}{{\left[8\right]}_{q_2}{\left[8\right]}_{q_1}}\left[\frac{q_{{}_2}^3{\left[6\right]}_{q_2}}{{\left[2\right]}_{q_2}}\mathsf{Ϝ}\left({\pi }_2,{\pi }_4+\frac{{\acute{O}}_2}{{\left[3\right]}_{q_2}}\right)+\frac{q_2^2{\left[6\right]}_{q_2}}{{\left[2\right]}_{q_2}}\mathsf{Ϝ}\left({\pi }_2,{\pi }_4+\frac{{\left[2\right]}_{q_2}{\acute{O}}_2}{{\left[3\right]}_{q_2}}\right)\right.\hfill \\ \hfill & \left.+q_2\mathsf{Ϝ}\left({\pi }_2,{\pi }_4\right)+\mathsf{Ϝ}\left({\pi }_2,{\pi }_4+{\acute{O}}_2\right)\right]+\frac{1}{{\left[8\right]}_{q_1}{\left[8\right]}_{q_2}}\left[\mathsf{Ϝ}\left({\pi }_2+{\acute{O}}_1,{\pi }_4\right)+\frac{q_{{}_1}^3{\left[6\right]}_{q_1}}{{\left[2\right]}_{q_1}}\right.\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \left.\left.\times \mathsf{Ϝ}\left({\pi }_2+\frac{{\acute{O}}_1}{{\left[3\right]}_{q_1}},{\pi }_4\right)+\frac{q_1^2{\left[6\right]}_{q1}}{{\left[2\right]}_{q_1}}\mathsf{Ϝ}\left({\pi }_2+\frac{{\left[2\right]}_{q_1}{\acute{O}}_1}{{\left[3\right]}_{q_1}},{\pi }_4\right)+\mathsf{Ϝ}\left({\pi }_2+{\acute{O}}_1,{\pi }_4+{\acute{O}}_2\right)\right]\right|\hfill \\ \hfill & \le q_1{q}_2{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2{\left[{\overline{A}}_1\left(q_1\right)\right]}^{\frac{1}{r_1}}{\left[{\overline{A}}_1\left(q_2\right)\right]}^{\frac{1}{r_1}}\hfill \\ \hfill & \times \left[{\check{D}}_1\left(q_1\right)\left({\check{D}}_1\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+{\check{D}}_2\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)\right.\hfill \\ \hfill & {\left.+{\check{D}}_2\left(q_1\right)\left({\check{D}}_1\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+{\check{D}}_2\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)\right]}^{\frac{1}{p_1}}\hfill \\ \hfill & +q_1{q}_2{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2{\left[{\overline{A}}_1\left(q_1\right)\right]}^{\frac{1}{r_1}}{\left[{\overline{A}}_2\left(q_2\right)\right]}^{\frac{1}{r_1}}\hfill \\ \hfill & \times \left[{\check{D}}_1\left(q_1\right)\left({\breve{E}}_1\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+{\breve{E}}_2\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)\right.\hfill \\ \hfill & {\left.+{\check{D}}_2\left(q_1\right)\left({\breve{E}}_1\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+{\breve{E}}_2\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)\right]}^{\frac{1}{p_1}}\hfill \\ \hfill & +q_1{q}_2{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2{\left[{\overline{A}}_1\left(q_1\right)\right]}^{\frac{1}{r_1}}{\left[{\overline{A}}_3\left(q_2\right)\right]}^{\frac{1}{r_1}}\hfill \\ \hfill & \times \left[{\check{D}}_1\left(q_1\right)\left({\breve{G}}_1\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+{\breve{G}}_2\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)\right.\hfill \\ \hfill & {\left.+{\check{D}}_2\left(q_1\right)\left({\breve{G}}_1\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+{\breve{G}}_2\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)\right]}^{\frac{1}{p_1}}\hfill \\ \hfill & +q_1{q}_2{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2{\left[{\overline{A}}_2\left(q_1\right)\right]}^{\frac{1}{r_1}}{\left[{\overline{A}}_1\left(q_2\right)\right]}^{\frac{1}{r_1}}\hfill \\ \hfill & \times \left[{\breve{E}}_1\left(q_1\right)\left({\check{D}}_1\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+{\check{D}}_2\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)\right.\hfill \\ \hfill & {\left.+{\breve{E}}_2\left(q_1\right)\left({\check{D}}_1\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+{\check{D}}_2\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)\right]}^{\frac{1}{p_1}}\hfill \\ \hfill & +q_1{q}_2{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2{\left[{\overline{A}}_2\left(q_1\right)\right]}^{\frac{1}{r_1}}{\left[{\overline{A}}_2\left(q_2\right)\right]}^{\frac{1}{r_1}}\hfill \\ \hfill & \times \left[{\breve{E}}_1\left(q_1\right)\left({\breve{E}}_1\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+{\breve{E}}_2\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)\right.\hfill \\ \hfill & {\left.+{\breve{E}}_2\left(q_1\right)\left({\breve{E}}_1\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+{\breve{E}}_2\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)\right]}^{\frac{1}{p_1}}\hfill \\ \hfill & +q_1{q}_2{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2{\left[{\overline{A}}_2\left(q_1\right)\right]}^{\frac{1}{r_1}}{\left[{\overline{A}}_3\left(q_2\right)\right]}^{\frac{1}{r_1}}\hfill \\ \hfill & \times \left[{\breve{E}}_1\left(q_1\right)\left({\breve{G}}_1\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+{\breve{G}}_2\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)\right.\hfill \\ \hfill & {\left.+{\breve{E}}_2\left(q_1\right)\left({\breve{G}}_1\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+{\breve{G}}_2\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)\right]}^{\frac{1}{p_1}}\hfill \\ \hfill & +q_1{q}_2{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2{\left[{\overline{A}}_3\left(q_1\right)\right]}^{\frac{1}{r_1}}{\left[{\overline{A}}_1\left(q_2\right)\right]}^{\frac{1}{r_1}}\hfill \\ \hfill & \times \left[{\breve{G}}_1\left(q_1\right)\left({\check{D}}_1\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+{\check{D}}_2\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)\right.\hfill \end{array}$$

$$\begin{array}{cc}\hfill & {\left.+{\breve{G}}_2\left(q_1\right)\left({\check{D}}_1\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+{\check{D}}_2\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)\right]}^{\frac{1}{p_1}}\hfill \\ \hfill & +q_1{q}_2{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2{\left[{\overline{A}}_3\left(q_1\right)\right]}^{\frac{1}{r_1}}{\left[{\overline{A}}_2\left(q_2\right)\right]}^{\frac{1}{r_1}}\hfill \\ \hfill & \times \left[{\breve{G}}_1\left(q_1\right)\left({\breve{E}}_1\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+{\breve{E}}_2\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)\right.\hfill \\ \hfill & {\left.+{\breve{G}}_2\left(q_1\right)\left({\breve{E}}_1\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+{\breve{E}}_2\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)\right]}^{\frac{1}{p_1}}\hfill \\ \hfill & +q_1{q}_2{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2{\left[{\overline{A}}_3\left(q_1\right)\right]}^{\frac{1}{r_1}}{\left[{\overline{A}}_3\left(q_2\right)\right]}^{\frac{1}{r_1}}\hfill \\ \hfill & \times \left[{\breve{G}}_1\left(q_1\right)\left({\breve{G}}_1\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+{\breve{G}}_2\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)\right.\hfill \\ \hfill & {\left.+{\breve{G}}_2\left(q_1\right)\left({\breve{G}}_1\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+{\breve{G}}_2\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)\right]}^{\frac{1}{p_1}},\hfill \end{array}$$

For $i\in \left\{1,2\right\}$ we have

$\begin{array}{cccc}{\overline{A}}_1\left(q_i\right)=\frac{q_i^{3r_1}{\left[5\right]}_{q_i}^{r_1}}{{\left[3\right]}_{q_i}^{r_1+1}{\left[2\right]}_{q_i}^{r_1}},\hfill & {\check{D}}_1\left(q_i\right)=\frac{1}{{\left[3\right]}_{q_i}^2{\left[2\right]}_{q_i}},\hfill & {\check{D}}_2\left(q_i\right)=\frac{{\left[3\right]}_{q_i}{\left[2\right]}_{q_i}-1}{{\left[3\right]}_{q_i}^2{\left[2\right]}_{q_i}},\hfill & \\ {\overline{A}}_2\left(q_i\right)=\frac{q_i^{r_1}{\left[2\right]}_{q_i}-q_i^{2r_1}}{{\left[3\right]}_{q_i}^{r_1+1}{\left[2\right]}_{q_i}^{r_1}},\hfill & {\breve{E}}_1\left(q_i\right)=\frac{q_i^2+2}{{\left[3\right]}_{q_i}^2{\left[2\right]}_{q_i}},\hfill & {\breve{E}}_2\left(q_i\right)=\frac{q_2{\left[3\right]}_{q_i}{\left[2\right]}_{q_i}-\left(q_i^2+2q_i\right)}{{\left[3\right]}_{q_i}^2{\left[2\right]}_{q_i}},\hfill & \\ {\overline{A}}_3\left(q_i\right)={\displaystyle \frac{q_i^{7r_1}}{{\left[8\right]}_{q_i}^{r_1}}}-\frac{{\left[2\right]}_{q_i}\left({\left[7\right]}_{q_i}{\left[3\right]}_{q_i}-{\left[8\right]}_{q_i}{\left[2\right]}_{q_i}\right)}{{\left[8\right]}_{q_i}^{r_1}{\left[3\right]}_{q_i}^{r_1+1}},\hfill & {\breve{G}}_1\left(q_i\right)=\frac{{\left[3\right]}_{q_i}^2-{\left[2\right]}_{q_i}^2}{{\left[3\right]}_{q_i}^2{\left[2\right]}_{q_i}},\hfill & {\breve{G}}_2\left(q_i\right)=\frac{q_i^2{\left[3\right]}_{q_i}{\left[2\right]}_{q_i}+{\left[2\right]}_{q_i}^2-{\left[3\right]}_{q_i}^2}{{\left[3\right]}_{q_i}^2{\left[2\right]}_{q_i}}.\hfill & \\ & & \end{array}$ where $0<q_i<1.$

Theorem 8.

We assume that the conditions of Lemma 3 hold. If ${\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2+t{\acute{O}}_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}$, where $p_1>1$ and ${\displaystyle \frac{1}{r_1}}+{\displaystyle \frac{1}{p_1}}=1$, is a $generalizedconvexfunction$ and integrable on Δ, then we have the following inequality:

$$\begin{array}{cc}\hfill & \left|\frac{1}{{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2}{\int }_{{\pi }_2+{\acute{O}}_1}^{{\pi }_2}{\int }_{{\pi }_4+{\acute{O}}_2}^{{\pi }_4}\mathsf{Ϝ}\left(x,y\right){}^{{\pi }_2}{d}_{q_1}\left(x\right){}^{{\pi }_4}{d}_{q_2}\left(y\right)-\frac{1}{{\stackrel{`}{O}}_1{\left[8\right]}_{q_2}}{\int }_{{\pi }_2+{\acute{O}}_1}^{{\pi }_2}\right.\hfill \\ \hfill & \times \left[q_{{}_2}\mathsf{Ϝ}\left(x,{\pi }_4\right)+\frac{q_2^3{\left[6\right]}_{q_2}}{{\left[2\right]}_{q_2}}\mathsf{Ϝ}\left(x,{\pi }_4+\frac{{\acute{O}}_2}{{\left[3\right]}_{q_2}}\right)+\frac{q_2^2{\left[6\right]}_{q_2}}{{\left[2\right]}_{q_2}}\mathsf{Ϝ}\left(x,{\pi }_4+\frac{{\left[2\right]}_{q_2}{\acute{O}}_2}{{\left[3\right]}_{q_2}}\right)+\mathsf{Ϝ}\left(x,{\pi }_4+{\acute{O}}_2\right)\right]d_{q_1}\left(x\right)\hfill \\ \hfill & -\frac{1}{{\stackrel{`}{O}}_2{\left[8\right]}_{q_1}}{\int }_{{\pi }_4+{\acute{O}}_2}^{{\pi }_4}\left[\mathsf{Ϝ}\left({\pi }_2+{\acute{O}}_2,y\right)+\frac{q_1^3{\left[6\right]}_{q_1}}{{\left[2\right]}_{q_1}}\mathsf{Ϝ}\left({\pi }_2+\frac{{\acute{O}}_1}{{\left[3\right]}_{q_1}},y\right)+q_1\mathsf{Ϝ}\left({\pi }_2,y\right)\right.\hfill \\ \hfill & \left.+\frac{q_{{}_1}^2{\left[6\right]}_{q_1}}{{\left[2\right]}_{q_1}}\mathsf{Ϝ}\left({\pi }_2+\frac{{\left[2\right]}_{q_1}{\acute{O}}_1}{{\left[3\right]}_{q_1}},y\right)\right]{}^{{\pi }_4}{d}_{q_2}\left(y\right)+\frac{1}{{\left[8\right]}_{q_2}{\left[8\right]}_{q_1}}\left[\frac{q_{{}_2}^3{\left[6\right]}_{q_2}}{{\left[2\right]}_{q_2}}\mathsf{Ϝ}\left({\pi }_2,{\pi }_4+\frac{{\acute{O}}_2}{{\left[3\right]}_{q_2}}\right)+\frac{q_2^2{\left[6\right]}_{q_2}}{{\left[2\right]}_{q_2}}\right.\hfill \\ \hfill & \left.\times \mathsf{Ϝ}\left({\pi }_2,{\pi }_4+\frac{{\left[2\right]}_{q_2}{\acute{O}}_2}{{\left[3\right]}_{q_2}}\right)+q_2\mathsf{Ϝ}\left({\pi }_2,{\pi }_4\right)+\mathsf{Ϝ}\left({\pi }_2,{\pi }_4+{\acute{O}}_2\right)\right]+\frac{1}{{\left[8\right]}_{q_1}{\left[8\right]}_{q_2}}\left[\mathsf{Ϝ}\left({\pi }_2+{\acute{O}}_1,{\pi }_4\right)\right.\hfill \\ \hfill & \left.\left.+\frac{q_1^2{\left[6\right]}_{q1}}{{\left[2\right]}_{q_1}}\mathsf{Ϝ}\left({\pi }_2+\frac{{\left[2\right]}_{q_1}{\acute{O}}_1}{{\left[3\right]}_{q_1}},{\pi }_4\right)+\frac{q_{{}_1}^3{\left[6\right]}_{q_1}}{{\left[2\right]}_{q_1}}\mathsf{Ϝ}\left({\pi }_2+\frac{{\acute{O}}_1}{{\left[3\right]}_{q_1}},{\pi }_4\right)+\mathsf{Ϝ}\left({\pi }_2+{\acute{O}}_1,{\pi }_4+{\acute{O}}_2\right)\right]\right|\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill & \le q_1{q}_2{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2{\left({\widehat{S}}_1\left(q_1\right)\right)}^{1-\frac{1}{p_1}}{\left({\widehat{S}}_1\left(q_2\right)\right)}^{1-\frac{1}{p_1}}\left[A_3\left(q_1\right)\left(A_3\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+B_3{(q}_2)\right.\right.\hfill \\ \hfill & {\left.\left.\times {\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)+B_3{(q}_1)\left(A_3{(q}_2){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+B_3{(q}_2){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)\right]}^{\frac{1}{p_1}}\hfill \\ \hfill & +q_1{q}_2{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2{\left({\widehat{S}}_1\left(q_1\right)\right)}^{1-\frac{1}{p_1}}{\left({\widehat{S}}_2\left(q_2\right)\right)}^{1-\frac{1}{p_1}}\left[A_3{(q}_1)\left(A_4\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+B_4{(q}_2)\right.\right.\hfill \\ \hfill & {\left.\left.\times {\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)+B_3{(q}_1)\left(A_4{(q}_2){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+B_4{(q}_2){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)\right]}^{\frac{1}{p_1}}\hfill \\ \hfill & +q_1{q}_2{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2{\left({\widehat{S}}_1\left(q_1\right)\right)}^{1-\frac{1}{p_1}}{\left({\widehat{S}}_3\left(q_3\right)\right)}^{1-\frac{1}{p_1}}\left[A_3\left(q_1\right)\left(A_5\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+B_5{(q}_2)\right.\right.\hfill \\ \hfill & {\left.\left.\times {\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)+B_3{(q}_1)\left(A_5{(q}_2){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+B_5{(q}_2){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)\right]}^{\frac{1}{p_1}}\hfill \\ \hfill & +q_1{q}_2{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2{\left({\widehat{S}}_2\left(q_1\right)\right)}^{1-\frac{1}{p_1}}{\left({\widehat{S}}_1\left(q_2\right)\right)}^{1-\frac{1}{p_1}}\left[A_4\left(q_1\right)\left(A_3\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+B_3{(q}_2)\right.\right.\hfill \\ \hfill & {\left.\left.\times {\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)+B_4{(q}_1)\left(A_3{(q}_2){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+B_3{(q}_2){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)\right]}^{\frac{1}{p_1}}\hfill \\ \hfill & +q_1{q}_2{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2{\left({\widehat{S}}_2\left(q_1\right)\right)}^{1-\frac{1}{p_1}}{\left({\widehat{S}}_2\left(q_2\right)\right)}^{1-\frac{1}{p_1}}\left[A_4\left(q_1\right)\left(A_4\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+B_4{(q}_2)\right.\right.\hfill \\ \hfill & {\left.\left.\times {\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)+B_4{(q}_1)\left(A_4{(q}_2){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+B_4{(q}_2){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)\right]}^{\frac{1}{p_1}}\hfill \\ \hfill & +q_1{q}_2{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2{\left({\widehat{S}}_2\left(q_1\right)\right)}^{1-\frac{1}{p_1}}{\left({\widehat{S}}_3\left(q_2\right)\right)}^{1-\frac{1}{p_1}}\left[A_4\left(q_1\right)\left(A_5\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+B_5{(q}_2)\right.\right.\hfill \\ \hfill & {\left.\left.\times {\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)+B_4{(q}_1)\left(A_5{(q}_2){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+B_5{(q}_2){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)\right]}^{\frac{1}{p_1}}\hfill \\ \hfill & +q_1{q}_2{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2{\left({\widehat{S}}_3\left(q_1\right)\right)}^{1-\frac{1}{p_1}}{\left({\widehat{S}}_1\left(q_2\right)\right)}^{1-\frac{1}{p_1}}\left[A_5\left(q_1\right)\left(A_3\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+B_3{(q}_2)\right.\right.\hfill \\ \hfill & {\left.\left.\times {\left|\frac{{}^{{\pi }_2,d}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)+B_5{(q}_1)\left(A_3{(q}_2){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+B_3{(q}_2){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)\right]}^{\frac{1}{p_1}}\hfill \\ \hfill & +q_1{q}_2{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2{\left({\widehat{S}}_3\left(q_1\right)\right)}^{1-\frac{1}{p_1}}{\left({\widehat{S}}_1\left(q_2\right)\right)}^{1-\frac{1}{p_1}}\left[A_5\left(q_1\right)\left(A_4\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+B_4{(q}_2)\right.\right.\hfill \\ \hfill & {\left.\left.\times {\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)+B_5{(q}_1)\left(A_4{(q}_2){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+B_4{(q}_2){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)\right]}^{\frac{1}{p_1}}\hfill \\ \hfill & +q_1{q}_2{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2{\left({\widehat{S}}_3\left(q_1\right)\right)}^{1-\frac{1}{p_1}}{\left({\widehat{S}}_3\left(q_2\right)\right)}^{1-\frac{1}{p_1}}\left[A_5\left(q_1\right)\left(A_5\left(q_2\right){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+B_5{(q}_2)\right.\right.\hfill \end{array}$$

$$\begin{array}{cc}\hfill & {\left.\left.\times {\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)+B_5{(q}_1)\left(A_5{(q}_2){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}+B_5{(q}_2){\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|}^{p_1}\right)\right]}^{\frac{1}{p_1}}\hfill \end{array}$$

where for $iϵ\left\{1,2\right\},$ we have

$$\begin{array}{cc}{\widehat{S}}_1\left(q_i\right)={\displaystyle \frac{2q_i}{{\left[8\right]}_{q_i}^2{\left[2\right]}_{q_i}}}+{\displaystyle \frac{{\left[8\right]}_{q_i}-{\left[3\right]}_{q_i}{\left[2\right]}_{q_i}}{{\left[3\right]}_{q_i}^2{\left[2\right]}_{q_i}{\left[8\right]}_{q_i}}},\hfill & {\widehat{S}}_2\left(q_i\right)={\displaystyle \frac{2q_i}{{\left[2\right]}_{q_i}^3}}+{\displaystyle \frac{q_i}{{\left[3\right]}_{q_i}^2{\left[2\right]}_{q_i}}}+{\displaystyle \frac{1-{\left[3\right]}_{q_i}{\left[2\right]}_{q_i}}{{\left[3\right]}_{q_i}^2{\left[2\right]}_{q_i}}},\hfill \\ {\widehat{S}}_3\left(q_i\right)=2{\displaystyle \frac{q_i{\left[7\right]}_{q_i}^2}{{\left[8\right]}_{q_i}^2{\left[2\right]}_{q_i}}}+{\displaystyle \frac{{\left[3\right]}_{q_i}^2+{\left[2\right]}_{q_i}^2}{{\left[2\right]}_{q_i}{\left[3\right]}_{q_i}^2}}-{\displaystyle \frac{{\left[7\right]}_{q_i}\left({\left[3\right]}_{q_i}+{\left[2\right]}_{q_i}\right)}{{\left[8\right]}_{q_i}{\left[3\right]}_{q_i}}}.\hfill \end{array}$$

Proof. 

Following the same arguments used in proof of Theorem 6, and by using Lemma 3, we have the required inequality (Equation (44)). □

5. Applications

In this section, we discuss some applications of our main findings.

5.1. Applications to Hypergeometric Functions

Now, we derive some more inequalities pertaining to hypergeometric functions and Mittag–Leffler functions.

For Equation (1), if we set $\mu =1,\nu =0$, and $\sigma ($ķ$)=\left({\left(\alpha \right)}_{ķ}{\left(\beta \right)}_{ķ}\right)/\left({\left(\gamma \right)}_{ķ}\right)$ for ķ= 0, 1, 2, 3, …, where $\alpha$, $\beta$, and $\gamma$ are parameters that may be real or complex values, provided that $\gamma \ne 0,-1,-2,\dots$ and ${\left(m\right)}_{ķ}$ is defined as

$${\left(m\right)}_{ķ}=\frac{\Gamma (m+ķ)}{\Gamma \left(m\right)}=m(m+1)\dots (m+ķ+1),ķ=0,1,\dots ,$$

and restrict the domain to $\left|x\right|\le 1\left(\mathrm{with}\mathrm{x}\in \mathbb{C}\right)$, then we find the following classical hypergeometric functions [30,32]:

$${\check{R}}_{\mu ,\nu ,\sigma }\left(\check{z}\right)=\check{R}\left(\alpha ,\beta ;\gamma ;\check{z}\right)=\sum _{ķ=0}^{\infty }\frac{{\left(\alpha \right)}_{ķ}{\left(\beta \right)}_{ķ}}{\Gamma {\left(\gamma \right)}_{ķ}ķ!}{\check{z}}^{ķ},$$

(45)

Using these substitutions and the assumption in our Riana functions will transform them into hypergeometric functions, and thus Equation (15) will be replaced by follows:

$$\begin{array}{ccc}\check{R}(\alpha ,\beta ;\gamma ;{\pi }_1-{\pi }_2)={Ħ}_1,& & \check{R}(\alpha ,\beta ;\gamma ;{\pi }_2-{\pi }_1)={}_1Ħ\\ \check{R}(\alpha ,\beta ;\gamma ;{\pi }_3-{\pi }_4)={Ħ}_2,& & \check{R}(\alpha ,\beta ;\gamma ;{\pi }_4-{\pi }_3)={}_2Ħ\end{array}$$

Therefore, our Theorem 3 can be changed as follows:

Theorem 9.

We assume that the conditions of Lemma 2 hold. If $\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1{q}_2}^2\mathsf{Ϝ}({\pi }_2+t{\acute{O}}_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|$ is a generalized convex function and integrable on Δ, then the following inequality holds for the right quantum integrals:

$$\begin{array}{cc}\hfill & \left|\frac{1}{{\left[6\right]}_{q_1}{\left[6\right]}_{q_2}}\left[q_1^2{\left[4\right]}_{q_1}\mathsf{Ϝ}\left({\pi }_2+\frac{{Ħ}_1}{{\left[2\right]}_{q_1}},{\pi }_4+{Ħ}_2\right)+q_1^2{q}_2{\left[4\right]}_{q_1}\mathsf{Ϝ}\left({\pi }_2+\frac{{Ħ}_1}{{\left[2\right]}_{q_1}},{\pi }_4\right)+q_1^2{q}_2^2{\left[4\right]}_{q_1}{\left[4\right]}_{q_2}\mathsf{Ϝ}\left({\pi }_2+\frac{{Ħ}_1}{{\left[2\right]}_{q_1}},{\pi }_4+\frac{{Ħ}_2}{{\left[2\right]}_{q2}}\right)\right.\right.\hfill \\ \hfill & \left.+q_1{q}_2^2{\left[4\right]}_{q_2}\mathsf{Ϝ}\left({\pi }_2,{\pi }_4+\frac{{Ħ}_2}{{\left[2\right]}_{q_2}}\right)+q_2^2{\left[4\right]}_{q_2}\mathsf{Ϝ}\left({\pi }_2+{Ħ}_1,{\pi }_4+\frac{{Ħ}_2}{{\left[2\right]}_{q2}}\right)\right]+\frac{1}{{\left[6\right]}_{q_1}{\left[6\right]}_{q_2}}\left[q_1\mathsf{Ϝ}\left({\pi }_2,{\pi }_4+{Ħ}_2\right)+q_2\mathsf{Ϝ}\left({\pi }_2+{Ħ}_1,{\pi }_4\right)\right.\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \left.+\mathsf{Ϝ}\left({\pi }_2+{Ħ}_1,{\pi }_4+{Ħ}_2\right)+q_1{q}_2\mathsf{Ϝ}({\pi }_2,{\pi }_4)\right]-{\int }_{{\pi }_2+{Ħ}_1}^{{\pi }_2}\frac{\left[q_2\mathsf{Ϝ}\left(x,{\pi }_4\right)+\mathsf{Ϝ}\left(x,{\pi }_4+{Ħ}_2\right)+q_2^2{\left[4\right]}_{q_2}\mathsf{Ϝ}\left(x,{\pi }_4+\frac{{Ħ}_2}{{\left[2\right]}_{q_2}}\right)\right]}{{}_1Ħ{\left[6\right]}_{q_2}}{}^{{\pi }_2}{d}_{q_1}\left(x\right)\hfill \\ \hfill & \left.-{\int }_{{\pi }_4+{Ħ}_2}^{{\pi }_4}\frac{\left[q_1\mathsf{Ϝ}\left({\pi }_2,y\right)+q_1^2{\left[4\right]}_{q_1}\mathsf{Ϝ}\left({\pi }_2+\frac{{Ħ}_1}{{\left[2\right]}_{q1}},y\right)+\mathsf{Ϝ}({\pi }_2+{Ħ}_1,y)\right]}{{}_2Ħ{\left[6\right]}_{q_1}}{}^{{\pi }_4}{d}_{q_2}\left(y\right)+{\int }_{{\pi }_2+{Ħ}_1}^{{\pi }_2}{\int }_{{\pi }_4+{Ħ}_2}^{{\pi }_4}{\frac{\mathsf{Ϝ}(x,y)}{{}_1{Ħ}_2Ħ}}^{{\pi }_2}{d}_{q_1}{\left(x\right)}^{{\pi }_4}{d}_{q_2}\left(y\right)\right|\hfill \end{array}$$

$$\begin{array}{cc}\hfill & ⩽{}_1{Ħ}_2Ħq_1{q}_2\left[\left(A_1\left(q_1\right)+A_2\left(q_1\right)\right)\left(A_1\left(q_2\right)+A_2\left(q_2\right)\right)\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|\right.\hfill \\ \hfill & +\left(B_1\left(q_2\right)+B_2\left(q_2\right)\right)\left(A_1\left(q_1\right)+A_2\left(q_1\right)\right)\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|+\left(B_1\left(q_1\right)+B_2\left(q_1\right)\right)\left(A_1\left(q_2\right)+A_2\left(q_2\right)\right)\hfill \\ \hfill & \times \left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|\left.+\left(B_1\left(q_1\right)+B_2\left(q_1\right)\right)\left(B_1\left(q_2\right)+B_2\left(q_2\right)\right)\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|\right].\hfill \end{array}$$

Similarly, one can find all inequalities of Section 4.1 and Section 4.2 involving a hypergeometric function.

5.2. Applications to Mittag–Leffler Functions

Furthermore, by considering $\sigma =(1,1,1,\dots ),\nu =1$, and $\mu =\Phi$ with $\check{R}(\Phi )>0$ in Equation (1), we arrive at the widely recognized Mittag–Leffler function:

$${\check{R}}_{\Phi }\left(\check{z}\right)=\sum _{ķ=0}^{\infty }\frac{1}{\Gamma \left(1+\Phi ķ\right)}{\check{z}}^{ķ},$$

Therefore, Theorem 3 can be changed as follows:

$$\begin{array}{ccc}{\check{R}}_{\Phi }({\pi }_1-{\pi }_2)={Ļ}_1,& & {\check{R}}_{\Phi }({\pi }_2-{\pi }_1)={}_1Ļ,\\ {\check{R}}_{\Phi }({\pi }_3-{\pi }_4)={Ļ}_2,& & {\check{R}}_{\Phi }({\pi }_4-{\pi }_3)={}_2Ļ.\end{array}$$

Theorem 10.

We assume that the conditions of Lemma 2 hold. If $\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1{q}_2}^2\mathsf{Ϝ}({\pi }_2+t{\acute{O}}_1,{\pi }_4+s{\acute{O}}_2)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|$ is a $generalizedconvexfunction$ and integrable on Δ, then the following inequality holds for the right quantum integrals:

$$\begin{array}{cc}\hfill & \left|\frac{1}{{\left[6\right]}_{q_1}{\left[6\right]}_{q_2}}\left[q_1^2{\left[4\right]}_{q_1}\mathsf{Ϝ}\left({\pi }_2+\frac{{Ļ}_1}{{\left[2\right]}_{q_1}},{\pi }_4+{Ļ}_2\right)+q_1^2{q}_2{\left[4\right]}_{q_1}\mathsf{Ϝ}\left({\pi }_2+\frac{{Ļ}_1}{{\left[2\right]}_{q_1}},{\pi }_4\right)+q_1^2{q}_2^2{\left[4\right]}_{q_1}{\left[4\right]}_{q_2}\mathsf{Ϝ}\left({\pi }_2+\frac{{Ļ}_1}{{\left[2\right]}_{q_1}},{\pi }_4+\frac{{Ļ}_2}{{\left[2\right]}_{q2}}\right)\right.\right.\hfill \\ \hfill & \left.+q_1{q}_2^2{\left[4\right]}_{q_2}\mathsf{Ϝ}\left({\pi }_2,{\pi }_4+\frac{{Ļ}_2}{{\left[2\right]}_{q_2}}\right)+q_2^2{\left[4\right]}_{q_2}\mathsf{Ϝ}\left({\pi }_2+{Ļ}_1,{\pi }_4+\frac{{Ļ}_2}{{\left[2\right]}_{q2}}\right)\right]+\frac{1}{{\left[6\right]}_{q_1}{\left[6\right]}_{q_2}}\left[q_1\mathsf{Ϝ}\left({\pi }_2,{\pi }_4+{Ļ}_2\right)+q_2\mathsf{Ϝ}\left({\pi }_2+{Ļ}_1,{\pi }_4\right)\right.\hfill \\ \hfill & \left.+\mathsf{Ϝ}\left({\pi }_2+{Ļ}_1,{\pi }_4+{Ļ}_2\right)+q_1{q}_2\mathsf{Ϝ}({\pi }_2,{\pi }_4)\right]-{\int }_{{\pi }_2+{Ļ}_1}^{{\pi }_2}\frac{\left[q_2\mathsf{Ϝ}\left(x,{\pi }_4\right)+\mathsf{Ϝ}\left(x,{\pi }_4+{Ļ}_2\right)+q_2^2{\left[4\right]}_{q_2}\mathsf{Ϝ}\left(x,{\pi }_4+\frac{{Ļ}_2}{{\left[2\right]}_{q_2}}\right)\right]}{{}_1Ļ{\left[6\right]}_{q_2}}{}^{{\pi }_2}{d}_{q_1}\left(x\right)\hfill \\ \hfill & \left.-{\int }_{{\pi }_4+{Ļ}_2}^{{\pi }_4}\frac{\left[q_1\mathsf{Ϝ}\left({\pi }_2,y\right)+q_1^2{\left[4\right]}_{q_1}\mathsf{Ϝ}\left({\pi }_2+\frac{{Ļ}_1}{{\left[2\right]}_{q1}},y\right)+\mathsf{Ϝ}({\pi }_2+{Ļ}_1,y)\right]}{{\left[6\right]}_{q_1}{}_2Ļ}{}^{{\pi }_4}{d}_{q_2}\left(y\right)+{\int }_{{\pi }_2+{Ļ}_1}^{{\pi }_2}{\int }_{{\pi }_4+{Ļ}_2}^{{\pi }_4}{\frac{\mathsf{Ϝ}(x,y)}{{}_1Ļ{}_2Ļ}}^{{\pi }_2}{d}_{q_1}{\left(x\right)}^{{\pi }_4}{d}_{q_2}\left(y\right)\right|\hfill \\ \hfill & ⩽{}_1{Ļ}_2Ļq_1{q}_2\left[\left(A_1\left(q_1\right)+A_2\left(q_1\right)\right)\left(A_1\left(q_2\right)+A_2\left(q_2\right)\right)\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|\right.\hfill \\ \hfill & +\left(B_1\left(q_2\right)+B_2\left(q_2\right)\right)\left(A_1\left(q_1\right)+A_2\left(q_1\right)\right)\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|+\left(B_1\left(q_1\right)+B_2\left(q_1\right)\right)\left(A_1\left(q_2\right)+A_2\left(q_2\right)\right)\hfill \\ \hfill & \times \left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|\left.+\left(B_1\left(q_1\right)+B_2\left(q_1\right)\right)\left(B_1\left(q_2\right)+B_2\left(q_2\right)\right)\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|\right].\hfill \end{array}$$

Similarly, one can find all inequalities of Section 4.1 and Section 4.2 involving Mittag–Leffler Functions.

Applications to Bounded Functions

We assume that the given conditions hold true:

$$\begin{array}{ccc}\left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|\le M_1,& & \left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_1,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|\le M_2,\\ \left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_3)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|\le M_3,& & \left|\frac{{}^{{\pi }_2,{\pi }_4}{\partial }_{q_1,q_2}^2\mathsf{Ϝ}({\pi }_2,{\pi }_4)}{{}^{{\pi }_2}{\partial }_{q_1}{\left(t\right)}^{{\pi }_4}{\partial }_{q_2}\left(s\right)}\right|\le M_4.\end{array}$$

In addition, $M_1,M_2,M_3,M_4$$\le M,$ where $\mathsf{Ϝ}$ is a function that is twice $\left(q_1,q_2\right)$ differentiable and bounded by the positive real number M.

Proposition 1.

Under the conditions of Theorem 3, we have

$$\begin{array}{cc}\hfill & \left|\frac{1}{{\left[6\right]}_{q_1}{\left[6\right]}_{q_2}}\left[q_1^2{\left[4\right]}_{q_1}\mathsf{Ϝ}\left({\pi }_2+\frac{{\acute{O}}_1}{{\left[2\right]}_{q_1}},{\pi }_4+{\acute{O}}_2\right)+q_1^2{q}_2{\left[4\right]}_{q_1}\mathsf{Ϝ}\left({\pi }_2+\frac{{\acute{O}}_1}{{\left[2\right]}_{q_1}},{\pi }_4\right)+q_1^2{q}_2^2{\left[4\right]}_{q_1}{\left[4\right]}_{q_2}\right.\right.\hfill \\ \hfill & \left.\times \mathsf{Ϝ}\left({\pi }_2+\frac{{\acute{O}}_1}{{\left[2\right]}_{q_1}},{\pi }_4+\frac{{\acute{O}}_2}{{\left[2\right]}_{q2}}\right)+q_1{q}_2^2{\left[4\right]}_{q_2}\mathsf{Ϝ}\left({\pi }_2,{\pi }_4+\frac{{\acute{O}}_2}{{\left[2\right]}_{q_2}}\right)+q_2^2{\left[4\right]}_{q_2}\mathsf{Ϝ}\left({\pi }_2+{\acute{O}}_1,{\pi }_4+\frac{{\acute{O}}_2}{{\left[2\right]}_{q2}}\right)\right]\hfill \\ \hfill & +\frac{\left[q_1\mathsf{Ϝ}\left({\pi }_2,{\pi }_4+{\acute{O}}_2\right)+q_2\mathsf{Ϝ}\left({\pi }_2+{\acute{O}}_1,{\pi }_4\right)+\mathsf{Ϝ}\left({\pi }_2+{\acute{O}}_1,{\pi }_4+{\acute{O}}_2\right)+q_1{q}_2\mathsf{Ϝ}({\pi }_2,{\pi }_4)\right]}{{\left[6\right]}_{q_1}{\left[6\right]}_{q_2}}\hfill \\ \hfill & -{\int }_{{\pi }_2+{\acute{O}}_1}^{{\pi }_2}\frac{q_2\mathsf{Ϝ}\left(x,{\pi }_4\right)+\mathsf{Ϝ}\left(x,{\pi }_4+{\acute{O}}_2\right)+q_2^2{\left[4\right]}_{q_2}\mathsf{Ϝ}\left(x,{\pi }_4+\frac{{\acute{O}}_2}{{\left[2\right]}_{q_2}}\right)}{{\left[6\right]}_{q_2}{\acute{O}}_1}{}^{{\pi }_2}{d}_{q_1}\left(x\right)\hfill \\ \hfill & -{\int }_{{\pi }_4+{\acute{O}}_2}^{{\pi }_4}\frac{q_1\mathsf{Ϝ}\left({\pi }_2,y\right)+q_1^2{\left[4\right]}_{q_1}\mathsf{Ϝ}\left({\pi }_2+\frac{{\acute{O}}_1}{{\left[2\right]}_{q1}},y\right)+\mathsf{Ϝ}({\pi }_2+{\acute{O}}_1,y)}{{\left[6\right]}_{q_1}{\stackrel{`}{O}}_2}{}^{{\pi }_4}{d}_{q_2}\left(y\right)\hfill \\ \hfill & \left.+\frac{1}{{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2}{\int }_{{\pi }_2+{\acute{O}}_1}^{{\pi }_2}{\int }_{{\pi }_4+{\acute{O}}_2}^{{\pi }_4}\mathsf{Ϝ}{(x,y)}^{{\pi }_2}{d}_{q_1}{\left(x\right)}^{{\pi }_4}{d}_{q_2}\left(y\right)\right|\hfill \\ \hfill & ⩽q_1{q}_2{\stackrel{`}{O}}_1{\stackrel{`}{O}}_2\left[\left(A_1\left(q_1\right)+A_2\left(q_1\right)\right)\left(A_1\left(q_2\right)+A_2\left(q_2\right)\right)M+\left(B_1\left(q_2\right)+B_2\left(q_2\right)\right)\left(A_1\left(q_1\right)+A_2\left(q_1\right)\right)M\right.\hfill \\ \hfill & +\left(B_1\left(q_1\right)+B_2\left(q_1\right)\right)\left(A_1\left(q_2\right)+A_2\left(q_2\right)\right)M\left.+\left(B_1\left(q_1\right)+B_2\left(q_1\right)\right)\left(B_1\left(q_2\right)+B_2\left(q_2\right)\right)M\right].\hfill \end{array}$$

6. Conclusions

The theory of inequalities has flourished at a fast rate in recent years due to its diverse range of applications. Undoubtedly, convexity and its general representation are the main tools for developing fundamental inequalities. Numerous inequalities involving convex functions have been retained in the literature. Various generalizations of Simpson-type inequalities have been proposed in the literature through quantum calculus, fractional calculus, and post-quantum calculus. To summarize, our work has established quantum integral identities and introduced new inequalities of the Simpson and Newton types for coordinated generalized convex functions in the context of right quantum integrals. By performing appropriate substitutions of $i\check{R}{{\mu }_i,{\nu }_i}^{{\sigma }_i}$ and using the limit $q_i\to 1$, where $i\in \{1,2\}$, we also obtained additional results. These coordinated formulations of novel and intriguing problems can prove useful for future research on identifying analogous inequalities for diverse forms of generalized convexity, which have numerous applications in various mathematical fields. In the future, we will develop some new variants of integral inequalities involving new generalizations of quantum calculus, fractional calculus, and fractional quantum calculus involving different kinds of convexity. We hope that this paper will open a new arena for research.