Some New Newton’s Type Integral Inequalities for Co-Ordinated Convex Functions in Quantum Calculus

Abstract

Some recent results have been found treating the famous Simpson’s rule in connection with the convexity property of functions and those called generalized convex. The purpose of this article is to address Newton-type integral inequalities by associating with them certain criteria of quantum calculus and the convexity of the functions of various variables. In this article, by using the concept of recently defined $q_1{q}_2$ -derivatives and integrals, some of Newton’s type inequalities for co-ordinated convex functions are revealed. We also employ the limits of $q_1,q_2\to 1^{-}$ in new results, and attain some new inequalities of Newton’s type for co-ordinated convex functions through ordinary integral. Finally, we provide a thorough application of the newly obtained key outcomes, these new consequences can be useful in the integral approximation study for symmetrical functions, or with some kind of symmetry.

1. Introduction

For numerical integration and approximations of definite integrals, Simpson’s rules are well-known techniques. Thomas Simpson (1710–1761) was the founder of these known techniques. These techniques are also called Kepler’s rules because Johannes Kepler used similar techniques for numerical integration about 100 years ago. Simpson’s rule contains the three-point Newton-Cotes quadrature rule, so estimates depended on the three-step quadratic core are sometimes called Newton-type results.

(1)

Simpson’s quadrature formula (Simpson’s $1/3$ rule)

$$\frac{1}{\epsilon -\alpha }{\int }_{\alpha }^{\epsilon }F\left(s\right)ds\approx \frac{1}{6}\left[F\left(\alpha \right)+4F\left(\frac{\alpha +\epsilon }{2}\right)+F\left(\epsilon \right)\right].$$

(2)

Simpson’s second formula or Newton–Cotes quadrature formula (Simpson’s $3/8$ rule).

$$\frac{1}{\epsilon -\alpha }{\int }_{\alpha }^{\epsilon }F\left(s\right)ds\approx \frac{1}{8}\left[F\left(\alpha \right)+3F\left(\frac{2\alpha +\epsilon }{3}\right)+3F\left(\frac{\alpha +2\epsilon }{3}\right)+F\left(\epsilon \right)\right].$$

There are numerous estimations correlated to these quadrature rules in the literature, one of them is the subsequent estimation identified as Simpson’s inequality:

Theorem 1.

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

$$\left|\frac{1}{3}\left[\frac{F\left(\alpha \right)+F\left(\epsilon \right)}{2}+2F\left(\frac{\alpha +\epsilon }{2}\right)\right]-\frac{1}{\epsilon -\alpha }{\int }_{\alpha }^{\epsilon }F\left(s\right)ds\right|\le \frac{1}{2880}{∥F^{\left(4\right)}∥}_{\infty }{\left(\epsilon -\alpha \right)}^4.$$

In recent years, especially over the past two decades, several authors have been engaged in the study of inequalities, including the Simpson’s various function classes (Symmetric or Asymmetric). Particularly, some mathematicians have dedicated the most to the study of Simpson and Newton-type consequences for functions with several kinds of generalized convexity, given that the theory of convexity it is an appropriate way to solve a huge number of problems appearing in different areas and subareas of applied and pure mathematics. For an instance, In Reference [1], Dragomir et al. proved some new inequalities of Simpson’s type and gave some application of numerical integration using the obtained results. In Reference [2], Alomari et al. used the notion of s–convexity and proved some new inequalities of Simpson’s type with the application of numerical integration. Afterward, Sarikaya et al. observed the variants of Simpson’s type inequalities based on convexity in Reference [3]. On the other hand, Özdemir et al. used the concept of co–ordinated convexity and proved Simpson’s type inequalities for double integrals in Reference [4]. In Reference [5,6], the authors utilized the concept of harmonic and p–harmonic convexities and gave some Newton-type inequalities. Moreover, Iftikhar et al. in Reference [7] proved some new inequalities of Newton’s type for the functions whose local fractional derivatives are generalized convex.

On the other hand, quantum calculus or q–calculus is sometimes referred to as calculus without limits. In this, we gain q–analogs of mathematical items that maybe got back as $q\to 1$. The Nalli-Ward-Al-Salam q–addition (NWA) and the Jackson-Hahn-Cigler q-addition (JHC) are two kinds of q-addition in this subject. The first one is commutative and associative, but at the same time, the second one is not. That’s why from time to time several q–analogs exist. These operators form the basis of the method which associations hypergeometric collection and q–hypergeometric collection and which gives numerous formulations of q–calculus in a usual form. The history of quantum calculus may be traced reverse to Euler $(1707-1783),$ who first added the q in the tracks of Newton’s infinite series. In recent decades, numerous researchers have revealed a keen hobby in investigating quantum calculus accordingly it emerges as an interdisciplinary subject. This is, of course, the quantum analysis is extremely useful in numerous fields and has vast applications in different areas of natural sciences such as computer science and particle physics and furthermore acts as a vital tool for researchers working with analytic number theory or in theoretical physics. Quantum calculus can be considered as a link between Mathematics and Physics. Several scientists who employ quantum calculus are physicists, as quantum calculus has numerous applications in quantum group theory. For some recent consequences in quantum calculus concerned readers are referred to References [8,9,10,11,12,13,14].

In recent years, because of the importance of convexity in numerous fields of applied and pure mathematics, it has been significantly investigated. The theory of convexity and inequalities are strongly connected to each other, therefore various inequalities can be established inside the literature which are proved for convex, generalized convex and differentiable convex functions of single and double variables, see References [15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32].

Inspired by these ongoing studies, we establish some new quantum analogs of Newton’s inequalities for q–differentiable co-ordinated convex functions. This is the primary motivation of this paper. The ideas and strategies of the paper may open new venues for the further research in this field.

Throughout in this paper the real numbers $q,q_1,q_2\in \left(0,1\right)$.

2. Preliminaries of q–Calculus and Some Inequalities

In this section we present some required definitions and related inequalities about q–calculus. For more information about q–calculus, one can refer to References [9,10,11].

Definition 1

([13]). For a continuous function $F:\left[a,\epsilon \right]\to \mathbb{R},$ the q–derivative of F at $s\in \left[a,\epsilon \right]$ is characterized by the expression

$$\begin{array}{c}{}_a{d}_{q}F\left(s\right)\end{array}=\frac{F\left(s\right)-F\left(qs+\left(1-q\right)a\right)}{\left(1-q\right)\left(s-a\right)},s\ne a.$$

(1)

The function F is said to be q–differentiable on $\left[a,\epsilon \right]$ if $\begin{array}{c}{}_a{d}_{q}F\left(s\right)\end{array}$ exists for all $s\in \left[a,\epsilon \right]$. If $a=0$ in (1), then $\begin{array}{c}{}_0{d}_{q}F\left(s\right)=d_{q}F\left(s\right)\end{array}$, where $\begin{array}{c}{d}_{q}F\left(s\right)\end{array}$ is the familiar q-derivative of F at $s\in \left[a,\epsilon \right]$ defined by the expression (see Reference [12]):

$$\begin{array}{c}{d}_{q}F\left(s\right)\end{array}=\frac{F\left(s\right)-F\left(qs\right)}{\left(1-q\right)s},s\ne 0.$$

(2)

Definition 2

([13]). Let $F:\left[a,\epsilon \right]\to \mathbb{R}$ be a continuous function. Then, the $q_a$-definite integral on $\left[a,\epsilon \right]$ is defined as

$$\underset{a}{\overset{s}{\int }}F\left(t\right)\begin{array}{c}{}_a{d}_{q}t\end{array}=\left(1-q\right)\left(s-a\right)\sum _{n=0}^{\infty }{q}^{n}F\left(q^{n}s+\left(1-q^n\right)a\right)$$

(3)

for $s\in \left[a,\epsilon \right]$.

We have to give the following notation which will be used many times in the next sections (see Reference [12]):

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

Moreover, we will need the following lemma in our main results:

Lemma 1

([33]). For $\alpha \in \mathbb{R}\setminus \left\{-1\right\}$ the following equality holds:

$$\underset{a}{\overset{\epsilon }{\int }}{\left(s-a\right)}^{\alpha }\begin{array}{c}{}_a{d}_{q}s\end{array}=\frac{{\left(\epsilon -a\right)}^{\alpha +1}}{{\left[\alpha +1\right]}_q}.$$

On the other hand, Bermudo et al. gave the following new definition and related Hermite– Hadamard type inequalities:

Definition 3

([17]). Let $F:\left[a,\epsilon \right]\to \mathbb{R}$ be a continuous function. Then, the $q^{\epsilon }$-definite integral on $\left[a,\epsilon \right]$ is defined as

$$\underset{s}{\overset{\epsilon }{\int }}F\left(t\right)\begin{array}{c}{}^{\epsilon }{d}_{q}t\end{array}=\left(1-q\right)\left(\epsilon -s\right)\sum _{n=0}^{\infty }{q}^{n}F\left(q^{n}s+\left(1-q^n\right)\epsilon \right),$$

for all $s\in \left[a,\epsilon \right]$.

Theorem 2

([17]). If $F:\left[a,\epsilon \right]\to \mathbb{R}$ is a convex differentiable function on $\left[a,\epsilon \right]$ and $0<q<1$. Then, we have the q-Hermite–Hadamard inequalities

$$F\left(\frac{a+q\epsilon }{{\left[2\right]}_q}\right)\le \frac{1}{\epsilon -a}\underset{a}{\overset{\epsilon }{\int }}F\left(s\right)\begin{array}{c}{}^{\epsilon }{d}_{q}s\hfill \end{array}\le \frac{F\left(a\right)+qF\left(\epsilon \right)}{{\left[2\right]}_q}.$$

(4)

In Reference [20], Latif defined $q_{ac}$-integral and partial q-derivatives for two variables functions as follows:

Definition 4.

Suppose that $F:\left[a,\epsilon \right]\times \left[c,d\right]\subseteq {\mathbb{R}}^2\to \mathbb{R}$ is continuous function. Then, the definite $q_{ac}$-integral on $\left[a,\epsilon \right]\times \left[c,d\right]$ is defined by

$$\begin{array}{ccccc}\hfill \underset{a}{\overset{s}{\int }}\underset{c}{\overset{y}{\int }}F\left(t,s\right)& {}_c{d}_{q_2}s& {}_a{d}_{q_1}t& =& \left(1-q_1\right)\left(1-q_2\right)\left(s-a\right)\left(y-c\right)\hfill \\ & & & & \times \sum _{n=0}^{\infty }\sum _{m=0}^{\infty }{q}_1{q}_{2}F\left(q_1^{n}s+\left(1-q_1^n\right)a,q_2^{m}y+\left(1-q_2^m\right)c\right),\hfill \end{array}$$

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

Lemma 2

([18]). If the assumptions of Definition 4 holds, then

$$\begin{array}{ccc}\hfill {\int }_{y_1}^y{\int }_{s_1}^{s}F\left(t,s\right){}_a{d}_{q_1}t{}_c{d}_{q_2}s& =& {\int }_{y_1}^y{\int }_a^{s}F\left(t,s\right){}_a{d}_{q_1}t{}_c{d}_{q_2}s-{\int }_{y_1}^y{\int }_a^{s_1}F\left(t,s\right){}_a{d}_{q_1}t{}_c{d}_{q_2}s\hfill \\ & =& {\int }_c^y{\int }_a^{s}F\left(t,s\right){}_a{d}_{q_1}t{}_c{d}_{q_2}s-{\int }_c^{y_1}{\int }_a^{s}F\left(t,s\right){}_a{d}_{q_1}t{}_c{d}_{q_2}s\hfill \\ & & -{\int }_c^y{\int }_a^{s_1}F\left(t,s\right){}_a{d}_{q_1}t{}_c{d}_{q_2}s+{\int }_c^{y_1}{\int }_a^{s_1}F\left(t,s\right){}_a{d}_{q_1}t{}_c{d}_{q_2}s.\hfill \end{array}$$

Definition 5

([20]). Let $F:\left[a,b\right]\times \left[c,d\right]\subseteq {\mathbb{R}}^2\to \mathbb{R}$ be a continuous function of two variables. Then the partial $q_1$-derivatives, $q_2$-derivatives and $q_1{q}_2$-derivatives at $\left(s,y\right)\in \left[a,b\right]\times \left[c,d\right]$ can be given as follows:

$$\begin{array}{ccc}\hfill \frac{{}_a{\partial }_{q_1}F\left(s,y\right)}{{}_a{\partial }_{q_1}s}& =& \frac{F\left(q_{1}s+\left(1-q_1\right)a,y\right)-F\left(s,y\right)}{\left(1-q_1\right)\left(s-a\right)},s\ne b;\hfill \\ \hfill \frac{{}_c{\partial }_{q_1}F\left(s,y\right)}{{}_c{\partial }_{q_2}y}& =& \frac{F\left(s,q_{2}y+\left(1-q_2\right)c\right)-F\left(s,y\right)}{\left(1-q_2\right)\left(y-c\right)},y\ne c;\hfill \\ \hfill \frac{{}_{a,c}{\partial }_{q_1,q_2}^{2}F\left(s,y\right)}{{}_a{\partial }_{q_1}s{}_c{\partial }_{q_2}y}& =& \frac{1}{\left(s-a\right)\left(y-c\right)\left(1-q_1\right)\left(1-q_2\right)}\left[F\left(q_{1}s+\left(1-q_1\right)a,q_{2}y+\left(1-q_2\right)c\right)\right.\hfill \\ & & \left.-F\left(q_{1}s+\left(1-q_1\right)a,y\right)-F\left(s,q_{2}y+\left(1-q_2\right)c\right)+F\left(s,y\right)\right],s\ne a,y\ne c.\hfill \end{array}$$

For more details related to q-integrals and derivatives for the functions of two variables one can see Reference [20].

On the other hand, Budak et al. gave the following definitions of $q_a^d$, $q_b^c$ and $q^{bd}$ integrals and related inequalities of Hermite–Hadamard type:

Definition 6

([34]). Suppose that $F:\left[a,b\right]\times \left[c,d\right]\subseteq {\mathbb{R}}^2\to \mathbb{R}$ is continuous function. Then the following $q_a^d,$$q_c^b$ and $q^{bd}$ integrals on $\left[a,b\right]\times \left[c,d\right]$ are defined by

$$\begin{array}{cc}\underset{a}{\overset{s}{\int }}\underset{y}{\overset{d}{\int }}F{\left(t,s\right)}^d{d}_{q_2}{s}_a{d}_{q_1}t=\hfill & \left(1-q_1\right)\left(1-q_2\right)\left(s-a\right)\left(d-y\right)\hfill \\ \hfill & \times {\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \sum _{m=0}^{\infty }}{q}_1{q}_{2}F\left(q_1^{n}s+\left(1-q_1^n\right)a,q_2^{m}y+\left(1-q_2^m\right)d\right),\hfill \end{array}$$

(5)

$$\begin{array}{ccccc}\hfill \underset{s}{\overset{b}{\int }}\underset{c}{\overset{y}{\int }}F\left(t,s\right)& {}_c{d}_{q_2}s& {}^b{d}_{q_1}t& =\hfill & \left(1-q_1\right)\left(1-q_2\right)\left(b-s\right)\left(y-c\right)\hfill \\ & & & & \times \sum _{n=0}^{\infty }\sum _{m=0}^{\infty }{q}_1{q}_{2}F\left(q_1^{n}s+\left(1-q_1^n\right)b,q_2^{m}y+\left(1-q_2^m\right)c\right)\hfill \end{array}$$

(6)

and

$$\begin{array}{ccccc}\hfill \underset{s}{\overset{b}{\int }}\underset{y}{\overset{d}{\int }}F\left(t,s\right)& {}^d{d}_{q_2}s& {}^b{d}_{q_1}t& =\hfill & \left(1-q_1\right)\left(1-q_2\right)\left(b-s\right)\left(d-y\right)\hfill \\ & & & & \times \sum _{n=0}^{\infty }\sum _{m=0}^{\infty }{q}_1{q}_{2}F\left(q_1^{n}s+\left(1-q_1^n\right)b,q_2^{m}y+\left(1-q_2^m\right)d\right),\hfill \end{array}$$

(7)

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

Theorem 3

([34]). Let $F:\left[a,b\right]\times \left[c,d\right]\subset {\mathbb{R}}^2\to \mathbb{R}$ be a coordinated convex function on $\left[a,b\right]\times \left[c,d\right]$. Then we have the following inequalities:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & F\left(\frac{q_{1}a+b}{{\left[2\right]}_{q_1}},\frac{c+q_{2}d}{{\left[2\right]}_{q_2}}\right)\hfill \\ \hfill & \le \frac{1}{2}\left[\frac{1}{b-a}\underset{a}{\overset{b}{\int }}F\left(s,\frac{c+q_{2}d}{{\left[2\right]}_{q_2}}\right)\begin{array}{c}{}_a{d}_{q_1}s\hfill \end{array}+\frac{1}{d-c}\underset{c}{\overset{d}{\int }}F\left(\frac{q_{1}a+b}{{\left[2\right]}_{q_1}},y\right)\begin{array}{c}{}^d{d}_{q_2}y\hfill \end{array}\right]\hfill \\ \hfill & \le \frac{1}{\left(b-a\right)\left(d-c\right)}\underset{a}{\overset{b}{\int }}\underset{c}{\overset{d}{\int }}F\left(s,y\right)\begin{array}{c}{}^d{d}_{q_2}y\hfill \end{array}\begin{array}{c}{}_a{d}_{q_1}s\hfill \end{array}\hfill \\ \hfill & \le \frac{q_1}{2{\left[2\right]}_{q_1}\left(d-c\right)}\underset{c}{\overset{d}{\int }}F\left(a,y\right)\begin{array}{c}{}^d{d}_{q_2}y\hfill \end{array}+\frac{1}{2{\left[2\right]}_{q_1}\left(d-c\right)}\underset{c}{\overset{d}{\int }}F\left(b,y\right)\begin{array}{c}{}^d{d}_{q_2}y\hfill \end{array}\hfill \\ \hfill & +\frac{1}{2{\left[2\right]}_{q_2}\left(b-a\right)}\underset{a}{\overset{b}{\int }}F\left(s,c\right)\begin{array}{c}{}_a{d}_{q_1}s\hfill \end{array}+\frac{q_2}{2{\left[2\right]}_{q_2}\left(b-a\right)}\underset{a}{\overset{b}{\int }}F\left(s,d\right)\begin{array}{c}{}_a{d}_{q_1}s\hfill \end{array}\hfill \\ \hfill & \le \frac{q_{1}F\left(a,c\right)+q_1{q}_{2}F\left(a,d\right)+F\left(b,c\right)+q_{2}F\left(b,d\right)}{{\left[2\right]}_{q_1}{\left[2\right]}_{q_2}}\hfill \end{array}\end{array}$$

(8)

for all $q_1,q_2\in \left(0,1\right).$

Theorem 4

([34]). Let $F:\left[a,b\right]\times \left[c,d\right]\subset {\mathbb{R}}^2\to \mathbb{R}$ be a coordinated convex function on $\left[a,b\right]\times \left[c,d\right]$. Then we have the following inequalities:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & F\left(\frac{a+q_{1}b}{{\left[2\right]}_{q_1}},\frac{q_{2}c+d}{{\left[2\right]}_{q_2}}\right)\hfill \\ \hfill & \le \frac{1}{2}\left[\frac{1}{b-a}\underset{a}{\overset{b}{\int }}F\left(x,\frac{q_{2}c+d}{{\left[2\right]}_{q_2}}\right)\begin{array}{c}{}^b{d}_{q_1}x\hfill \end{array}+\frac{1}{d-c}\underset{c}{\overset{d}{\int }}F\left(\frac{a+q_{1}b}{{\left[2\right]}_{q_1}},y\right)\begin{array}{c}{}_c{d}_{q_2}y\hfill \end{array}\right]\hfill \\ \hfill & \le \frac{1}{\left(b-a\right)\left(d-c\right)}\underset{a}{\overset{b}{\int }}\underset{c}{\overset{d}{\int }}F\left(x,y\right)\begin{array}{c}{}_c{d}_{q_2}y\hfill \end{array}\begin{array}{c}{}^b{d}_{q_1}x\hfill \end{array}\hfill \\ \hfill & \le \frac{1}{2{\left[2\right]}_{q_1}\left(d-c\right)}\underset{c}{\overset{d}{\int }}F\left(a,y\right)\begin{array}{c}{}_c{d}_{q_2}y\hfill \end{array}+\frac{q_1}{2{\left[2\right]}_{q_1}\left(d-c\right)}\underset{c}{\overset{d}{\int }}F\left(b,y\right)\begin{array}{c}{}_c{d}_{q_2}y\hfill \end{array}\hfill \\ \hfill & +\frac{q_2}{2{\left[2\right]}_{q_2}\left(b-a\right)}\underset{a}{\overset{b}{\int }}F\left(x,c\right)\begin{array}{c}{}^b{d}_{q_1}x\hfill \end{array}+\frac{1}{2{\left[2\right]}_{q_2}\left(b-a\right)}\underset{a}{\overset{b}{\int }}F\left(x,d\right)\begin{array}{c}{}^b{d}_{q_1}x\hfill \end{array}\hfill \\ \hfill & \le \frac{q_{2}F\left(a,c\right)+F\left(a,d\right)+q_1{q}_{2}F\left(b,c\right)+q_{1}F\left(b,d\right)}{{\left[2\right]}_{q_1}{\left[2\right]}_{q_2}}\hfill \end{array}\end{array}$$

(9)

for all $q_1,q_2\in \left(0,1\right).$

Theorem 5

([34]). Let $F:\left[a,b\right]\times \left[c,d\right]\subset {\mathbb{R}}^2\to \mathbb{R}$ be a coordinated convex function on $\left[a,b\right]\times \left[c,d\right]$. Then we have the following inequalities:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & F\left(\frac{a+q_{1}b}{{\left[2\right]}_{q_1}},\frac{c+q_{2}d}{{\left[2\right]}_{q_2}}\right)\hfill \\ \hfill & \le \frac{1}{2}\left[\frac{1}{b-a}\underset{a}{\overset{b}{\int }}F\left(x,\frac{c+q_{2}d}{{\left[2\right]}_{q_2}}\right)\begin{array}{c}{}^b{d}_{q_1}x\hfill \end{array}+\frac{1}{d-c}\underset{c}{\overset{d}{\int }}F\left(\frac{a+q_{1}b}{{\left[2\right]}_{q_1}}y,\right)\begin{array}{c}{}^d{d}_{q_2}y\hfill \end{array}\right]\hfill \\ \hfill & \le \frac{1}{\left(b-a\right)\left(d-c\right)}\underset{a}{\overset{b}{\int }}\underset{c}{\overset{d}{\int }}F\left(x,y\right)\begin{array}{c}{}^d{d}_{q_2}y\hfill \end{array}\begin{array}{c}{}^b{d}_{q_1}x\hfill \end{array}\hfill \\ \hfill & \le \frac{1}{2{\left[2\right]}_{q_1}\left(d-c\right)}\underset{c}{\overset{d}{\int }}F\left(a,y\right)\begin{array}{c}{}^d{d}_{q_2}y\hfill \end{array}+\frac{q_1}{2{\left[2\right]}_{q_1}\left(d-c\right)}\underset{c}{\overset{d}{\int }}F\left(b,y\right)\begin{array}{c}{}^d{d}_{q_2}y\hfill \end{array}\hfill \\ \hfill & +\frac{1}{2{\left[2\right]}_{q_2}\left(b-a\right)}\underset{a}{\overset{b}{\int }}F\left(x,c\right)\begin{array}{c}{}^d{d}_{q_2}y\hfill \end{array}+\frac{q_2}{2{\left[2\right]}_{q_2}\left(b-a\right)}\underset{a}{\overset{b}{\int }}F\left(x,d\right)\begin{array}{c}{}^b{d}_{q_1}x\hfill \end{array}\hfill \\ \hfill & \le \frac{F\left(a,c\right)+q_{2}F\left(a,d\right)+q_{1}F\left(b,c\right)+q_1{q}_{2}F\left(b,d\right)}{{\left[2\right]}_{q_1}{\left[2\right]}_{q_2}}\hfill \end{array}\end{array}$$

(10)

for all $q_1,q_2\in \left(0,1\right).$

Theorem 6.

($q_1{q}_2$-Hölder’s inequality for two variables functions, [20]). Let $x,y>0,$$0<q_1,q_2<1,$$p_1>1$ such that $\frac{1}{p_1}+\frac{1}{r_1}=1.$ Then

$${\int }_0^x{\int }_0^y\left|F\left(x,y\right)G\left(x,y\right)\right|d_{q_1}x{d}_{q_2}y\le {\left({\int }_0^x{\int }_0^y{\left|F\left(x,y\right)\right|}^{p_1}{d}_{q_1}x{d}_{q_2}y\right)}^{\frac{1}{p_1}}{\left({\int }_0^x{\int }_0^y{\left|G\left(x,y\right)\right|}^{r_1}{d}_{q_1}x{d}_{q_2}y\right)}^{\frac{1}{r_1}}.$$

3. New q–Derivatives for the Functions of Two Variables

In this section, some new partial q-derivatives for functions of two variables are given.

Definition 7

([35]). Let $F:\left[a,b\right]\times \left[c,d\right]\subseteq {\mathbb{R}}^2\to \mathbb{R}$ be a continuous function of two variables. Then the partial $q_1$-derivatives, $q_2$-derivatives and $q_1{q}_2$-derivatives at $\left(x,y\right)\in \left[a,b\right]\times \left[c,d\right]$ can be given as follows:

$$\begin{array}{ccc}\hfill \frac{{}^b{\partial }_{q_1}F\left(x,y\right)}{{}^b{\partial }_{q_1}x}& =& \frac{F\left(q_{1}x+\left(1-q_1\right)b,y\right)-F\left(x,y\right)}{\left(1-q_1\right)\left(b-x\right)},x\ne b;\hfill \\ \hfill \frac{{}^d{\partial }_{q_1}F\left(x,y\right)}{{}^b{\partial }_{q_2}y}& =& \frac{F\left(x,q_{2}y+\left(1-q_2\right)d\right)-F\left(x,y\right)}{\left(1-q_2\right)\left(d-y\right)},d\ne y;\hfill \\ \hfill \frac{{}_a^d{\partial }_{q_1,q_2}^{2}F\left(x,y\right)}{{}_a{\partial }_{q_1}x{}^d{\partial }_{q_2}y}& =& \frac{1}{\left(x-a\right)\left(d-y\right)\left(1-q_1\right)\left(1-q_2\right)}\left[F\left(q_{1}x+\left(1-q_1\right)a,q_{2}y+\left(1-q_2\right)d\right)\right.\hfill \\ & & \left.-F\left(q_{1}x+\left(1-q_1\right)a,y\right)-F\left(x,q_{2}y+\left(1-q_2\right)d\right)+F\left(x,y\right)\right],x\ne a,y\ne d;\hfill \\ \hfill \frac{{}_c^b{\partial }_{q_1,q_2}^{2}F\left(x,y\right)}{{}^b{\partial }_{q_1}{x}_c{\partial }_{q_2}y}& =& \frac{1}{\left(b-x\right)\left(y-c\right)\left(1-q_1\right)\left(1-q_2\right)}\left[F\left(q_{1}x+\left(1-q_1\right)b,q_{2}y+\left(1-q_2\right)c\right)\right.\hfill \\ & & \left.-F\left(q_{1}x+\left(1-q_1\right)b,y\right)-F\left(x,q_{2}y+\left(1-q_2\right)c\right)+F\left(x,y\right)\right],x\ne b,y\ne c;\hfill \\ \hfill \frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(x,y\right)}{{}^b{\partial }_{q_1}x{}_{}^d{\partial }_{q_2}y}& =& \frac{1}{\left(b-x\right)\left(d-y\right)\left(1-q_1\right)\left(1-q_2\right)}\left[F\left(q_{1}x+\left(1-q_1\right)b,q_{2}y+\left(1-q_2\right)d\right)\right.\hfill \\ & & \left.-F\left(q_{1}x+\left(1-q_1\right)b,y\right)-F\left(x,q_{2}y+\left(1-q_2\right)d\right)+F\left(x,y\right)\right],x\ne b,y\ne d.\hfill \end{array}$$

4. New Identity

We deal with an identity which is necessary to attain our main estimations in this section.

Let’s start with the following useful Lemma 3:

Lemma 3.

Let $F:\Delta \subseteq {\mathbb{R}}^2\to \mathbb{R}$ be a twice partially $q_1{q}_2$-differentiable function on ${\Delta }^{\circ }$. If the partial $q_1{q}_2$-derivative $\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(t,s\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}$ is continuous and integrable on $\left[a,b\right]\times \left[c,d\right]\subseteq {\Delta }^{\circ },$ then the following identity holds for $q_1{q}_2$-integrals:

$$\begin{array}{cc}& {}^{b,d}{\mathcal{I}}_{q_1,q_2}\hfill \\ =\hfill & \left(b-a\right)\left(d-c\right){\int }_0^1{\int }_0^1{\mathsf{\Lambda }}_{q_1}\left(t\right){\mathsf{\Lambda }}_{q_2}\left(s\right)\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}{d}_{q_1}t{d}_{q_2}s,\hfill \end{array}$$

(11)

where

$$\begin{array}{ccc}\hfill {}^{b,d}{\mathcal{I}}_{q_1,q_2}& =& \frac{3}{64}\left[F\left(\frac{a+2b}{3},d\right)+F\left(b,\frac{c+2d}{3}\right)+3F\left(\frac{a+2b}{3},\frac{c+2d}{3}\right)+3F\left(\frac{a+2b}{3},\frac{2c+d}{3}\right)\right.\hfill \\ & & +F\left(\frac{2a+b}{3},d\right)+3F\left(\frac{2a+b}{3},\frac{c+2d}{3}\right)+F\left(b,\frac{2c+d}{3}\right)+3F\left(\frac{2a+b}{3},\frac{2c+d}{3}\right)\hfill \\ & & \left.+F\left(\frac{a+2b}{3},c\right)+F\left(\frac{2a+b}{3},c\right)+F\left(a,\frac{c+2d}{3}\right)+F\left(a,\frac{2c+d}{3}\right)\right]\hfill \\ & & +\frac{F\left(a,c\right)+F\left(a,d\right)+F\left(b,c\right)+F\left(b,d\right)}{64}\hfill \\ & & -\frac{1}{8\left(b-a\right)}{\int }_a^b\left[F\left(x,c\right)+3F\left(x,\frac{c+2d}{3}\right)+3F\left(x,\frac{2c+d}{3}\right)+F\left(x,d\right)\right]{}^b{d}_{q_1}x\hfill \\ & & -\frac{1}{8\left(d-c\right)}{\int }_c^d\left[F\left(a,y\right)+3F\left(\frac{a+2b}{3},y\right)+3F\left(\frac{2a+b}{3},y\right)+F\left(b,y\right)\right]{}^d{d}_{q_2}y\hfill \\ & & +\frac{1}{\left(b-a\right)\left(d-c\right)}{\int }_a^b{\int }_c^{d}F\left(x,y\right){}^b{d}_{q_1}x{}^d{d}_{q_2}y\hfill \end{array}$$

and

$${\mathsf{\Lambda }}_{q_1}\left(t\right)=\left\{\begin{array}{c}{q}_{1}t-\frac{1}{8},t\in \left[0,\frac{1}{3}\right)\\ q_{1}t-\frac{1}{2},t\in \left[\frac{1}{3},\frac{2}{3}\right)\\ q_{1}t-\frac{7}{8},t\in \left[\frac{2}{3},1\right],\end{array}\right.$$

$${\mathsf{\Lambda }}_{q_2}\left(s\right)=\left\{\begin{array}{c}{q}_{2}s-\frac{1}{8},s\in \left[0,\frac{1}{3}\right)\\ q_{2}s-\frac{1}{2},s\in \left[\frac{1}{3},\frac{2}{3}\right)\\ q_{2}s-\frac{7}{8},s\in \left[\frac{2}{3},1\right].\end{array}\right.$$

Proof. 

Using Lemma 2, the definition of ${\mathsf{\Lambda }}_{q_1}\left(t\right)$ and ${\mathsf{\Lambda }}_{q_2}\left(s\right),$ it is easy to see that

$$\begin{array}{cc}& {\int }_0^1{\int }_0^1{\mathsf{\Lambda }}_{q_1}\left(t\right){\mathsf{\Lambda }}_{q_2}\left(s\right)\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}{d}_{q_1}t{d}_{q_2}s\hfill \\ =\hfill & \frac{9}{64}\left[{\int }_0^{\frac{1}{3}}{\int }_0^{\frac{1}{3}}\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}{d}_{q_1}t{d}_{q_2}s\right.\hfill \\ & +{\int }_0^{\frac{1}{3}}{\int }_0^{\frac{2}{3}}\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}{d}_{q_1}t{d}_{q_2}s\hfill \\ & +{\int }_0^{\frac{2}{3}}{\int }_0^{\frac{1}{3}}\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}{d}_{q_1}t{d}_{q_2}s\hfill \\ & \left.+{\int }_0^{\frac{2}{3}}{\int }_0^{\frac{2}{3}}\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}{d}_{q_1}t{d}_{q_2}s\right]\hfill \\ & +\frac{3}{8}\left[{\int }_0^{\frac{1}{3}}{\int }_0^1\left(q_{2}s-\frac{7}{8}\right)\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}{d}_{q_1}t{d}_{q_2}s\right.\hfill \\ & +{\int }_0^{\frac{2}{3}}{\int }_0^1\left(q_{2}s-\frac{7}{8}\right)\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}{d}_{q_1}t{d}_{q_2}s\hfill \\ & +{\int }_0^1{\int }_0^{\frac{1}{3}}\left(q_{1}t-\frac{7}{8}\right)\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}{d}_{q_1}t{d}_{q_2}s\hfill \\ & \left.+{\int }_0^1{\int }_0^{\frac{2}{3}}\left(q_{1}t-\frac{7}{8}\right)\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}{d}_{q_1}t{d}_{q_2}s\right]\hfill \\ & +{\int }_0^1{\int }_0^1\left(q_{1}t-\frac{7}{8}\right)\left(q_{1}t-\frac{7}{8}\right)\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}{d}_{q_1}t{d}_{q_2}s\hfill \\ =\hfill & I_1+I_2+I_3+I_4+I_5+I_6+I_7+I_8+I_9.\hfill \end{array}$$

(12)

From Definition 7, we have

$$\begin{array}{cc}& \frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\hfill \\ =& \frac{1}{\left(1-q_1\right)\left(1-q_2\right)\left(b-a\right)\left(d-c\right)ts}\left[F\left(t{q}_{1}a+\left(1-t{q}_1\right)b,s{q}_{2}c+\left(1-s{q}_2\right)d\right)\right.\hfill \\ & -F\left(t{q}_{1}a+\left(1-t{q}_1\right)b,sc+\left(1-s\right)d\right)-F\left(ta+\left(1-t\right)b,s{q}_{2}c+\left(1-s{q}_2\right)d\right)\hfill \\ & \left.+F\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)\right].\hfill \end{array}$$

We need to calculate the integrals in the right side of (12) in order to finish this proof. By using the definition of $q_1{q}_2$-integrals, we obtain that

$$\begin{array}{cc}& {\int }_0^{\frac{1}{3}}{\int }_0^{\frac{1}{3}}\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}{d}_{q_1}t{d}_{q_2}s\hfill \\ =\hfill & \frac{1}{\left(1-q_1\right)\left(1-q_2\right)\left(b-a\right)\left(d-c\right)}{\int }_0^{\frac{1}{3}}{\int }_0^{\frac{1}{3}}\frac{1}{ts}\left[F\left(t{q}_{1}a+\left(1-t{q}_1\right)b,s{q}_{2}c+\left(1-s{q}_2\right)d\right)\right.\hfill \\ & -F\left(t{q}_{1}a+\left(1-t{q}_1\right)b,sc+\left(1-s\right)d\right)-F\left(ta+\left(1-t\right)b,s{q}_{2}c+\left(1-s{q}_2\right)d\right)\hfill \\ & \left.+F\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)\right]d_{q_1}t{d}_{q_2}s\hfill \\ =\hfill & \frac{1}{\left(b-a\right)\left(d-c\right)}\left\{\sum _{n=0}^{\infty }\sum _{m=0}^{\infty }F\left(\frac{q_1^{n+1}}{3}a+\left(1-\frac{q_1^{n+1}}{3}\right)b,\frac{q_2^{m+1}}{3}c+\left(1-\frac{q_2^{m+1}}{3}\right)d\right)\right.\hfill \\ & -\sum _{n=0}^{\infty }\sum _{m=0}^{\infty }F\left(\frac{q_1^{n+1}}{3}a+\left(1-\frac{q_1^{n+1}}{3}\right)b,\frac{q_2^m}{3}c+\left(1-\frac{q_2^m}{3}\right)d\right)\hfill \\ & -\sum _{n=0}^{\infty }\sum _{m=0}^{\infty }F\left(\frac{q_1^n}{3}a+\left(1-\frac{q_1^n}{3}\right)b,\frac{q_2^{m+1}}{3}c+\left(1-\frac{q_2^{m+1}}{3}\right)d\right)\hfill \\ & +\sum _{n=0}^{\infty }\sum _{m=0}^{\infty }F\left(\frac{q_1^n}{3}a+\left(1-\frac{q_1^n}{3}\right)b,\frac{q_2^m}{3}c+\left(1-\frac{q_2^m}{3}\right)d\right)\hfill \\ =\hfill & \frac{1}{\left(b-a\right)\left(d-c\right)}\left[F\left(b,d\right)-F\left(\frac{a+2b}{3},d\right)-F\left(b,\frac{c+2d}{3}\right)+F\left(\frac{a+2b}{3},\frac{c+2d}{3}\right)\right].\hfill \end{array}$$

(13)

Similarly, we get

$$\begin{array}{cc}& {\int }_0^{\frac{1}{3}}{\int }_0^{\frac{2}{3}}\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}{d}_{q_1}t{d}_{q_2}s\hfill \\ =\hfill & \frac{1}{\left(b-a\right)\left(d-c\right)}\left[F\left(b,d\right)-F\left(\frac{a+2b}{3},d\right)-F\left(b,\frac{2c+d}{3}\right)+F\left(\frac{a+2b}{3},\frac{2c+d}{3}\right)\right],\hfill \end{array}$$

(14)

$$\begin{array}{cc}& {\int }_0^{\frac{2}{3}}{\int }_0^{\frac{1}{3}}\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}{d}_{q_1}t{d}_{q_2}s\hfill \\ =\hfill & \frac{1}{\left(b-a\right)\left(d-c\right)}\left[F\left(b,d\right)-F\left(\frac{2a+b}{3},d\right)-F\left(b,\frac{c+2d}{3}\right)+F\left(\frac{2a+b}{3},\frac{c+2d}{3}\right)\right],\hfill \end{array}$$

(15)

$$\begin{array}{cc}& {\int }_0^{\frac{2}{3}}{\int }_0^{\frac{2}{3}}\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}{d}_{q_1}t{d}_{q_2}s\hfill \\ =\hfill & \frac{1}{\left(b-a\right)\left(d-c\right)}\left[F\left(b,d\right)-F\left(\frac{2a+b}{3},d\right)-F\left(b,\frac{2c+d}{3}\right)+F\left(\frac{2a+b}{3},\frac{2c+d}{3}\right)\right],\hfill \end{array}$$

(16)

$$\begin{array}{cc}& {\int }_0^{\frac{1}{3}}{\int }_0^1\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}{d}_{q_1}t{d}_{q_2}s\hfill \\ =\hfill & \frac{1}{\left(b-a\right)\left(d-c\right)}\left[F\left(b,d\right)-F\left(\frac{a+2b}{3},d\right)-F\left(b,c\right)+F\left(\frac{a+2b}{3},c\right)\right],\hfill \end{array}$$

(17)

$$\begin{array}{cc}& {\int }_0^1{\int }_0^{\frac{1}{3}}\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}{d}_{q_1}t{d}_{q_2}s\hfill \\ =\hfill & \frac{1}{\left(b-a\right)\left(d-c\right)}\left[F\left(b,d\right)-F\left(b,\frac{c+2d}{3}\right)-F\left(a,d\right)+F\left(a,\frac{c+2d}{3}\right)\right],\hfill \end{array}$$

(18)

$$\begin{array}{cc}& {\int }_0^{\frac{2}{3}}{\int }_0^1\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}{d}_{q_1}t{d}_{q_2}s\hfill \\ =\hfill & \frac{1}{\left(b-a\right)\left(d-c\right)}\left[F\left(b,d\right)-F\left(\frac{2a+b}{3},d\right)-F\left(b,c\right)+F\left(\frac{2a+b}{3},c\right)\right],\hfill \end{array}$$

(19)

$$\begin{array}{cc}& {\int }_0^1{\int }_0^{\frac{2}{3}}\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}{d}_{q_1}t{d}_{q_2}s\hfill \\ =\hfill & \frac{1}{\left(b-a\right)\left(d-c\right)}\left[F\left(b,d\right)-F\left(a,d\right)-F\left(b,\frac{2c+d}{3}\right)+F\left(a,\frac{2c+d}{3}\right)\right],\hfill \end{array}$$

(20)

$$\begin{array}{cc}& {\int }_0^1{\int }_0^1\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}{d}_{q_1}t{d}_{q_2}s\hfill \\ =\hfill & \frac{1}{\left(b-a\right)\left(d-c\right)}\left[F\left(b,d\right)-F\left(a,d\right)-F\left(b,c\right)+F\left(a,c\right)\right].\hfill \end{array}$$

(21)

Now from Definition 6, we obtain the following

$$\begin{array}{cc}& {\int }_0^{\frac{1}{3}}{\int }_0^{1}s\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}{d}_{q_1}t{d}_{q_2}s\hfill \\ =\hfill & \frac{1}{\left(1-q_1\right)\left(1-q_2\right)\left(b-a\right)\left(d-c\right)}{\int }_0^{\frac{1}{3}}{\int }_0^1\frac{1}{t}\left[F\left(t{q}_{1}a+\left(1-t{q}_1\right)b,s{q}_{2}c+\left(1-s{q}_2\right)d\right)\right.\hfill \\ & -F\left(t{q}_{1}a+\left(1-t{q}_1\right)b,sc+\left(1-s\right)d\right)-F\left(ta+\left(1-t\right)b,s{q}_{2}c+\left(1-s{q}_2\right)d\right)\hfill \\ & \left.+F\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)\right]d_{q_1}t{d}_{q_2}s\hfill \end{array}$$

(22)

$$\begin{array}{cc}=& \frac{1}{\left(b-a\right)\left(d-c\right)}\left\{\sum _{n=0}^{\infty }\sum _{m=0}^{\infty }{q}_2^m\left(F\left(\frac{q_1^{n+1}}{3}a+\left(1-\frac{q_1^{n+1}}{3}\right)b,q_2^{m+1}c+\left(1-q_2^{m+1}\right)d\right)\right)\right.\hfill \\ & -\sum _{n=0}^{\infty }\sum _{m=0}^{\infty }{q}_2^m\left(F\left(\frac{q_1^{n+1}}{3}a+\left(1-\frac{q_1^{n+1}}{3}\right)b,q_2^{m}c+\left(1-q_2^m\right)d\right)\right)\hfill \\ & -\sum _{n=0}^{\infty }\sum _{m=0}^{\infty }{q}_2^m\left(F\left(\frac{q_1^n}{3}a+\left(1-\frac{q_1^n}{3}\right)b,q_2^{m+1}c+\left(1-q_2^{m+1}\right)d\right)\right)\hfill \\ & \left.+\sum _{n=0}^{\infty }\sum _{m=0}^{\infty }{q}_2^m\left(F\left(\frac{q_1^n}{3}a+\left(1-\frac{q_1^n}{3}\right)b,q_2^{m}c+\left(1-q_2^m\right)d\right)\right)\right\}\hfill \\ =& \frac{1}{\left(b-a\right)\left(d-c\right)}\left\{\sum _{m=0}^{\infty }{q}_2^m\left[\sum _{n=0}^{\infty }F\left(\frac{q_1^{n+1}}{3}a+\left(1-\frac{q_1^{n+1}}{3}\right)b,q_2^{m+1}c+\left(1-q_2^{m+1}\right)d\right)\right.\right.\hfill \\ & \left.-\sum _{n=0}^{\infty }\left(F\left(\frac{q_1^n}{3}a+\left(1-\frac{q_1^n}{3}\right)b,q_2^{m+1}c+\left(1-q_2^{m+1}\right)d\right)\right)\right]\hfill \\ & +\sum _{m=0}^{\infty }{q}_2^m\left[\sum _{n=0}^{\infty }F\left(\frac{q_1^n}{3}a+\left(1-\frac{q_1^n}{3}\right)b,q_2^{m}c+\left(1-q_2^m\right)d\right)\right.\hfill \\ & \left.-\left.\sum _{n=0}^{\infty }F\left(\frac{q_1^{n+1}}{3}a+\left(1-\frac{q_1^{n+1}}{3}\right)b,q_2^{m}c+\left(1-q_2^m\right)d\right)\right]\right\}\hfill \\ =& \frac{1}{\left(b-a\right)\left(d-c\right)}\left\{\sum _{m=0}^{\infty }{q}_2^{m}F\left(b,q_2^{m+1}c+\left(1-q_2^{m+1}\right)d\right)-\sum _{m=0}^{\infty }{q}_2^{m}F\left(b,q_2^{m}c+\left(1-q_2^m\right)d\right)\right.\hfill \\ & \left.+\sum _{m=0}^{\infty }{q}_2^{m}F\left(\frac{a+2b}{3},q_2^{m}c+\left(1-q_2^m\right)d\right)-\sum _{m=0}^{\infty }{q}_2^{m}F\left(\frac{a+2b}{3},q_2^{m+1}c+\left(1-q_2^{m+1}\right)d\right)\right\}\hfill \\ =& \frac{1}{\left(b-a\right)\left(d-c\right)}\left\{\frac{1-q_2}{q_2}\sum _{m=0}^{\infty }{q}_2^{m}F\left(b,q_2^{m}c+\left(1-q_2^m\right)d\right)-\frac{1}{q_2}F\left(b,c\right)\right.\hfill \\ & -\frac{1-q_2}{q_2}\sum _{m-0}^{\infty }{q}_2^{m}F\left(\frac{a+2b}{3},q_2^{m}c+\left(1-q_2^m\right)d\right)+\frac{1}{q_2}F\left(\frac{a+2b}{3},c\right)\hfill \\ =& \frac{1}{\left(b-a\right)\left(d-c\right)}\left[\frac{1}{q_2\left(d-c\right)}{\int }_c^{d}F\left(b,y\right){}^d{d}_{q_2}y-\frac{1}{q_2\left(d-c\right)}{\int }_c^{d}F\left(\frac{a+2b}{3},y\right){}^d{d}_{q_2}y\right.\hfill \\ & \left.-\frac{1}{q_2}F\left(b,c\right)+\frac{1}{q_2}F\left(\frac{a+2b}{3},c\right)\right].\hfill \end{array}$$

Similarly, we get

$$\begin{array}{cc}& {\int }_0^{\frac{2}{3}}{\int }_0^{1}s\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}{d}_{q_1}t{d}_{q_2}s\hfill \\ =\hfill & \frac{1}{\left(b-a\right)\left(d-c\right)}\left[\frac{1}{q_2\left(d-c\right)}{\int }_c^{d}F\left(b,y\right){}^d{d}_{q_2}y-\frac{1}{q_2\left(d-c\right)}{\int }_c^{d}F\left(\frac{2a+b}{3},y\right){}^d{d}_{q_2}y\right.\hfill \\ & \left.-\frac{1}{q_2}F\left(b,c\right)+\frac{1}{q_2}F\left(\frac{2a+b}{3},c\right)\right],\hfill \end{array}$$

(23)

$$\begin{array}{cc}& {\int }_0^1{\int }_0^{\frac{1}{3}}t\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}{d}_{q_1}t{d}_{q_2}s\hfill \\ =\hfill & \frac{1}{\left(b-a\right)\left(d-c\right)}\left[\frac{1}{q_1\left(b-a\right)}{\int }_a^{b}F\left(x,d\right){}^b{d}_{q_1}x-\frac{1}{q_1\left(b-a\right)}{\int }_a^{b}F\left(x,\frac{c+2d}{3}\right){}^b{d}_{q_1}x\right.\hfill \\ & \left.-F\left(a,d\right)+F\left(a,\frac{c+2d}{3}\right)\right],\hfill \end{array}$$

(24)

$$\begin{array}{cc}& {\int }_0^1{\int }_0^{\frac{2}{3}}t\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}{d}_{q_1}t{d}_{q_2}s\hfill \\ =\hfill & \frac{1}{\left(b-a\right)\left(d-c\right)}\left[\frac{1}{q_1\left(b-a\right)}{\int }_a^{b}F\left(x,d\right){}^b{d}_{q_1}x-\frac{1}{q_1\left(b-a\right)}{\int }_a^{b}F\left(x,\frac{2c+2}{3}\right){}^b{d}_{q_1}x\right.\hfill \\ & \left.-F\left(a,d\right)+F\left(a,\frac{2c+d}{3}\right)\right].\hfill \end{array}$$

(25)

Also, we have

$$\begin{array}{cc}& {\int }_0^1{\int }_0^{1}s\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}{d}_{q_1}t{d}_{q_2}s\hfill \\ =\hfill & \frac{1}{\left(b-a\right)\left(d-c\right)}\left\{\frac{1}{q_2\left(d-c\right)}{\int }_c^{d}F\left(b,y\right){}^d{d}_{q_2}y-\frac{1}{q_2\left(d-c\right)}{\int }_c^{d}F\left(a,y\right){}^d{d}_{q_2}y\right.\hfill \\ & \left.-\frac{1}{q_2}F\left(b,c\right)+\frac{1}{q_2}F\left(a,c\right)\right\},\hfill \end{array}$$

(26)

$$\begin{array}{cc}& {\int }_0^1{\int }_0^{1}t\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}{d}_{q_1}t{d}_{q_2}s\hfill \\ =\hfill & \frac{1}{\left(b-a\right)\left(d-c\right)}\left\{\frac{1}{q_1\left(b-a\right)}{\int }_a^{b}F\left(x,d\right){}^b{d}_{q_1}x-\frac{1}{q_1\left(b-a\right)}{\int }_a^{b}F\left(x,c\right){}^b{d}_{q_1}x\right.\hfill \\ & \left.-\frac{1}{q_1}F\left(a,d\right)+\frac{1}{q_1}F\left(a,c\right)\right\}\hfill \end{array}$$

(27)

and

$$\begin{array}{cc}& {\int }_0^1{\int }_0^{1}ts\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}{d}_{q_1}t{d}_{q_2}s\hfill \\ =\hfill & \frac{1}{\left(b-a\right)\left(d-c\right)}\left\{\sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{q}_1^n{q}_2^{m}F\left(q_1^{n+1}a+\left(1-q_1^{n+1}\right)b,q_2^{m+1}c+\left(1-q_2^{m+1}\right)d\right)\right.\hfill \\ & -\sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{q}_1^n{q}_2^{m}F\left(q_1^{n+1}a+\left(1-q_1^{n+1}\right)b,q_2^{m}c+\left(1-q_2^m\right)d\right)\hfill \\ & -\sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{q}_1^n{q}_2^{m}F\left(q_1^{n}a+\left(1-q_1^n\right)b,q_2^{m+1}c+\left(1-q_2^{m+1}\right)d\right)\hfill \\ & \left.+\sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{q}_1^n{q}_2^{m}F\left(q_1^{n}a+\left(1-q_1^n\right)b,q_2^{m}c+\left(1-q_2^m\right)d\right)\right\}\hfill \end{array}$$

(28)

$$\begin{array}{cc}=\hfill & \frac{1}{\left(b-a\right)\left(d-c\right)}\left\{\frac{1}{q_1{q}_2}\left[\sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{q}_1^n{q}_2^{m}F\left(q_1^{n}a+\left(1-q_1^n\right)b,q_2^{m}c+\left(1-q_2^m\right)d\right)\right.\right.\hfill \\ & \left.-\sum _{m=0}^{\infty }{q}_2^{m}F\left(a,q_2^{m}c+\left(1-q_2^m\right)d\right)-\sum _{n=0}^{\infty }{q}_1^{n}F\left(q_1^{n}a+\left(1-q_1^n\right)b,c\right)+F\left(a,c\right)\right]\hfill \\ & -\frac{1}{q_1}\left[\sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{q}_1^n{q}_2^{m}F\left(q_1^{n}a+\left(1-q_1^n\right)b,q_2^{m}c+\left(1-q_2^m\right)d\right)\right.\hfill \\ & \left.-\sum _{m=0}^{\infty }{q}_2^{m}F\left(a,q_2^{m}c+\left(1-q_2^m\right)d\right)\right]\hfill \\ & -\frac{1}{q_2}\left[\sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{q}_1^n{q}_2^{m}F\left(q_1^{n}a+\left(1-q_1^n\right)b,q_2^{m}c+\left(1-q_2^m\right)d\right)\right.\hfill \\ & \left.\left.-\sum _{n=0}^{\infty }{q}_1^{n}F\left(q_1^{n}a+\left(1-q_1^n\right)b,c\right)\right]+\sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{q}_1^n{q}_2^{m}F\left(q_1^{n}a+\left(1-q_1^n\right)b,q_2^{m}c+\left(1-q_2^m\right)d\right)\right\}\hfill \\ =\hfill & \frac{1}{\left(b-a\right)\left(d-c\right)}\left\{\frac{\left(1-q_1\right)\left(1-q_2\right)}{q_1{q}_2}\sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{q}_1^n{q}_2^{m}F\left(q_1^{n}a+\left(1-q_1^n\right)b,q_2^{m}c+\left(1-q_2^m\right)d\right)\right.\hfill \\ & -\frac{1-q_2}{q_1{q}_2}\sum _{m=0}^{\infty }{q}_2^{m}F\left(a,q_2^{m}c+\left(1-q_2^m\right)d\right)\hfill \\ & \left.-\frac{1-q_1}{q_1{q}_2}\sum _{n=0}^{\infty }{q}_1^{n}F\left(q_1^{n}a+\left(1-q_1^n\right)b,c\right)+\frac{1}{q_1{q}_2}F\left(a,c\right)\right\}\hfill \\ =\hfill & \frac{1}{\left(b-a\right)\left(d-c\right)}\left\{\frac{1}{q_1{q}_2\left(b-a\right)\left(d-c\right)}{\int }_a^b{\int }_c^{d}F\left(x,y\right){}^b{d}_{q_1}x{}^d{d}_{q_2}y\right.\hfill \\ & \left.-\frac{1}{q_1{q}_2\left(b-a\right)}{\int }_a^{b}F\left(x,c\right){}^b{d}_{q_1}x-\frac{1}{q_1{q}_2\left(d-c\right)}{\int }_c^{d}F\left(a,x\right){}^d{d}_{q_2}y+\frac{1}{q_1{q}_2}F\left(a,c\right)\right\}.\hfill \end{array}$$

(29)

From (13)–(29), we attain that

$$I_1=\frac{9}{64\left(b-a\right)\left(d-c\right)}\left[F\left(b,d\right)-F\left(\frac{a+2b}{3},d\right)-F\left(b,\frac{c+2d}{3}\right)+F\left(\frac{a+2b}{3},\frac{c+2d}{3}\right)\right],$$

$$I_2=\frac{9}{64\left(b-a\right)\left(d-c\right)}\left[F\left(b,d\right)-F\left(\frac{a+2b}{3},d\right)-F\left(b,\frac{2c+d}{3}\right)+F\left(\frac{a+2b}{3},\frac{2c+d}{3}\right)\right],$$

$$\begin{array}{ccc}\hfill I_3& =& \frac{3}{8\left(b-a\right)\left(d-c\right)}\left[\frac{1}{d-c}{\int }_c^{d}F\left(b,y\right){}^d{d}_{q_2}y-\frac{1}{d-c}{\int }_c^{d}F\left(\frac{a+2b}{3},y\right){}^d{d}_{q_2}y\right.\hfill \\ & & \left.-F\left(b,c\right)+F\left(\frac{a+2b}{3},c\right)\right]-\frac{21}{64\left(b-a\right)\left(d-c\right)}\left[F\left(b,d\right)-F\left(\frac{a+2b}{3},d\right)-F\left(b,c\right)\right.\hfill \\ & & +\left.F\left(\frac{a+2b}{3},c\right)\right],\hfill \end{array}$$

$$I_4=\frac{9}{64\left(b-a\right)\left(d-c\right)}\left[F\left(b,d\right)-F\left(\frac{2a+b}{3},d\right)-F\left(b,\frac{c+2d}{3}\right)+F\left(\frac{2a+b}{3},\frac{c+2d}{3}\right)\right],$$

$$I_5=\frac{9}{64\left(b-a\right)\left(d-c\right)}\left[F\left(b,d\right)-F\left(\frac{2a+b}{3},d\right)-F\left(b,\frac{2c+d}{3}\right)+F\left(\frac{2a+b}{3},\frac{2c+d}{3}\right)\right],$$

$$\begin{array}{ccc}\hfill I_6& =& \frac{3}{8\left(b-a\right)\left(d-c\right)}\left[\frac{1}{d-c}{\int }_c^{d}F\left(b,y\right){}^d{d}_{q_2}y-\frac{1}{d-c}{\int }_c^{d}F\left(\frac{2a+b}{3},y\right){}^d{d}_{q_2}y\right.\hfill \\ & & \left.-F\left(b,c\right)+F\left(\frac{2a+b}{3},c\right)\right]-\frac{21}{64\left(b-a\right)\left(d-c\right)}\left[F\left(b,d\right)-F\left(\frac{2a+b}{3},d\right)-F\left(b,c\right)\right.\hfill \\ & & +\left.F\left(\frac{2a+b}{3},c\right)\right],\hfill \end{array}$$

$$\begin{array}{ccc}\hfill I_7& =& \frac{3}{8\left(b-a\right)\left(d-c\right)}\left[\frac{1}{b-a}{\int }_a^{b}F\left(x,d\right){}^b{d}_{q_1}x-\frac{1}{b-a}{\int }_a^{b}F\left(x,\frac{c+2d}{3}\right){}^b{d}_{q_1}x-F\left(a,d\right)\right.\hfill \\ & & \left.+F\left(a,\frac{c+2d}{3}\right)\right]-\frac{21}{64\left(b-a\right)\left(d-c\right)}\left[F\left(b,d\right)-F\left(b,\frac{c+2d}{3}\right)-F\left(a,d\right)\right.\hfill \\ & & \left.+F\left(a,\frac{c+2d}{3}\right)\right],\hfill \end{array}$$

$$\begin{array}{ccc}\hfill I_8& =& \frac{3}{8\left(b-a\right)\left(d-c\right)}\left[\frac{1}{b-a}{\int }_a^{b}F\left(x,d\right){}^b{d}_{q_1}x-\frac{1}{b-a}{\int }_a^{b}F\left(x,\frac{2c+d}{3}\right){}^b{d}_{q_1}x-F\left(a,d\right)\right.\hfill \\ & & \left.+F\left(a,\frac{2c+d}{3}\right)\right]-\frac{21}{64\left(b-a\right)\left(d-c\right)}\left[F\left(b,d\right)-F\left(b,\frac{2c+d}{3}\right)-F\left(a,d\right)\right.\hfill \\ & & \left.+F\left(a,\frac{c+2d}{3}\right)\right],\hfill \\ \hfill I_9& =& \frac{1}{\left(b-a\right)\left(d-c\right)}\left\{\frac{1}{\left(b-a\right)\left(d-c\right)}{\int }_a^b{\int }_c^{d}F\left(x,y\right){}^b{d}_{q_1}x{}^d{d}_{q_2}y\right.\hfill \\ & & \left.-\frac{1}{b-a}{\int }_a^{b}F\left(x,c\right){}^b{d}_{q_1}x-\frac{1}{d-c}{\int }_c^{d}F\left(a,y\right){}^d{d}_{q_2}y+F\left(a,c\right)\right\}\hfill \\ & & -\frac{7}{8\left(b-a\right)\left(d-c\right)}\left\{\frac{1}{b-a}{\int }_a^{b}F\left(x,d\right){}^b{d}_{q_1}x-\frac{1}{b-a}{\int }_a^{b}F\left(x,c\right){}^b{d}_{q_1}x\right.\hfill \\ & & \left.+F\left(a,c\right)-F\left(a,d\right)\right\}-\frac{7}{8\left(b-a\right)\left(d-c\right)}\left\{\frac{1}{d-c}{\int }_c^{d}F\left(b,y\right){}^d{d}_{q_2}y\right.\hfill \\ & & \left.-\frac{1}{d-c}{\int }_c^{d}F\left(a,y\right){}^d{d}_{q_2}y+F\left(a,c\right)-F\left(b,c\right)\right\}\hfill \\ & & +\frac{49}{64\left(b-a\right)\left(d-c\right)}\left[F\left(b,d\right)-F\left(a,d\right)-F\left(b,c\right)+F\left(a,c\right)\right].\hfill \end{array}$$

Now using the calculated integrals $\left(I_1\right)-\left(I_9\right)$ in (12) and multiplying the resulting one with $\left(b-a\right)\left(d-c\right)$, we have the desired equality (11) which accomplishes the proof. □

Remark 1.

Under the given conditions of Lemma 3 with $q_1,q_2\to 1^{-},$ we have the following new identity:

$$\begin{array}{cc}& \mathcal{I}\left(F\right)\hfill \\ =\hfill & \left(b-a\right)\left(d-c\right){\int }_0^1{\int }_0^1\mathsf{\Lambda }\left(t\right)\mathsf{\Lambda }\left(s\right)\frac{{\partial }^{2}F\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)}{\partial t\partial s}dtds,\hfill \end{array}$$

(30)

where

$$\begin{array}{ccc}\hfill \mathcal{I}\left(F\right)& =& \frac{3}{64}\left[F\left(\frac{a+2b}{3},d\right)+F\left(b,\frac{c+2d}{3}\right)+3F\left(\frac{a+2b}{3},\frac{c+2d}{3}\right)+3F\left(\frac{a+2b}{3},\frac{2c+d}{3}\right)\right.\hfill \\ & & +F\left(\frac{2a+b}{3},d\right)+3F\left(\frac{2a+b}{3},\frac{c+2d}{3}\right)+F\left(b,\frac{2c+d}{3}\right)+3F\left(\frac{2a+b}{3},\frac{2c+d}{3}\right)\hfill \\ & & \left.+F\left(\frac{a+2b}{3},c\right)+F\left(\frac{2a+b}{3},c\right)+F\left(a,\frac{c+2d}{3}\right)+F\left(a,\frac{2c+d}{3}\right)\right]\hfill \\ & & +\frac{F\left(a,c\right)+F\left(a,d\right)+F\left(b,c\right)+F\left(b,d\right)}{64}\hfill \\ & & -\frac{1}{8\left(b-a\right)}{\int }_a^b\left[F\left(x,c\right)+3F\left(x,\frac{c+2d}{3}\right)+3F\left(x,\frac{2c+d}{3}\right)+F\left(x,d\right)\right]dx\hfill \\ & & -\frac{1}{8\left(d-c\right)}{\int }_c^d\left[F\left(a,y\right)+3F\left(\frac{a+2b}{3},y\right)+3F\left(\frac{2a+b}{3},y\right)+F\left(b,y\right)\right]dy\hfill \\ & & +\frac{1}{\left(b-a\right)\left(d-c\right)}{\int }_a^b{\int }_c^{d}F\left(x,y\right)dxdy\hfill \end{array}$$

and

$$\mathsf{\Lambda }\left(t\right)=\left\{\begin{array}{c}t-\frac{1}{8},t\in \left[0,\frac{1}{3}\right)\\ t-\frac{1}{2},t\in \left[\frac{1}{3},\frac{2}{3}\right)\\ t-\frac{7}{8},t\in \left[\frac{2}{3},1\right],\end{array}\right.$$

$$\mathsf{\Lambda }\left(s\right)=\left\{\begin{array}{c}s-\frac{1}{8},s\in \left[0,\frac{1}{3}\right)\\ s-\frac{1}{2},s\in \left[\frac{1}{3},\frac{2}{3}\right)\\ s-\frac{7}{8},s\in \left[\frac{2}{3},1\right].\end{array}\right.$$

5. Some New $q_1{q}_2$–Newton’s Type Inequalities

For brevity, we give some calculated integrals before giving new estimates.

$${\mathsf{\Psi }}_1\left(q\right)=\underset{0}{\overset{\frac{1}{3}}{\int }}\left|qt-\frac{1}{8}\right|t{d}_{q}t=\left\{\begin{array}{cc}\frac{3-5q-5q^2}{216\left(1+q\right)\left(1+q+q^2\right)},& 0<q<\frac{3}{8}\\ \\ \frac{160q^2+160q-69}{6912\left(1+q\right)\left(1+q+q^2\right)},& \frac{3}{8}\le q<1,\end{array}\right.$$

(31)

$${\mathsf{\Psi }}_2\left(q\right)=\underset{0}{\overset{\frac{1}{3}}{\int }}\left|qt-\frac{1}{8}\right|\left(1-t\right)d_{q}t=\left\{\begin{array}{cc}\frac{6-q-q^2-15q^3}{216\left(1+q\right)\left(1+q+q^2\right)},& 0<q<\frac{3}{8}\\ \\ \frac{480q^3+248q^2+248q-3}{6912\left(1+q\right)\left(1+q+q^2\right)},& \frac{3}{8}\le q<1,\end{array}\right.$$

(32)

$${\mathsf{\Psi }}_3\left(q\right)=\underset{\frac{1}{3}}{\overset{\frac{2}{3}}{\int }}\left|qt-\frac{1}{2}\right|t{d}_{q}t=\left\{\begin{array}{cc}\frac{9-5q-5q^2}{54\left(1+q\right)\left(1+q+q^2\right)},& 0<q<\frac{3}{4}\\ \\ \frac{6q^2+6q-3}{108\left(1+q\right)\left(1+q+q^2\right)},& \frac{3}{4}\le q<1,\end{array}\right.$$

(33)

$${\mathsf{\Psi }}_4\left(q\right)=\underset{\frac{1}{3}}{\overset{\frac{2}{3}}{\int }}\left|qt-\frac{1}{2}\right|\left(1-t\right)d_{q}t=\left\{\begin{array}{cc}\frac{5q+5q^2-9q^3}{54\left(1+q\right)\left(1+q+q^2\right)},& 0<q<\frac{3}{4}\\ \\ \frac{6q^3+3}{108\left(1+q\right)\left(1+q+q^2\right)},& \frac{3}{4}\le q<1,\end{array}\right.$$

(34)

$${\mathsf{\Psi }}_5\left(q\right)=\underset{\frac{2}{3}}{\overset{1}{\int }}\left|qt-\frac{7}{8}\right|t{d}_{q}t=\left\{\begin{array}{cc}\frac{105-47q-47q^2}{216\left(1+q\right)\left(1+q+q^2\right)},& 0<q<\frac{7}{8}\\ \\ \frac{224q^2+224q+525}{6912\left(1+q\right)\left(1+q+q^2\right)},& \frac{7}{8}\le q<1,\end{array}\right.$$

(35)

$${\mathsf{\Psi }}_6\left(q\right)=\underset{\frac{2}{3}}{\overset{1}{\int }}\left|qt-\frac{7}{8}\right|\left(1-t\right)d_{q}t=\left\{\begin{array}{cc}\frac{-42+53q+53q^2-57q^3}{216\left(1+q\right)\left(1+q+q^2\right)},& 0<q<\frac{7}{8}\\ \\ \frac{-96q^3+184q^2+184q-21}{6912\left(1+q\right)\left(1+q+q^2\right)},& \frac{7}{8}\le q<1,\end{array}\right.$$

(36)

$${\mathsf{\Psi }}_7\left(q\right)=\underset{0}{\overset{\frac{1}{3}}{\int }}\left|qt-\frac{1}{8}\right|d_{q}t=\left\{\begin{array}{cc}\frac{3-5q}{72\left(1+q\right)},& 0<q<\frac{3}{8}\\ \\ \frac{20q-3}{288\left(1+q\right)},& \frac{3}{8}\le q<1,\end{array}\right.$$

(37)

$${\mathsf{\Psi }}_8\left(q\right)=\underset{\frac{1}{3}}{\overset{\frac{2}{3}}{\int }}\left|qt-\frac{1}{2}\right|d_{q}t=\left\{\begin{array}{cc}\frac{3-3q}{18\left(1+q\right)},& 0<q<\frac{3}{4}\\ \\ \frac{q}{18\left(1+q\right)},& \frac{3}{4}\le q<1\end{array}\right.$$

(38)

and

$${\mathsf{\Psi }}_9\left(q\right)=\underset{\frac{2}{3}}{\overset{1}{\int }}\left|qt-\frac{7}{8}\right|d_{q}t=\left\{\begin{array}{cc}\frac{21-19q}{72\left(1+q\right)},& 0<q<\frac{7}{8}\\ \\ \frac{21-4q}{288\left(1+q\right)},& \frac{7}{8}\le q<1.\end{array}\right.$$

(39)

Now we give some new quantum estimates by using the identity in Lemma 3.

Theorem 7.

Let $F:\Delta \subseteq {\mathbb{R}}^2\to \mathbb{R}$ be a twice partially $q_1{q}_2$-differentiable function on ${\Delta }^{\circ }$ such that partial $q_1{q}_2$-derivative $\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(t,s\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}$ is continuous and integrable on $\left[a,b\right]\times \left[c,d\right]\subseteq {\Delta }^{\circ }$. If $\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(t,s\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|$ is convex on $\left[a,b\right]\times \left[c,d\right],$ then we have the following inequality:

$$\begin{array}{c}\hfill \begin{array}{c}\left|{}^{b,d}{\mathcal{I}}_{q_1,q_2}\right|\le \left(b-a\right)\left(d-c\right)\hfill \\ \times \left[\left({\mathsf{\Psi }}_1\left(q_1\right)+{\mathsf{\Psi }}_3\left(q_1\right)+{\mathsf{\Psi }}_5\left(q_1\right)\right)\left({\mathsf{\Psi }}_1\left(q_2\right)+{\mathsf{\Psi }}_3\left(q_2\right)+{\mathsf{\Psi }}_5\left(q_2\right)\right)\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(a,c\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|\right.\hfill \\ +\left({\mathsf{\Psi }}_1\left(q_1\right)+{\mathsf{\Psi }}_3\left(q_1\right)+{\mathsf{\Psi }}_5\left(q_1\right)\right)\left({\mathsf{\Psi }}_2\left(q_2\right)+{\mathsf{\Psi }}_4\left(q_2\right)+{\mathsf{\Psi }}_6\left(q_2\right)\right)\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(a,d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|\hfill \\ +\left({\mathsf{\Psi }}_2\left(q_1\right)+{\mathsf{\Psi }}_4\left(q_1\right)+{\mathsf{\Psi }}_6\left(q_1\right)\right)\left({\mathsf{\Psi }}_1\left(q_2\right)+{\mathsf{\Psi }}_3\left(q_2\right)+{\mathsf{\Psi }}_5\left(q_2\right)\right)\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(b,c\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|\hfill \\ \left.+\left({\mathsf{\Psi }}_2\left(q_1\right)+{\mathsf{\Psi }}_4\left(q_1\right)+{\mathsf{\Psi }}_6\left(q_1\right)\right)\left({\mathsf{\Psi }}_2\left(q_2\right)+{\mathsf{\Psi }}_4\left(q_2\right)+{\mathsf{\Psi }}_6\left(q_2\right)\right)\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(b,d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|\right],\hfill \end{array}\end{array}$$

(40)

where $q_1,q_2\in \left(0,1\right).$

Proof. 

Taking properties of modulus in Lemma 3, we find that

$$\begin{array}{cc}& \left|{}^{b,d}{\mathcal{I}}_{q_1,q_2}\right|\hfill \\ \le \hfill & \left(b-a\right)\left(d-c\right){\int }_0^1{\int }_0^1\left|{\mathsf{\Lambda }}_{q_1}\left(t\right){\mathsf{\Lambda }}_{q_2}\left(s\right)\right|\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|d_{q_1}t{d}_{q_2}s\hfill \end{array}$$

(41)

Now using the convexity of $\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(t,s\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|$, (41) becomes

$$\begin{array}{cc}& \left|{}^{b,d}{\mathcal{I}}_{q_1,q_2}\right|\hfill \\ \le \hfill & \left(b-a\right)\left(d-c\right){\int }_0^1{\mathsf{\Lambda }}_{q_2}\left(s\right)\left[{\int }_0^1{\mathsf{\Lambda }}_{q_1}\left(t\right)\left\{t\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(a,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|\right.\right.\hfill \\ & \left.\left.+\left(1-t\right)\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|\right\}{d}_{q_1}t\right]d_{q_2}s.\hfill \end{array}$$

(42)

Now we compute the integrals that appear in right side of inequality (42).

$$\begin{array}{cc}& \hfill {\int }_0^1{\mathsf{\Lambda }}_{q_1}\left(t\right)\left\{t\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(a,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|+\left(1-t\right)\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|\right\}{d}_{q_1}t\\ =\hfill & {\int }_0^{\frac{1}{3}}t\left|q_{1}t-\frac{1}{8}\right|\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(a,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|d_{q_1}t\hfill \\ & +{\int }_{\frac{1}{3}}^{\frac{2}{3}}t\left|q_{1}t-\frac{1}{2}\right|\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(a,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|d_{q_1}t\hfill \\ & +{\int }_{\frac{2}{3}}^{1}t\left|q_{1}t-\frac{7}{8}\right|\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(a,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|d_{q_1}t\hfill \\ & +{\int }_0^{\frac{1}{3}}\left(1-t\right)\left|q_{1}t-\frac{1}{8}\right|\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|d_{q_1}t\hfill \\ & +{\int }_{\frac{1}{3}}^{\frac{2}{3}}\left(1-t\right)\left|q_{1}t-\frac{1}{2}\right|\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|d_{q_1}t\hfill \\ & +{\int }_{\frac{2}{3}}^1\left(1-t\right)\left|q_{1}t-\frac{7}{8}\right|\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|d_{q_1}t\hfill \end{array}$$

From (31)–(36), we obtain

$$\begin{array}{cc}& {\int }_0^1{\mathsf{\Lambda }}_{q_1}\left(t\right)\left\{t\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(a,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|+\left(1-t\right)\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|\right\}{d}_{q_1}t\hfill \\ =& \left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(a,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|\left({\mathsf{\Psi }}_1\left(q_1\right)+{\mathsf{\Psi }}_3\left(q_1\right)+{\mathsf{\Psi }}_5\left(q_1\right)\right)\hfill \\ & +\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|\left({\mathsf{\Psi }}_2\left(q_1\right)+{\mathsf{\Psi }}_4\left(q_1\right)+{\mathsf{\Psi }}_6\left(q_1\right)\right).\hfill \end{array}$$

Thus, we have

$$\begin{array}{ccc}\hfill \left|{}^{b,d}{\mathcal{I}}_{q_1,q_2}\right|& \le & \left(b-a\right)\left(d-c\right){\int }_0^1{\mathsf{\Lambda }}_{q_2}\left(s\right)\left[\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(a,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|\left({\mathsf{\Psi }}_1\left(q_1\right)+{\mathsf{\Psi }}_3\left(q_1\right)+{\mathsf{\Psi }}_5\left(q_1\right)\right)\right.\hfill \\ & & \left.+\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|\left({\mathsf{\Psi }}_2\left(q_1\right)+{\mathsf{\Psi }}_4\left(q_1\right)+{\mathsf{\Psi }}_6\left(q_1\right)\right)\right]d_{q_2}s\hfill \end{array}$$

$$\begin{array}{cc}\le & \left(b-a\right)\left(d-c\right){\int }_0^1{\mathsf{\Lambda }}_{q_2}\left(s\right)\left[\left\{s\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(a,c\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|+\left(1-s\right)\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(a,d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|\right.\right.\hfill \\ & \left.\times \left({\mathsf{\Psi }}_1\left(q_1\right)+{\mathsf{\Psi }}_3\left(q_1\right)+{\mathsf{\Psi }}_5\left(q_1\right)\right)\right\}+\left\{s\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(b,c\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|+\left(1-s\right)\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(b,d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|\right.\hfill \\ & \left.\left.\times \left({\mathsf{\Psi }}_2\left(q_1\right)+{\mathsf{\Psi }}_4\left(q_1\right)+{\mathsf{\Psi }}_6\left(q_1\right)\right)\right\}\right]d_{q_2}s\hfill \\ =& \left(b-a\right)\left(d-c\right)\left({\mathsf{\Psi }}_1\left(q_1\right)+{\mathsf{\Psi }}_3\left(q_1\right)+{\mathsf{\Psi }}_5\left(q_1\right)\right)\left[\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(a,c\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|\left\{{\int }_0^{\frac{1}{3}}s\left|q_{2}s-\frac{1}{8}\right|d_{q_2}s\right.\right.\hfill \\ & \left.+{\int }_{\frac{1}{3}}^{\frac{2}{3}}s\left|q_{2}s-\frac{1}{2}\right|d_{q_2}s+{\int }_{\frac{2}{3}}^{1}s\left|q_{2}s-\frac{7}{8}\right|d_{q_2}s\right\}+\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(a,d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|\left\{{\int }_0^{\frac{1}{3}}\left(1-s\right)\left|q_{2}s-\frac{1}{8}\right|d_{q_2}s\right.\hfill \\ & \left.\left.+{\int }_{\frac{1}{3}}^{\frac{2}{3}}\left(1-s\right)\left|q_{2}s-\frac{1}{2}\right|d_{q_2}s+{\int }_{\frac{2}{3}}^1\left(1-s\right)\left|q_{2}s-\frac{7}{8}\right|d_{q_2}s\right\}\right]\hfill \\ & +\left(b-a\right)\left(d-c\right)\left({\mathsf{\Psi }}_2\left(q_1\right)+{\mathsf{\Psi }}_4\left(q_1\right)+{\mathsf{\Psi }}_6\left(q_1\right)\right)\left[\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(b,c\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|\left\{{\int }_0^{\frac{1}{3}}s\left|q_{2}s-\frac{1}{8}\right|d_{q_2}s\right.\right.\hfill \\ & \left.+{\int }_{\frac{1}{3}}^{\frac{2}{3}}s\left|q_{2}s-\frac{1}{2}\right|d_{q_2}s+{\int }_{\frac{2}{3}}^{1}s\left|q_{2}s-\frac{7}{8}\right|d_{q_2}s\right\}+\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(b,d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|\left\{{\int }_0^{\frac{1}{3}}\left(1-s\right)\left|q_{2}s-\frac{1}{8}\right|d_{q_2}s\right.\hfill \\ & \left.\left.+{\int }_{\frac{1}{3}}^{\frac{2}{3}}\left(1-s\right)\left|q_{2}s-\frac{1}{2}\right|d_{q_2}s+{\int }_{\frac{2}{3}}^1\left(1-s\right)\left|q_{2}s-\frac{7}{8}\right|d_{q_2}s\right\}\right].\hfill \end{array}$$

From (31)–(36), we have

$$\begin{array}{cc}& \left|{}^{b,d}{\mathcal{I}}_{q_1,q_2}\right|\hfill \\ \le \hfill & \left(b-a\right)\left(d-c\right)\hfill \\ & \times \left[\left({\mathsf{\Psi }}_1\left(q_1\right)+{\mathsf{\Psi }}_3\left(q_1\right)+{\mathsf{\Psi }}_5\left(q_1\right)\right)\left({\mathsf{\Psi }}_1\left(q_2\right)+{\mathsf{\Psi }}_3\left(q_2\right)+{\mathsf{\Psi }}_5\left(q_2\right)\right)\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(a,c\right)}{{}^b{\partial }_{q_1}^{2}t{}^d{\partial }_{q_2}s}\right|\right.\hfill \\ & +\left({\mathsf{\Psi }}_1\left(q_1\right)+{\mathsf{\Psi }}_3\left(q_1\right)+{\mathsf{\Psi }}_5\left(q_1\right)\right)\left({\mathsf{\Psi }}_2\left(q_2\right)+{\mathsf{\Psi }}_4\left(q_2\right)+{\mathsf{\Psi }}_6\left(q_2\right)\right)\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(a,d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|\hfill \\ & +\left({\mathsf{\Psi }}_2\left(q_1\right)+{\mathsf{\Psi }}_4\left(q_1\right)+{\mathsf{\Psi }}_6\left(q_1\right)\right)\left({\mathsf{\Psi }}_1\left(q_2\right)+{\mathsf{\Psi }}_3\left(q_2\right)+{\mathsf{\Psi }}_5\left(q_2\right)\right)\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(b,c\right)}{{}^b{\partial }_{q_1}^{2}t{}^d{\partial }_{q_2}s}\right|\hfill \\ & \left.+\left({\mathsf{\Psi }}_2\left(q_1\right)+{\mathsf{\Psi }}_4\left(q_1\right)+{\mathsf{\Psi }}_6\left(q_1\right)\right)\left({\mathsf{\Psi }}_2\left(q_2\right)+{\mathsf{\Psi }}_4\left(q_2\right)+{\mathsf{\Psi }}_6\left(q_2\right)\right)\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(b,d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|\right],\hfill \end{array}$$

which completes the proof. □

Remark 2.

Under the given conditions of Theorem 7 with $q_1,q_2\to 1^{-},$ we attain the following new inequality:

$$\begin{array}{c}\left|\mathcal{I}\left(F\right)\right|\le \frac{625\left(b-a\right)\left(d-c\right)}{576}\left[\frac{\left|\frac{{\partial }^{2}F\left(a,c\right)}{\partial t\partial s}\right|+\left|\frac{{\partial }^{2}F\left(a,d\right)}{\partial t\partial s}\right|+\left|\frac{{\partial }^{2}F\left(b,c\right)}{\partial t\partial s}\right|+\left|\frac{{\partial }^{2}F\left(b,d\right)}{\partial t\partial s}\right|}{576}\right].\hfill \end{array}$$

Corollary 1.

Under the given conditions of Theorem 7 if $\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(t,s\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|\le K,$ we get

$$\begin{array}{c}\left|{}^{b,d}{\mathcal{I}}_{q_1,q_2}\right|\le K\left(b-a\right)\left(d-c\right)\hfill \\ \times \left[\left({\mathsf{\Psi }}_1\left(q_1\right)+{\mathsf{\Psi }}_3\left(q_1\right)+{\mathsf{\Psi }}_5\left(q_1\right)\right)\left({\mathsf{\Psi }}_1\left(q_2\right)+{\mathsf{\Psi }}_3\left(q_2\right)+{\mathsf{\Psi }}_5\left(q_2\right)\right)\right.\hfill \\ +\left({\mathsf{\Psi }}_1\left(q_1\right)+{\mathsf{\Psi }}_3\left(q_1\right)+{\mathsf{\Psi }}_5\left(q_1\right)\right)\left({\mathsf{\Psi }}_2\left(q_2\right)+{\mathsf{\Psi }}_4\left(q_2\right)+{\mathsf{\Psi }}_6\left(q_2\right)\right)\hfill \\ +\left({\mathsf{\Psi }}_2\left(q_1\right)+{\mathsf{\Psi }}_4\left(q_1\right)+{\mathsf{\Psi }}_6\left(q_1\right)\right)\left({\mathsf{\Psi }}_1\left(q_2\right)+{\mathsf{\Psi }}_3\left(q_2\right)+{\mathsf{\Psi }}_5\left(q_2\right)\right)\hfill \\ \left.+\left({\mathsf{\Psi }}_2\left(q_1\right)+{\mathsf{\Psi }}_4\left(q_1\right)+{\mathsf{\Psi }}_6\left(q_1\right)\right)\left({\mathsf{\Psi }}_2\left(q_2\right)+{\mathsf{\Psi }}_4\left(q_2\right)+{\mathsf{\Psi }}_6\left(q_2\right)\right)\right].\hfill \end{array}$$

Theorem 8.

Let $F:\Delta \subseteq {\mathbb{R}}^2\to \mathbb{R}$ be a twice partially $q_1{q}_2$-differentiable function on ${\Delta }^{\circ }$ such that the partial $q_1{q}_2$-derivative $\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(t,s\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}$ is continuous and integrable on $\left[a,b\right]\times \left[c,d\right]\subseteq {\Delta }^{\circ }$. If ${\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(t,s\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^{p_1}$ is convex on $\left[a,b\right]\times \left[c,d\right]$ for some $p_1>1$ and $\frac{1}{r_1}+\frac{1}{p_1}=1,$ then we have the following inequality:

$$\begin{array}{c}\begin{array}{c}\left|{}^{b,d}{\mathcal{I}}_{q_1,q_2}\right|\le \left(b-a\right)\left(d-c\right){\left({\int }_0^1{\int }_0^1{\left|{\mathsf{\Lambda }}_{q_1}\left(t\right){\mathsf{\Lambda }}_{q_2}\left(s\right)\right|}^{r_1}{d}_{q_1}t{d}_{q_2}s\right)}^{\frac{1}{r_1}}\hfill \\ \times \left[\frac{1}{{\left[2\right]}_{q_1}{\left[2\right]}_{q_2}}{\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F(a,c)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^{p_1}+\frac{q_2}{{\left[2\right]}_{q_1}{\left[2\right]}_{q_2}}{\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F(a,d)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^{p_1}\right.\hfill \\ {\left.+\frac{q_1}{{\left[2\right]}_{q_1}{\left[2\right]}_{q_2}}{\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F(b,c)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^{p_1}+\frac{q_1{q}_2}{{\left[2\right]}_{q_1}{\left[2\right]}_{q_2}}{\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F(b,d)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^{p_1}\right]}^{\frac{1}{p_1}},\hfill \end{array}\hfill \end{array}$$

(43)

where $q_1,q_2\in (0,1).$

Proof. 

Applying well–known Hölder’s inequality for $q_1{q}_2$-integrals to the integrals in right side of (41), it is found that

$$\begin{array}{cc}& \left|{}^{b,d}{\mathcal{I}}_{q_1,q_2}\right|\hfill \\ \le \hfill & \left(b-a\right)\left(d-c\right)\left[{\left({\int }_0^1{\int }_0^1{\left|{\mathsf{\Lambda }}_{q_1}\left(t\right){\mathsf{\Lambda }}_{q_2}\left(s\right)\right|}^{r_1}{d}_{q_1}t{d}_{q_2}s\right)}^{\frac{1}{r_1}}\right.\hfill \\ & \left.\times {\left({\int }_0^1{\int }_0^1{\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^{p_1}{d}_{q_1}t{d}_{q_2}s\right)}^{\frac{1}{p_1}}\right].\hfill \end{array}$$

(44)

By applying convexity of ${\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(t,s\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^{p_1},$ (44) becomes

$$\begin{array}{cc}& \left|{}^{b,d}{\mathcal{I}}_{q_1,q_2}\right|\hfill \\ \le \hfill & \left(b-a\right)\left(d-c\right)\left[{\left({\int }_0^1{\int }_0^1{\left|{\mathsf{\Lambda }}_{q_1}\left(t\right){\mathsf{\Lambda }}_{q_2}\left(s\right)\right|}^{r_1}{d}_{q_1}t{d}_{q_2}s\right)}^{\frac{1}{r_1}}\right.\hfill \\ & \times \left({\int }_0^1{\int }_0^1\left[ts{\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F(a,c)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^{p_1}+t\left(1-s\right){\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(a,d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^{p_1}\right.\right.\hfill \\ & \left.{\left.\left.+\left(1-t\right)s{\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(b,c\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^{p_1}+\left(1-t\right)\left(1-s\right){\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(b,d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^{p_1}\right]d_{q_1}t{d}_{q_2}s\right)}^{\frac{1}{p_1}}\right].\hfill \end{array}$$

(45)

Now, if we apply the concept of Lemma 1 for $a=0$ to the above quantum integrals, we attain

$$\begin{array}{ccc}\hfill {\int }_0^1{\int }_0^{1}ts{d}_{q_1}t{d}_{q_2}s& =\hfill & \left({\int }_0^{1}t{d}_{q_1}t\right)\left({\int }_0^{1}s{d}_{q_2}s\right)\hfill \\ & =\hfill & \frac{1}{{\left[2\right]}_{q_1}{\left[2\right]}_{q_2}},\hfill \end{array}$$

(46)

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

(47)

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

(48)

$${\int }_0^1{\int }_0^1\left(1-t\right)\left(1-s\right)d_{q_1}t{d}_{q_2}s=\frac{q_1{q}_2}{{\left[2\right]}_{q_1}{\left[2\right]}_{q_2}}.$$

(49)

By substituting the calculated integrals (45)–(49) in (45), we obtain the desired inequality (43) which finishes the proof. □

Corollary 2.

Under the given conditions of Theorem 8 if $\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(t,s\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|\le K,$ we get

$$\begin{array}{c}\left|{}^{b,d}{\mathcal{I}}_{q_1,q_2}\right|\le \frac{K\left(b-a\right)\left(d-c\right)}{\sqrt[p_1]{{\left[2\right]}_{q_1}{\left[2\right]}_{q_2}}}{\left({\int }_0^1{\int }_0^1{\left|{\mathsf{\Lambda }}_{q_1}\left(t\right){\mathsf{\Lambda }}_{q_2}\left(s\right)\right|}^{r_1}{d}_{q_1}t{d}_{q_2}s\right)}^{\frac{1}{r_1}}.\hfill \end{array}$$

Theorem 9.

Let $F:\Delta \subseteq {\mathbb{R}}^2\to \mathbb{R}$ be a twice partially $q_1{q}_2$-differentiable function on ${\Delta }^{\circ }$ such that the partial $q_1{q}_2$-derivative $\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(t,s\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}$ is continuous and integrable on $\left[a,b\right]\times \left[c,d\right]\subseteq {\Delta }^{\circ }$. If ${\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(t,s\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p$ is convex on $\left[a,b\right]\times \left[c,d\right]$ for some $p\ge 1$. Then, we have following inequality

$$\begin{array}{cc}& \left|{}^{b,d}{\mathcal{I}}_{q_1,q_2}\left(F\right)\right|\hfill \\ \le \hfill & \left(b-a\right)\left(d-c\right)\left[{\mathsf{\Psi }}_7^{1-\frac{1}{p}}\left(q_1\right){\mathsf{\Psi }}_7^{1-\frac{1}{p}}\left(q_2\right)\right.\hfill \\ & \times \left\{{\mathsf{\Psi }}_1\left(q_1\right)\left({\mathsf{\Psi }}_1\left(q_2\right){\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(a,c\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p+{\mathsf{\Psi }}_2\left(q_2\right){\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(a,d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p\right)\right.\hfill \\ & {\left.+{\mathsf{\Psi }}_2\left(q_1\right)\left({\mathsf{\Psi }}_1\left(q_2\right){\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(b,c\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p+{\mathsf{\Psi }}_2\left(q_2\right){\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(b,d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p\right)\right\}}^{\frac{1}{p}}\hfill \\ & +{\mathsf{\Psi }}_7^{1-\frac{1}{p}}\left(q_1\right){\mathsf{\Psi }}_8^{1-\frac{1}{p}}\left(q_2\right)\hfill \\ & \times \left\{{\mathsf{\Psi }}_1\left(q_1\right)\left({\mathsf{\Psi }}_3\left(q_2\right){\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(a,c\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p+{\mathsf{\Psi }}_4\left(q_2\right){\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(a,d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p\right)\right.\hfill \\ & {\left.+{\mathsf{\Psi }}_2\left(q_1\right)\left({\mathsf{\Psi }}_3\left(q_2\right){\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(b,c\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p+{\mathsf{\Psi }}_4\left(q_2\right){\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(b,d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p\right)\right\}}^{\frac{1}{p}}\hfill \\ & +{\mathsf{\Psi }}_7^{1-\frac{1}{p}}\left(q_1\right){\mathsf{\Psi }}_9^{1-\frac{1}{p}}\left(q_2\right)\hfill \\ & \times \left\{{\mathsf{\Psi }}_1\left(q_1\right)\left({\mathsf{\Psi }}_5\left(q_2\right){\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(a,c\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p+{\mathsf{\Psi }}_6\left(q_2\right){\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(a,d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p\right)\right.\hfill \\ & {\left.+{\mathsf{\Psi }}_2\left(q_1\right)\left({\mathsf{\Psi }}_5\left(q_2\right){\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(b,c\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p+{\mathsf{\Psi }}_6\left(q_2\right){\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(b,d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p\right)\right\}}^{\frac{1}{p}}\hfill \\ & +{\mathsf{\Psi }}_8^{1-\frac{1}{p}}\left(q_1\right){\mathsf{\Psi }}_7^{1-\frac{1}{p}}\left(q_2\right)\hfill \\ & \times \left\{{\mathsf{\Psi }}_3\left(q_1\right)\left({\mathsf{\Psi }}_1\left(q_2\right){\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(a,c\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p+{\mathsf{\Psi }}_2\left(q_2\right){\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(a,d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p\right)\right.\hfill \\ & \hfill {\left.+{\mathsf{\Psi }}_4\left(q_1\right)\left({\mathsf{\Psi }}_1\left(q_2\right){\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(b,c\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p+{\mathsf{\Psi }}_2\left(q_2\right){\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(b,d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p\right)\right\}}^{\frac{1}{p}}\\ & +{\mathsf{\Psi }}_8^{1-\frac{1}{p}}\left(q_1\right){\mathsf{\Psi }}_8^{1-\frac{1}{p}}\left(q_2\right)\hfill \\ & \times \left\{{\mathsf{\Psi }}_3\left(q_1\right)\left({\mathsf{\Psi }}_3\left(q_2\right){\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(a,c\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p+{\mathsf{\Psi }}_4\left(q_2\right){\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(a,d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p\right)\right.\hfill \\ & {\left.+{\mathsf{\Psi }}_4\left(q_1\right)\left({\mathsf{\Psi }}_3\left(q_2\right){\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(b,c\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p+{\mathsf{\Psi }}_4\left(q_2\right){\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(b,d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p\right)\right\}}^{\frac{1}{p}}\hfill \\ & +{\mathsf{\Psi }}_8^{1-\frac{1}{p}}\left(q_1\right){\mathsf{\Psi }}_9^{1-\frac{1}{p}}\left(q_2\right)\hfill \\ & \times \left\{{\mathsf{\Psi }}_3\left(q_1\right)\left({\mathsf{\Psi }}_5\left(q_2\right){\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(a,c\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p+{\mathsf{\Psi }}_6\left(q_2\right){\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(a,d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p\right)\right.\hfill \\ & {\left.+{\mathsf{\Psi }}_4\left(q_1\right)\left({\mathsf{\Psi }}_5\left(q_2\right){\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(b,c\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p+{\mathsf{\Psi }}_6\left(q_2\right){\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(b,d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p\right)\right\}}^{\frac{1}{p}}\hfill \\ & +{\mathsf{\Psi }}_9^{1-\frac{1}{p}}\left(q_1\right){\mathsf{\Psi }}_7^{1-\frac{1}{p}}\left(q_2\right)\hfill \\ & \times \left\{{\mathsf{\Psi }}_5\left(q_1\right)\left({\mathsf{\Psi }}_1\left(q_2\right){\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(a,c\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p+{\mathsf{\Psi }}_2\left(q_2\right){\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(a,d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p\right)\right.\hfill \\ & {\left.+{\mathsf{\Psi }}_6\left(q_1\right)\left({\mathsf{\Psi }}_1\left(q_2\right){\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(b,c\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p+{\mathsf{\Psi }}_2\left(q_2\right){\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(b,d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p\right)\right\}}^{\frac{1}{p}}\hfill \\ & +{\mathsf{\Psi }}_9^{1-\frac{1}{p}}\left(q_1\right){\mathsf{\Psi }}_8^{1-\frac{1}{p}}\left(q_2\right)\hfill \\ & \times \left\{{\mathsf{\Psi }}_5\left(q_1\right)\left({\mathsf{\Psi }}_3\left(q_2\right){\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(a,c\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p+{\mathsf{\Psi }}_4\left(q_2\right){\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(a,d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p\right)\right.\hfill \\ & {\left.+{\mathsf{\Psi }}_6\left(q_1\right)\left({\mathsf{\Psi }}_3\left(q_2\right){\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(b,c\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p+{\mathsf{\Psi }}_4\left(q_2\right){\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(b,d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p\right)\right\}}^{\frac{1}{p}}\hfill \\ & +{\mathsf{\Psi }}_9^{1-\frac{1}{p}}\left(q_1\right){\mathsf{\Psi }}_9^{1-\frac{1}{p}}\left(q_2\right)\hfill \\ & \times \left\{{\mathsf{\Psi }}_5\left(q_1\right)\left({\mathsf{\Psi }}_5\left(q_2\right){\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(a,c\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p+{\mathsf{\Psi }}_6\left(q_2\right){\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(a,d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p\right)\right.\hfill \\ & {\left.+{\mathsf{\Psi }}_6\left(q_1\right)\left({\mathsf{\Psi }}_5\left(q_2\right){\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(b,c\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p+{\mathsf{\Psi }}_6\left(q_2\right){\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(b,d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p\right)\right\}}^{\frac{1}{p}},\hfill \end{array}$$

(50)

where $0<q_1,q_2<1.$

Proof. 

Applying well-known power mean inequality for $q_1{q}_2$-integrals to the integrals in right side of (41), it is found that

$$\begin{array}{cc}& \left|{}^{b,d}{\mathcal{I}}_{q_1,q_2}\right|\hfill \\ \le \hfill & \left(b-a\right)\left(d-c\right)\left[{\left({\int }_0^{\frac{1}{3}}{\int }_0^{\frac{1}{3}}\left|q_{1}t-\frac{1}{8}\right|\left|q_{2}s-\frac{1}{8}\right|d_{q_1}t{d}_{q_2}s\right)}^{1-\frac{1}{p}}\right.\hfill \\ & \times \left({\int }_0^{\frac{1}{3}}{\int }_0^{\frac{1}{3}}\left|q_{1}t-\frac{1}{8}\right|\left|q_{2}s-\frac{1}{8}\right|\right.\hfill \\ & {\left.\times {\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p{d}_{q_1}t{d}_{q_2}s\right)}^{\frac{1}{p}}\hfill \\ & +{\left({\int }_0^{\frac{1}{3}}{\int }_{\frac{1}{3}}^{\frac{2}{3}}\left|q_{1}t-\frac{1}{8}\right|\left|q_{2}s-\frac{1}{2}\right|d_{q_1}t{d}_{q_2}s\right)}^{1-\frac{1}{p}}\hfill \\ & \times \left({\int }_0^{\frac{1}{3}}{\int }_{\frac{1}{3}}^{\frac{2}{3}}\left|q_{1}t-\frac{1}{8}\right|\left|q_{2}s-\frac{1}{2}\right|\right.\hfill \\ & \hfill {\left.\times {\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^{p_1}{d}_{q_1}t{d}_{q_2}s\right)}^{\frac{1}{p}}\\ & +{\left({\int }_0^{\frac{1}{3}}{\int }_{\frac{2}{3}}^1\left|q_{1}t-\frac{1}{8}\right|\left|q_{2}s-\frac{7}{8}\right|d_{q_1}t{d}_{q_2}s\right)}^{1-\frac{1}{p}}\hfill \\ & \times \left({\int }_0^{\frac{1}{3}}{\int }_{\frac{2}{3}}^1\left|q_{1}t-\frac{1}{8}\right|\left|q_{2}s-\frac{7}{8}\right|\right.\hfill \\ & {\left.\times {\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p{d}_{q_1}t{d}_{q_2}s\right)}^{\frac{1}{p}}\hfill \\ & +{\left({\int }_{\frac{1}{3}}^{\frac{2}{3}}{\int }_0^{\frac{1}{3}}\left|q_{1}t-\frac{1}{2}\right|\left|q_{2}s-\frac{1}{8}\right|d_{q_1}t{d}_{q_2}s\right)}^{1-\frac{1}{p}}\hfill \\ & \times \left({\int }_{\frac{1}{3}}^{\frac{2}{3}}{\int }_0^{\frac{1}{3}}\left|q_{1}t-\frac{1}{2}\right|\left|q_{2}s-\frac{1}{8}\right|\right.\hfill \\ & {\left.\times {\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p{d}_{q_1}t{d}_{q_2}s\right)}^{\frac{1}{p}}\hfill \\ & +{\left({\int }_{\frac{1}{3}}^{\frac{2}{3}}{\int }_{\frac{1}{3}}^{\frac{2}{3}}\left|q_{1}t-\frac{1}{2}\right|\left|q_{2}s-\frac{1}{2}\right|d_{q_1}t{d}_{q_2}s\right)}^{1-\frac{1}{p}}\hfill \\ & \times \left({\int }_{\frac{1}{3}}^{\frac{2}{3}}{\int }_{\frac{1}{3}}^{\frac{2}{3}}\left|q_{1}t-\frac{1}{2}\right|\left|q_{2}s-\frac{1}{2}\right|\right.\hfill \\ & {\left.\times {\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p{d}_{q_1}t{d}_{q_2}s\right)}^{\frac{1}{p}}\hfill \\ & +{\left({\int }_{\frac{1}{3}}^{\frac{2}{3}}{\int }_{\frac{2}{3}}^1\left|q_{1}t-\frac{1}{2}\right|\left|q_{2}s-\frac{7}{8}\right|d_{q_1}t{d}_{q_2}s\right)}^{1-\frac{1}{p}}\hfill \\ & \times \left({\int }_{\frac{1}{3}}^{\frac{2}{3}}{\int }_{\frac{2}{3}}^1\left|q_{1}t-\frac{1}{2}\right|\left|q_{2}s-\frac{7}{8}\right|\right.\hfill \\ & {\left.\times {\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p{d}_{q_1}t{d}_{q_2}s\right)}^{\frac{1}{p}}\hfill \\ & +{\left({\int }_{\frac{2}{3}}^1{\int }_0^{\frac{1}{3}}\left|q_{1}t-\frac{7}{8}\right|\left|q_{2}s-\frac{1}{8}\right|d_{q_1}t{d}_{q_2}s\right)}^{1-\frac{1}{p}}\hfill \\ & \times \left({\int }_{\frac{2}{3}}^1{\int }_0^{\frac{1}{3}}\left|q_{1}t-\frac{7}{8}\right|\left|q_{2}s-\frac{1}{8}\right|\right.\hfill \\ & {\left.\times {\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p{d}_{q_1}t{d}_{q_2}s\right)}^{\frac{1}{p}}\hfill \\ & +{\left({\int }_{\frac{2}{3}}^1{\int }_{\frac{1}{3}}^{\frac{2}{3}}\left|q_{1}t-\frac{7}{8}\right|\left|q_{2}s-\frac{1}{2}\right|d_{q_1}t{d}_{q_2}s\right)}^{1-\frac{1}{p}}\hfill \\ & \times \left({\int }_{\frac{2}{3}}^1{\int }_{\frac{1}{3}}^{\frac{2}{3}}\left|q_{1}t-\frac{7}{8}\right|\left|q_{2}s-\frac{1}{2}\right|\right.\hfill \\ & {\left.\times {\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p{d}_{q_1}t{d}_{q_2}s\right)}^{\frac{1}{p}}\hfill \\ & +{\left({\int }_{\frac{2}{3}}^1{\int }_{\frac{2}{3}}^1\left|q_{1}t-\frac{7}{8}\right|\left|q_{2}s-\frac{7}{8}\right|d_{q_1}t{d}_{q_2}s\right)}^{1-\frac{1}{p}}\hfill \\ & \times \left({\int }_{\frac{2}{3}}^1{\int }_{\frac{2}{3}}^1\left|q_{1}t-\frac{7}{8}\right|\left|q_{2}s-\frac{7}{8}\right|\right.\hfill \\ & \left.{\times \left.{\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p{d}_{q_1}t{d}_{q_2}s\right)}^{\frac{1}{p}}\right].\hfill \end{array}$$

(51)

By applying convexity of ${\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(t,s\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p,$ we have

$$\begin{array}{cc}& {\left({\int }_0^{\frac{1}{3}}{\int }_0^{\frac{1}{3}}\left|q_{1}t-\frac{1}{8}\right|\left|q_{2}s-\frac{1}{8}\right|d_{q_1}t{d}_{q_2}s\right)}^{1-\frac{1}{p}}\hfill \\ & \times \left({\int }_0^{\frac{1}{3}}{\int }_0^{\frac{1}{3}}\left|q_{1}t-\frac{1}{8}\right|\left|q_{2}s-\frac{1}{8}\right|\right.\hfill \\ & {\left.\times {\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p{d}_{q_1}t{d}_{q_2}s\right)}^{\frac{1}{p}}\hfill \\ =& {\mathsf{\Psi }}_7^{1-\frac{1}{p}}\left(q_1\right){\mathsf{\Psi }}_7^{1-\frac{1}{p}}\left(q_2\right)\hfill \\ & \times \left[{\int }_0^{\frac{1}{3}}\left|q_{2}s-\frac{1}{8}\right|\left\{{\int }_0^{\frac{1}{3}}\left|q_{1}t-\frac{1}{8}\right|\right.\right.\hfill \\ & {\left.\left.\times {\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p{d}_{q_1}t\right\}{d}_{q_2}s\right]}^{\frac{1}{p}}\hfill \\ \le & {\mathsf{\Psi }}_7^{1-\frac{1}{p}}\left(q_1\right){\mathsf{\Psi }}_7^{1-\frac{1}{p}}\left(q_2\right)\hfill \\ & \times \left[{\int }_0^{\frac{1}{3}}\left|q_{2}s-\frac{1}{8}\right|\left\{{\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(a,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p{\int }_0^{\frac{1}{3}}t\left|q_{1}t-\frac{1}{8}\right|\right.\right.\hfill \\ & {\left.\left.+{\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p{\int }_0^{\frac{1}{3}}\left(1-t\right)\left|q_{1}t-\frac{1}{8}\right|\right\}\right]}^{\frac{1}{p}}\hfill \\ =& {\mathsf{\Psi }}_7^{1-\frac{1}{p}}\left(q_1\right){\mathsf{\Psi }}_7^{1-\frac{1}{p}}\left(q_2\right)\hfill \\ & \times \left[{\int }_0^{\frac{1}{3}}\left|q_{2}s-\frac{1}{8}\right|\left\{{\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(a,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p{\mathsf{\Psi }}_1\left(q_1\right)\right.\right.\hfill \\ & {\left.\left.+{\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(b,sc+\left(1-s\right)d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p{\mathsf{\Psi }}_2\left(q_1\right)\right\}\right]}^{\frac{1}{p}}\hfill \\ \le & {\mathsf{\Psi }}_7^{1-\frac{1}{p}}\left(q_1\right){\mathsf{\Psi }}_7^{1-\frac{1}{p}}\left(q_2\right)\hfill \\ & \times \left[{\mathsf{\Psi }}_1\left(q_1\right)\left\{{\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(a,c\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p{\int }_0^{\frac{1}{3}}s\left|q_{2}s-\frac{1}{8}\right|+{\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(a,d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p{\int }_0^{\frac{1}{3}}\left(1-s\right)\left|q_{2}s-\frac{1}{8}\right|\right\}\right.\hfill \\ & {\left.+{\mathsf{\Psi }}_1\left(q_1\right)\left\{{\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(b,c\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p{\int }_0^{\frac{1}{3}}s\left|q_{2}s-\frac{1}{8}\right|+{\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(b,d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p{\int }_0^{\frac{1}{3}}\left(1-s\right)\left|q_{2}s-\frac{1}{8}\right|\right\}\right]}^{\frac{1}{p}}\hfill \end{array}$$

$$\begin{array}{cc}=& {\mathsf{\Psi }}_7^{1-\frac{1}{p}}\left(q_1\right){\mathsf{\Psi }}_7^{1-\frac{1}{p}}\left(q_2\right)\hfill \\ & \times \left[{\mathsf{\Psi }}_1\left(q_1\right)\left\{{\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(a,c\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p{\mathsf{\Psi }}_1\left(q_2\right)+{\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(a,d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p{\mathsf{\Psi }}_2\left(q_2\right)\right\}\right.\hfill \\ & {\left.+{\mathsf{\Psi }}_1\left(q_1\right)\left\{{\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(b,c\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p{\mathsf{\Psi }}_1\left(q_2\right)+{\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F\left(b,d\right)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|}^p{\mathsf{\Psi }}_2\left(q_2\right)\right\}\right]}^{\frac{1}{p}}.\hfill \end{array}$$

By using similar operations we can compute the remaining integrals that involved in (51). Thus, the proof is finished. □

Remark 3.

Under the given conditions of Theorem 9 with $q_1,q_2\to 1^{-},$ we attain the following new inequality

$$\begin{array}{cc}& \left|\mathcal{I}\left(F\right)\right|\hfill \\ \le \hfill & \frac{\left(b-a\right)\left(d-c\right)}{1296}\left[{\left(\frac{289}{256}\right)}^{1-\frac{1}{p}}\left\{\frac{251}{1152}\left(\frac{251}{1152}{\left|\frac{{\partial }^{2}F\left(a,c\right)}{\partial t\partial s}\right|}^p+\frac{973}{1152}{\left|\frac{{\partial }^{2}F\left(a,d\right)}{\partial t\partial s}\right|}^p\right)\right.\right.\hfill \\ & {\left.+\frac{973}{1152}\left(\frac{251}{1152}{\left|\frac{{\partial }^{2}F\left(b,c\right)}{\partial t\partial s}\right|}^p+\frac{973}{1152}{\left|\frac{{\partial }^{2}F\left(b,d\right)}{\partial t\partial s}\right|}^p\right)\right\}}^{\frac{1}{p}}\hfill \\ & +{\left(\frac{17}{16}\right)}^{1-\frac{1}{p}}\left\{\frac{251}{1152}\left(\frac{1}{2}{\left|\frac{{\partial }^{2}F\left(a,c\right)}{\partial t\partial s}\right|}^p+\frac{1}{2}{\left|\frac{{\partial }^{2}F\left(a,d\right)}{\partial t\partial s}\right|}^p\right)\right.\hfill \\ & {\left.+\frac{973}{1152}\left(\frac{1}{2}{\left|\frac{{\partial }^{2}F\left(b,c\right)}{\partial t\partial s}\right|}^p+\frac{1}{2}{\left|\frac{{\partial }^{2}F\left(b,d\right)}{\partial t\partial s}\right|}^p\right)\right\}}^{\frac{1}{p}}\hfill \\ & +{\left(\frac{289}{56}\right)}^{1-\frac{1}{p}}\left\{\frac{251}{1152}\left(\frac{973}{1152}{\left|\frac{{\partial }^{2}F\left(a,c\right)}{\partial t\partial s}\right|}^p+\frac{251}{1152}{\left|\frac{{\partial }^{2}F\left(a,d\right)}{\partial t\partial s}\right|}^p\right)\right.\hfill \\ & {\left.+\frac{973}{1152}\left(\frac{973}{1152}{\left|\frac{{\partial }^{2}F\left(b,c\right)}{\partial t\partial s}\right|}^p+\frac{251}{1152}{\left|\frac{{\partial }^{2}F\left(b,d\right)}{\partial t\partial s}\right|}^p\right)\right\}}^{\frac{1}{p}}\hfill \\ & +{\left(\frac{17}{16}\right)}^{1-\frac{1}{p}}\left\{\frac{1}{2}\left(\frac{251}{1152}{\left|\frac{{\partial }^{2}F\left(a,c\right)}{\partial t\partial s}\right|}^p+\frac{973}{1152}{\left|\frac{{\partial }^{2}F\left(a,d\right)}{\partial t\partial s}\right|}^p\right)\right.\hfill \\ & {\left.+\frac{1}{2}\left(\frac{251}{1152}{\left|\frac{{\partial }^{2}F\left(b,c\right)}{\partial t\partial s}\right|}^p+\frac{973}{1152}{\left|\frac{{\partial }^{2}F\left(b,d\right)}{\partial t\partial s}\right|}^p\right)\right\}}^{\frac{1}{p}}\hfill \\ & +\left\{\frac{1}{2}\left(\frac{1}{2}{\left|\frac{{\partial }^{2}F\left(a,c\right)}{\partial t\partial s}\right|}^p+\frac{1}{2}{\left|\frac{{\partial }^{2}F\left(a,d\right)}{\partial t\partial s}\right|}^p\right)\right.\hfill \\ & {\left.+\frac{1}{2}\left(\frac{1}{2}{\left|\frac{{\partial }^{2}F\left(a,c\right)}{\partial t\partial s}\right|}^p+\frac{1}{2}{\left|\frac{{\partial }^{2}F\left(a,d\right)}{\partial t\partial s}\right|}^p\right)\right\}}^{\frac{1}{p}}\hfill \\ & +{\left(\frac{17}{16}\right)}^{1-\frac{1}{p}}\left\{\frac{1}{2}\left(\frac{973}{1152}{\left|\frac{{\partial }^{2}F\left(b,c\right)}{\partial t\partial s}\right|}^p+\frac{251}{1152}{\left|\frac{{\partial }^{2}F\left(b,d\right)}{\partial t\partial s}\right|}^p\right)\right.\hfill \\ & {\left.+\frac{1}{2}\left(\frac{973}{1152}{\left|\frac{{\partial }^{2}F\left(b,c\right)}{\partial t\partial s}\right|}^p+\frac{251}{1152}{\left|\frac{{\partial }^{2}F\left(b,d\right)}{\partial t\partial s}\right|}^p\right)\right\}}^{\frac{1}{p}}\hfill \\ & +{\left(\frac{289}{256}\right)}^{1-\frac{1}{p}}\left\{\frac{973}{1152}\left(\frac{251}{1152}{\left|\frac{{\partial }^{2}F\left(a,c\right)}{\partial t\partial s}\right|}^p+\frac{973}{1152}{\left|\frac{{\partial }^{2}F\left(a,d\right)}{\partial t\partial s}\right|}^p\right)\right.\hfill \\ & {\left.+\frac{251}{1152}\left(\frac{251}{1152}{\left|\frac{{\partial }^{2}F\left(b,c\right)}{\partial t\partial s}\right|}^p+\frac{973}{1152}{\left|\frac{{\partial }^{2}F\left(b,d\right)}{\partial t\partial s}\right|}^p\right)\right\}}^{\frac{1}{p}}\hfill \\ & +{\left(\frac{17}{16}\right)}^{1-\frac{1}{p}}\left\{\frac{973}{1152}\left(\frac{1}{2}{\left|\frac{{\partial }^{2}F\left(a,c\right)}{\partial t\partial s}\right|}^p+\frac{1}{2}{\left|\frac{{\partial }^{2}F\left(a,d\right)}{\partial t\partial s}\right|}^p\right)\right.\hfill \\ & {\left.+\frac{251}{1152}\left(\frac{1}{2}{\left|\frac{{\partial }^{2}F\left(b,c\right)}{\partial t\partial s}\right|}^p+\frac{1}{2}{\left|\frac{{\partial }^{2}F\left(b,d\right)}{\partial t\partial s}\right|}^p\right)\right\}}^{\frac{1}{p}}\hfill \\ & +{\left(\frac{289}{256}\right)}^{1-\frac{1}{p}}\left\{\frac{973}{1152}\left(\frac{973}{1152}{\left|\frac{{\partial }^{2}F\left(a,c\right)}{\partial t\partial s}\right|}^p+\frac{251}{1152}{\left|\frac{{\partial }^{2}F\left(a,d\right)}{\partial t\partial s}\right|}^p\right)\right.\hfill \\ & {\left.+\frac{251}{1152}\left(\frac{973}{1152}{\left|\frac{{\partial }^{2}F\left(b,c\right)}{\partial t\partial s}\right|}^p+\frac{251}{1152}{\left|\frac{{\partial }^{2}F\left(b,d\right)}{\partial t\partial s}\right|}^p\right)\right\}}^{\frac{1}{p}}.\hfill \end{array}$$

Corollary 3.

Under the given conditions of Theorem 9 if $\left|\frac{{}^{b,d}{\partial }_{q_1,q_2}^{2}F(t,s)}{{}^b{\partial }_{q_1}t{}^d{\partial }_{q_2}s}\right|\le K,$ we obtain

$$\begin{array}{cc}& \left|{}^{b,d}{\mathcal{I}}_{q_1,q_2}\left(F\right)\right|\hfill \\ \le \hfill & K\left(b-a\right)\left(d-c\right)\left[{\mathsf{\Psi }}_7^{1-\frac{1}{p}}\left(q_1\right){\mathsf{\Psi }}_7^{1-\frac{1}{p}}\left(q_2\right)\right.\hfill \\ & \times {\left\{\left({\mathsf{\Psi }}_1\left(q_1\right)+{\mathsf{\Psi }}_2\left(q_1\right)\right)\left({\mathsf{\Psi }}_1\left(q_2\right)+{\mathsf{\Psi }}_2\left(q_2\right)\right)\right\}}^{\frac{1}{p}}\hfill \end{array}$$

$$\begin{array}{c}\hfill \begin{array}{c}+{\mathsf{\Psi }}_7^{1-\frac{1}{p}}\left(q_1\right){\mathsf{\Psi }}_8^{1-\frac{1}{p}}\left(q_2\right){\left\{\left({\mathsf{\Psi }}_1\left(q_1\right)+{\mathsf{\Psi }}_2\left(q_1\right)\right)\left({\mathsf{\Psi }}_3\left(q_2\right)+{\mathsf{\Psi }}_4\left(q_2\right)\right)\right\}}^{\frac{1}{p}}\hfill \\ +{\mathsf{\Psi }}_7^{1-\frac{1}{p}}\left(q_1\right){\mathsf{\Psi }}_9^{1-\frac{1}{p}}\left(q_2\right){\left\{\left({\mathsf{\Psi }}_1\left(q_1\right)+{\mathsf{\Psi }}_2\left(q_1\right)\right)\left({\mathsf{\Psi }}_5\left(q_2\right)+{\mathsf{\Psi }}_6\left(q_2\right)\right)\right\}}^{\frac{1}{p}}\hfill \\ +{\mathsf{\Psi }}_8^{1-\frac{1}{p}}\left(q_1\right){\mathsf{\Psi }}_7^{1-\frac{1}{p}}\left(q_2\right){\left\{\left({\mathsf{\Psi }}_3\left(q_1\right)+{\mathsf{\Psi }}_4\left(q_1\right)\right)\left({\mathsf{\Psi }}_1\left(q_2\right)+{\mathsf{\Psi }}_2\left(q_2\right)\right)\right\}}^{\frac{1}{p}}\hfill \\ +{\mathsf{\Psi }}_8^{1-\frac{1}{p}}\left(q_1\right){\mathsf{\Psi }}_8^{1-\frac{1}{p}}\left(q_2\right){\left\{\left({\mathsf{\Psi }}_3\left(q_1\right)+{\mathsf{\Psi }}_4\left(q_1\right)\right)\left({\mathsf{\Psi }}_3\left(q_2\right)+{\mathsf{\Psi }}_4\left(q_2\right)\right)\right\}}^{\frac{1}{p}}\hfill \\ +{\mathsf{\Psi }}_8^{1-\frac{1}{p}}\left(q_1\right){\mathsf{\Psi }}_9^{1-\frac{1}{p}}\left(q_2\right){\left\{\left({\mathsf{\Psi }}_3\left(q_1\right)+{\mathsf{\Psi }}_4\left(q_1\right)\right)\left({\mathsf{\Psi }}_5\left(q_2\right)+{\mathsf{\Psi }}_6\left(q_2\right)\right)\right\}}^{\frac{1}{p}}\hfill \\ +{\mathsf{\Psi }}_9^{1-\frac{1}{p}}\left(q_1\right){\mathsf{\Psi }}_7^{1-\frac{1}{p}}\left(q_2\right){\left\{\left({\mathsf{\Psi }}_5\left(q_1\right)+{\mathsf{\Psi }}_6\left(q_1\right)\right)\left({\mathsf{\Psi }}_1\left(q_2\right)+{\mathsf{\Psi }}_2\left(q_2\right)\right)\right\}}^{\frac{1}{p}}\hfill \\ +{\mathsf{\Psi }}_9^{1-\frac{1}{p}}\left(q_1\right){\mathsf{\Psi }}_8^{1-\frac{1}{p}}\left(q_2\right){\left\{\left({\mathsf{\Psi }}_5\left(q_1\right)+{\mathsf{\Psi }}_6\left(q_1\right)\right)\left({\mathsf{\Psi }}_3\left(q_2\right)+{\mathsf{\Psi }}_4\left(q_2\right)\right)\right\}}^{\frac{1}{p}}\hfill \\ \left.+{\mathsf{\Psi }}_9^{1-\frac{1}{p}}\left(q_1\right){\mathsf{\Psi }}_9^{1-\frac{1}{p}}\left(q_2\right){\left\{\left({\mathsf{\Psi }}_5\left(q_1\right)+{\mathsf{\Psi }}_6\left(q_1\right)\right)\left({\mathsf{\Psi }}_5\left(q_2\right)+{\mathsf{\Psi }}_6\left(q_2\right)\right)\right\}}^{\frac{1}{p}}\right],\hfill \end{array}\end{array}$$

where $0<q_1,q_2<1.$

6. Applications

In this section, using our main results we can derive some applications for the following two convex functions:

$$F(t,s)=t^m{s}^n,\mathrm{where}m,n>1\mathrm{and}F(t,s)=e^{t+s}.$$

Since

$$\begin{array}{ccc}\hfill \mathcal{I}\left(F\right)& =& \frac{3}{64}\left[F\left(\frac{a+2b}{3},d\right)+F\left(b,\frac{c+2d}{3}\right)+3F\left(\frac{a+2b}{3},\frac{c+2d}{3}\right)+3F\left(\frac{a+2b}{3},\frac{2c+d}{3}\right)\right.\hfill \\ & & +F\left(\frac{2a+b}{3},d\right)+3F\left(\frac{2a+b}{3},\frac{c+2d}{3}\right)+F\left(b,\frac{2c+d}{3}\right)+3F\left(\frac{2a+b}{3},\frac{2c+d}{3}\right)\hfill \\ & & \left.+F\left(\frac{a+2b}{3},c\right)+F\left(\frac{2a+b}{3},c\right)+F\left(a,\frac{c+2d}{3}\right)+F\left(a,\frac{2c+d}{3}\right)\right]\hfill \\ & & +\frac{F\left(a,c\right)+F\left(a,d\right)+F\left(b,c\right)+F\left(b,d\right)}{64}\hfill \\ & & -\frac{1}{8\left(b-a\right)}{\int }_a^b\left[F\left(x,c\right)+3F\left(x,\frac{c+2d}{3}\right)+3F\left(x,\frac{2c+d}{3}\right)+F\left(x,d\right)\right]dx\hfill \\ & & -\frac{1}{8\left(d-c\right)}{\int }_c^d\left[F\left(a,y\right)+3F\left(\frac{a+2b}{3},y\right)+3F\left(\frac{2a+b}{3},y\right)+F\left(b,y\right)\right]dy\hfill \\ & & +\frac{1}{\left(b-a\right)\left(d-c\right)}{\int }_a^b{\int }_c^{d}F\left(x,y\right)dxdy,\hfill \end{array}$$

we need to calculate first each one of the above expressions for both convex functions, respectively.

$$F\left(\frac{a+2b}{3},d\right)={\left(\frac{a+2b}{3}\right)}^m{d}^n,F\left(b,\frac{c+2d}{3}\right)=b^m{\left(\frac{c+2d}{3}\right)}^n,$$

$$F\left(\frac{a+2b}{3},\frac{c+2d}{3}\right)={\left(\frac{a+2b}{3}\right)}^m{\left(\frac{c+2d}{3}\right)}^n,F\left(\frac{a+2b}{3},\frac{2c+d}{3}\right)={\left(\frac{a+2b}{3}\right)}^m{\left(\frac{2c+d}{3}\right)}^n,$$

$$F\left(\frac{2a+b}{3},d\right)={\left(\frac{2a+b}{3}\right)}^m{d}^n,F\left(\frac{2a+b}{3},\frac{c+2d}{3}\right)={\left(\frac{2a+b}{3}\right)}^m{\left(\frac{c+2d}{3}\right)}^n,$$

$$F\left(b,\frac{2c+d}{3}\right)=b^m{\left(\frac{2c+d}{3}\right)}^n,F\left(\frac{2a+b}{3},\frac{2c+d}{3}\right)={\left(\frac{2a+b}{3}\right)}^m{\left(\frac{2c+d}{3}\right)}^n,$$

$$F\left(\frac{a+2b}{3},c\right)={\left(\frac{a+2b}{3}\right)}^m{c}^n,F\left(\frac{2a+b}{3},c\right)={\left(\frac{2a+b}{3}\right)}^m{c}^n,$$

$$F\left(a,\frac{c+2d}{3}\right)=a^m{\left(\frac{c+2d}{3}\right)}^n,F\left(a,\frac{2c+d}{3}\right)=a^m{\left(\frac{2c+d}{3}\right)}^n,$$

$$F\left(a,c\right)=a^m{c}^n,F\left(a,d\right)=a^m{d}^n,F\left(b,c\right)=b^m{c}^n,F\left(b,d\right)=b^m{d}^n,$$

$${\int }_a^{b}F\left(x,c\right)dx={\int }_a^b{x}^m{c}^{n}dx=c^n\left(\frac{b^{m+1}-a^{m+1}}{m+1}\right),$$

$${\int }_a^{b}F\left(x,\frac{c+2d}{3}\right)dx={\int }_a^b{x}^m{\left(\frac{c+2d}{3}\right)}^{n}dx={\left(\frac{c+2d}{3}\right)}^n\left(\frac{b^{m+1}-a^{m+1}}{m+1}\right),$$

$${\int }_a^{b}F\left(x,\frac{2c+d}{3}\right)dx={\int }_a^b{x}^m{\left(\frac{2c+d}{3}\right)}^{n}dx={\left(\frac{2c+d}{3}\right)}^n\left(\frac{b^{m+1}-a^{m+1}}{m+1}\right),$$

$${\int }_a^{b}F\left(x,d\right)dx={\int }_a^b{x}^m{d}^{n}dx=d^n\left(\frac{b^{m+1}-a^{m+1}}{m+1}\right),$$

$${\int }_c^{d}F\left(a,y\right)dy={\int }_c^d{a}^m{y}^{n}dy=a^m\left(\frac{d^{n+1}-c^{n+1}}{n+1}\right),$$

$${\int }_c^{d}F\left(\frac{a+2b}{3},y\right)dy={\int }_c^d{\left(\frac{a+2b}{3}\right)}^m{y}^{n}dy={\left(\frac{a+2b}{3}\right)}^m\left(\frac{d^{n+1}-c^{n+1}}{n+1}\right),$$

$${\int }_c^{d}F\left(\frac{2a+b}{3},y\right)dy={\int }_c^d{\left(\frac{2a+b}{3}\right)}^m{y}^{n}dy={\left(\frac{2a+b}{3}\right)}^m\left(\frac{d^{n+1}-c^{n+1}}{n+1}\right),$$

$${\int }_c^{d}F\left(b,y\right)dy={\int }_c^d{b}^m{y}^{n}dy=b^m\left(\frac{d^{n+1}-c^{n+1}}{n+1}\right),$$

$${\int }_a^b{\int }_c^{d}F\left(x,y\right)dxdy={\int }_a^b{\int }_c^d{x}^m{y}^{n}dxdy=\left(\frac{b^{m+1}-a^{m+1}}{m+1}\right)\left(\frac{d^{n+1}-c^{n+1}}{n+1}\right);$$

and

$$F\left(\frac{a+2b}{3},d\right)=e^{\frac{a+2b}{3}+d},F\left(b,\frac{c+2d}{3}\right)=e^{b+\frac{c+2d}{3}},$$

$$F\left(\frac{a+2b}{3},\frac{c+2d}{3}\right)=e^{\frac{a+2b}{3}+\frac{c+2d}{3}},F\left(\frac{a+2b}{3},\frac{2c+d}{3}\right)=e^{\frac{a+2b}{3}+\frac{2c+d}{3}},$$

$$F\left(\frac{2a+b}{3},d\right)=e^{\frac{2a+b}{3}+d},F\left(\frac{2a+b}{3},\frac{c+2d}{3}\right)=e^{\frac{2a+b}{3}+\frac{c+2d}{3}},$$

$$F\left(b,\frac{2c+d}{3}\right)=e^{b+\frac{2c+d}{3}},F\left(\frac{2a+b}{3},\frac{2c+d}{3}\right)=e^{\frac{2a+b}{3}+\frac{2c+d}{3}},$$

$$F\left(\frac{a+2b}{3},c\right)=e^{\frac{a+2b}{3}+c},F\left(\frac{2a+b}{3},c\right)=e^{\frac{2a+b}{3}+c},$$

$$F\left(a,\frac{c+2d}{3}\right)=e^{a+\frac{c+2d}{3}},F\left(a,\frac{2c+d}{3}\right)=e^{a+\frac{2c+d}{3}},$$

$$F\left(a,c\right)=e^{a+c},F\left(a,d\right)=e^{a+d},F\left(b,c\right)=e^{b+c},F\left(b,d\right)=e^{b+d},$$

$${\int }_a^{b}F\left(x,c\right)dx={\int }_a^b{e}^{x+c}dx=e^{b+c}-e^{a+c}=e^c\left(e^b-e^a\right),$$

$${\int }_a^{b}F\left(x,\frac{c+2d}{3}\right)dx={\int }_a^b{e}^{x+\frac{c+2d}{3}}dx=e^{b+\frac{c+2d}{3}}-e^{a+\frac{c+2d}{3}}=e^{\frac{c+2d}{3}}\left(e^b-e^a\right),$$

$${\int }_a^{b}F\left(x,\frac{2c+d}{3}\right)dx={\int }_a^b{e}^{x+\frac{2c+d}{3}}dx=e^{b+\frac{2c+d}{3}}-e^{a+\frac{2c+d}{3}}=e^{\frac{2c+d}{3}}\left(e^b-e^a\right),$$

$${\int }_a^{b}F\left(x,d\right)dx={\int }_a^b{e}^{x+d}dx=e^{b+d}-e^{a+d}=e^d\left(e^b-e^a\right),$$

$${\int }_c^{d}F\left(a,y\right)dy={\int }_c^d{e}^{a+y}dy=e^{a+d}-e^{a+c}=e^a\left(e^d-e^c\right),$$

$${\int }_c^{d}F\left(\frac{a+2b}{3},y\right)dy={\int }_c^d{e}^{\frac{a+2b}{3}+y}dy=e^{\frac{a+2b}{3}+d}-e^{\frac{a+2b}{3}+c}=e^{\frac{a+2b}{3}}\left(e^d-e^c\right),$$

$${\int }_c^{d}F\left(\frac{2a+b}{3},y\right)dy={\int }_c^d{e}^{\frac{2a+b}{3}+y}dy=e^{\frac{2a+b}{3}+d}-e^{\frac{2a+b}{3}+c}=e^{\frac{2a+b}{3}}\left(e^d-e^c\right),$$

$${\int }_c^{d}F\left(b,y\right)dy={\int }_c^d{e}^{b+y}dy=e^{b+d}-e^{b+c}=e^b\left(e^d-e^c\right),$$

$${\int }_a^b{\int }_c^{d}F\left(x,y\right)dxdy={\int }_a^b{\int }_c^d{e}^{x+y}dxdy=\left(e^b-e^a\right)\left(e^d-e^c\right).$$

Proposition 1.

Let $0<a<b$ and $0<c<d,$ where $m,n>1.$ Then, we get

$$\begin{array}{c}\left|\mathcal{I}\left(F\right)\right|\le \frac{625mn\left(b-a\right)\left(d-c\right)}{576}\left[\frac{\left(a^{m-1}+b^{m-1}\right)\left(c^{n-1}+d^{n-1}\right)}{576}\right].\hfill \end{array}$$

(52)

Proof. 

Using Theorem 7 with $q_1,q_2\to 1^{-}$ and taking $F(t,s)=t^m{s}^n,$ where $m,n>1,$ the desire result is obtained. □

Proposition 2.

Let $a,b,c,d\in \mathbb{R}.$ Then, we have

$$\begin{array}{c}\left|\mathcal{I}\left(F\right)\right|\le \frac{625\left(b-a\right)\left(d-c\right)}{576}\left[\frac{\left(e^a+e^b\right)\left(e^c+e^d\right)}{576}\right].\hfill \end{array}$$

(53)

Proof. 

Using Theorem 7 with $q_1,q_2\to 1^{-}$ and choosing $F(t,s)=e^{t+s},$ the above result is captured. □

Proposition 3.

Let $0<a<b$ and $0<c<d,$ where $m,n>1.$ Then for $p\ge 1,$ we get

$$\begin{array}{cc}& \left|\mathcal{I}\left(F\right)\right|\hfill \\ \le \hfill & \frac{mn\left(b-a\right)\left(d-c\right)}{1296}\left[{\left(\frac{289}{256}\right)}^{1-\frac{1}{p}}\left\{\frac{251}{1152}\left(\frac{251}{1152}{a}^{p(m-1)}{c}^{p(n-1)}+\frac{973}{1152}{a}^{p(m-1)}{d}^{p(n-1)}\right)\right.\right.\hfill \\ & {\left.+\frac{973}{1152}\left(\frac{251}{1152}{b}^{p(m-1)}{c}^{p(n-1)}+\frac{973}{1152}{b}^{p(m-1)}{d}^{p(n-1)}\right)\right\}}^{\frac{1}{p}}\hfill \\ & +{\left(\frac{17}{16}\right)}^{1-\frac{1}{p}}\left\{\frac{251}{1152}\left(\frac{1}{2}{a}^{p(m-1)}{c}^{p(n-1)}+\frac{1}{2}{a}^{p(m-1)}{d}^{p(n-1)}\right)\right.\hfill \\ & {\left.+\frac{973}{1152}\left(\frac{1}{2}{b}^{p(m-1)}{c}^{p(n-1)}+\frac{1}{2}{b}^{p(m-1)}{d}^{p(n-1)}\right)\right\}}^{\frac{1}{p}}\hfill \\ & +{\left(\frac{289}{56}\right)}^{1-\frac{1}{p}}\left\{\frac{251}{1152}\left(\frac{973}{1152}{a}^{p(m-1)}{c}^{p(n-1)}+\frac{251}{1152}{a}^{p(m-1)}{d}^{p(n-1)}\right)\right.\hfill \\ & {\left.+\frac{973}{1152}\left(\frac{973}{1152}{b}^{p(m-1)}{c}^{p(n-1)}+\frac{251}{1152}{b}^{p(m-1)}{d}^{p(n-1)}\right)\right\}}^{\frac{1}{p}}\hfill \\ & +{\left(\frac{17}{16}\right)}^{1-\frac{1}{p}}\left\{\frac{1}{2}\left(\frac{251}{1152}{a}^{p(m-1)}{c}^{p(n-1)}+\frac{973}{1152}{a}^{p(m-1)}{d}^{p(n-1)}\right)\right.\hfill \\ & {\left.+\frac{1}{2}\left(\frac{251}{1152}{b}^{p(m-1)}{c}^{p(n-1)}+\frac{973}{1152}{b}^{p(m-1)}{d}^{p(n-1)}\right)\right\}}^{\frac{1}{p}}\hfill \\ & +\left\{\frac{1}{2}\left(\frac{1}{2}{a}^{p(m-1)}{c}^{p(n-1)}+\frac{1}{2}{a}^{p(m-1)}{d}^{p(n-1)}\right)\right.\hfill \\ & {\left.+\frac{1}{2}\left(\frac{1}{2}{a}^{p(m-1)}{c}^{p(n-1)}+\frac{1}{2}{a}^{p(m-1)}{d}^{p(n-1)}\right)\right\}}^{\frac{1}{p}}\hfill \\ & +{\left(\frac{17}{16}\right)}^{1-\frac{1}{p}}\left\{\frac{1}{2}\left(\frac{973}{1152}{b}^{p(m-1)}{c}^{p(n-1)}+\frac{251}{1152}{b}^{p(m-1)}{d}^{p(n-1)}\right)\right.\hfill \\ & {\left.+\frac{1}{2}\left(\frac{973}{1152}{b}^{p(m-1)}{c}^{p(n-1)}+\frac{251}{1152}{b}^{p(m-1)}{d}^{p(n-1)}\right)\right\}}^{\frac{1}{p}}\hfill \\ & +{\left(\frac{289}{256}\right)}^{1-\frac{1}{p}}\left\{\frac{973}{1152}\left(\frac{251}{1152}{a}^{p(m-1)}{c}^{p(n-1)}+\frac{973}{1152}{a}^{p(m-1)}{d}^{p(n-1)}\right)\right.\hfill \\ & {\left.+\frac{251}{1152}\left(\frac{251}{1152}{b}^{p(m-1)}{c}^{p(n-1)}+\frac{973}{1152}{b}^{p(m-1)}{d}^{p(n-1)}\right)\right\}}^{\frac{1}{p}}\hfill \\ & +{\left(\frac{17}{16}\right)}^{1-\frac{1}{p}}\left\{\frac{973}{1152}\left(\frac{1}{2}{a}^{p(m-1)}{c}^{p(n-1)}+\frac{1}{2}{a}^{p(m-1)}{d}^{p(n-1)}\right)\right.\hfill \\ & {\left.+\frac{251}{1152}\left(\frac{1}{2}{b}^{p(m-1)}{c}^{p(n-1)}+\frac{1}{2}{b}^{p(m-1)}{d}^{p(n-1)}\right)\right\}}^{\frac{1}{p}}\hfill \\ & +{\left(\frac{289}{256}\right)}^{1-\frac{1}{p}}\left\{\frac{973}{1152}\left(\frac{973}{1152}{a}^{p(m-1)}{c}^{p(n-1)}+\frac{251}{1152}{a}^{p(m-1)}{d}^{p(n-1)}\right)\right.\hfill \\ & \left.{\left.+\frac{251}{1152}\left(\frac{973}{1152}{b}^{p(m-1)}{c}^{p(n-1)}+\frac{251}{1152}{b}^{p(m-1)}{d}^{p(n-1)}\right)\right\}}^{\frac{1}{p}}\right].\hfill \end{array}$$

Proof. 

Using Theorem 9 with $q_1,q_2\to 1^{-}$ and taking $F(t,s)=t^m{s}^n,$ where $m,n>1,$ the desire result is obtained. □

Proposition 4.

Let $a,b,c,d\in \mathbb{R}.$ Then for $p\ge 1,$ we have

$$\begin{array}{cc}& \left|\mathcal{I}\left(F\right)\right|\hfill \\ \le \hfill & \frac{\left(b-a\right)\left(d-c\right)}{1296}\left[{\left(\frac{289}{256}\right)}^{1-\frac{1}{p}}\left\{\frac{251}{1152}\left(\frac{251}{1152}{e}^{p(a+c)}+\frac{973}{1152}{e}^{p(a+d)}\right)\right.\right.\hfill \\ & {\left.+\frac{973}{1152}\left(\frac{251}{1152}{e}^{p(b+c)}+\frac{973}{1152}{e}^{p(b+d)}\right)\right\}}^{\frac{1}{p}}\hfill \\ & +{\left(\frac{17}{16}\right)}^{1-\frac{1}{p}}\left\{\frac{251}{1152}\left(\frac{1}{2}{e}^{p(a+c)}+\frac{1}{2}{e}^{p(a+d)}\right)\right.\hfill \\ & {\left.+\frac{973}{1152}\left(\frac{1}{2}{e}^{p(b+c)}+\frac{1}{2}{e}^{p(b+d)}\right)\right\}}^{\frac{1}{p}}\hfill \\ & +{\left(\frac{289}{56}\right)}^{1-\frac{1}{p}}\left\{\frac{251}{1152}\left(\frac{973}{1152}{e}^{p(a+c)}+\frac{251}{1152}{e}^{p(a+d)}\right)\right.\hfill \\ & {\left.+\frac{973}{1152}\left(\frac{973}{1152}{e}^{p(b+c)}+\frac{251}{1152}{e}^{p(b+d)}\right)\right\}}^{\frac{1}{p}}\hfill \\ & +{\left(\frac{17}{16}\right)}^{1-\frac{1}{p}}\left\{\frac{1}{2}\left(\frac{251}{1152}{e}^{p(a+c)}+\frac{973}{1152}{e}^{p(a+d)}\right)\right.\hfill \\ & {\left.+\frac{1}{2}\left(\frac{251}{1152}{e}^{p(b+c)}+\frac{973}{1152}{e}^{p(b+d)}\right)\right\}}^{\frac{1}{p}}\hfill \\ & +\left\{\frac{1}{2}\left(\frac{1}{2}{e}^{p(a+c)}+\frac{1}{2}{e}^{p(a+d)}\right)\right.\hfill \\ & {\left.+\frac{1}{2}\left(\frac{1}{2}{e}^{p(a+c)}+\frac{1}{2}{e}^{p(a+d)}\right)\right\}}^{\frac{1}{p}}\hfill \\ & +{\left(\frac{17}{16}\right)}^{1-\frac{1}{p}}\left\{\frac{1}{2}\left(\frac{973}{1152}{e}^{p(b+c)}+\frac{251}{1152}{e}^{p(b+d)}\right)\right.\hfill \\ & {\left.+\frac{1}{2}\left(\frac{973}{1152}{e}^{p(b+c)}+\frac{251}{1152}{e}^{p(b+d)}\right)\right\}}^{\frac{1}{p}}\hfill \\ & +{\left(\frac{289}{256}\right)}^{1-\frac{1}{p}}\left\{\frac{973}{1152}\left(\frac{251}{1152}{e}^{p(a+c)}+\frac{973}{1152}{e}^{p(a+d)}\right)\right.\hfill \\ & {\left.+\frac{251}{1152}\left(\frac{251}{1152}{e}^{p(b+c)}+\frac{973}{1152}{e}^{p(b+d)}\right)\right\}}^{\frac{1}{p}}\hfill \\ & +{\left(\frac{17}{16}\right)}^{1-\frac{1}{p}}\left\{\frac{973}{1152}\left(\frac{1}{2}{e}^{p(a+c)}+\frac{1}{2}{e}^{p(a+d)}\right)\right.\hfill \\ & {\left.+\frac{251}{1152}\left(\frac{1}{2}{e}^{p(b+c)}+\frac{1}{2}{e}^{p(b+d)}\right)\right\}}^{\frac{1}{p}}\hfill \\ & +{\left(\frac{289}{256}\right)}^{1-\frac{1}{p}}\left\{\frac{973}{1152}\left(\frac{973}{1152}{e}^{p(a+c)}+\frac{251}{1152}{e}^{p(a+d)}\right)\right.\hfill \\ & \left.{\left.+\frac{251}{1152}\left(\frac{973}{1152}{e}^{p(b+c)}+\frac{251}{1152}{e}^{p(b+d)}\right)\right\}}^{\frac{1}{p}}\right].\hfill \end{array}$$

Proof. 

Using Theorem 9 with $q_1,q_2\to 1^{-}$ and choosing $F(t,s)=e^{t+s},$ the above result is captured. □

All the results presented and demonstrated in this article find application both in the mathematical field, from the point of view of numerical analysis, and in that of the physical sciences and engineering. They can even be used in the area of approximation and evaluation of integrals under the work scheme used in mathematical engineering.

7. Conclusions

In this paper, using the notion of $q_1{q}_2$-derivatives and integrals, some of Newton’s type inequalities for co-ordinated convex functions are given. Special cases of the key results of the present paper are also discussed and new inequalities are obtained. To the best of our knowledge, these results are new in the literature. As future directions, one may find similar inequalities through different types of convexities, with emphasis on functions with some type of symmetry.