On some new quantum trapezoid-type inequalities for q-differentiable coordinated convex functions

Abstract

In this paper, we establish several new inequalities for q-differentiable coordinated convex functions that are related to the right side of Hermite–Hadamard inequalities for coordinated convex functions. We also show that the inequalities proved in this paper generalize the results given in earlier works. Moreover, we give some examples in order to demonstrate our main results.

1 Introduction

The Hermite–Hadamard inequality is a classical inequality stated as: If $f:[a,b]\rightarrow \mathbb{R}$ is a convex function, then

$$\begin{aligned} f \biggl( \frac{a+b}{2} \biggr) \leq \frac{1}{b-a} \int _{a}^{b}{f(x)} \,dx \leq \frac{f(a)+f(b)}{2}. \end{aligned}$$

(1.1)

The double inequality (1.1) was introduced by Hermite [1] in 1883 and was investigated by Hadamard [2] in 1893.

Definition 1.1

([3])

A function $f : \bigtriangleup \to \mathbb{R}$ is said to be coordinated convex, if the partial mappings

$$\begin{aligned} f_{x} : [c,d]\ni v \mapsto f(x,v)\in \mathbb{R}\quad \text{and}\quad f_{y} : [a,b]\ni u \mapsto f(u,y)\in \mathbb{R} \end{aligned}$$

are convex for all $x \in (a,b)$ and $y\in (c,d)$.

A formal definition for coordinated convex functions may be stated as follows:

Definition 1.2

A function $f:\bigtriangleup \rightarrow \mathbb{R}$ is said to be coordinated convex, if

$$\begin{aligned} f \bigl(tx+(1-t)z,\lambda y+(1-\lambda )w \bigr)& \leq t\lambda f(x,y)+t(1- \lambda )f(x,w)+(1-t)\lambda f(z,y) \\ & \quad{}+(1-t) (1-\lambda )f(z,w) \end{aligned}$$

(1.2)

holds for all $t,\lambda \in {}[ 0,1]$ and $(x,y), (z,w)\in \bigtriangleup$.

Dragomir [3] presented the Hermite–Hadamard-type inequalities for coordinated convex functions in 2001 as follows:

Theorem 1.1

If $f:\bigtriangleup \to \mathbb{R}$ is a coordinated convex function, then we have

$$\begin{aligned} &f \biggl( \frac {a+b}{2},\frac {c+d}{2} \biggr) \\ &\quad \leq \frac {1}{2} \biggl[ \frac {1}{b-a} \int _{a}^{b}{f \biggl( x, \frac {c+d}{2} \biggr) } \,dx +\frac {1}{d-c} \int _{c}^{d}{f \biggl( \frac {a+b}{2},y \biggr) } \,dy \biggr] \\ &\quad \leq \frac {1}{(b-a)(d-c)} \int _{a}^{b} \int _{c}^{d}{f(x,y)} \,dy \,dx \\ &\quad \leq \frac{1}{4} \biggl[\frac {1}{(b-a)} \int _{a}^{b}{f ( x,c )+f ( x,d ) } \,dx \\ &\quad \quad{}+ \frac {1}{d-c} \int _{c}^{d}{f ( a,y )+f ( b,y ) } \,dy \biggr] \\ &\quad \leq \frac {1}{4} \bigl[ f(a,c)+f(b,c)+f(a,d)+f(b,d) \bigr] . \end{aligned}$$

(1.3)

In 2012, Sarikaya and Set [4] proved some inequalities that give estimations between the middle and the rightmost terms in (1.3).

Theorem 1.2

Let $f:\bigtriangleup \to \mathbb{R}$ be a partially differentiable function on $(a,b)\times (a,b)$. If $\vert \frac {\partial ^{2}{}f}{\partial t \partial s} \vert$ is coordinated convex on △, then the following inequality holds:

$$\begin{aligned} & \biggl\vert \frac{f(a,c)+f(a,d)+f(b,c)+f(b,d)}{4}+ \frac{1}{(b-a)(d-c)} \int _{a}^{b} \int _{c}^{d} f(x,y) \,ds \,dt -A \biggr\vert \\ &\quad \leq \frac{(b-a)(d-c)}{16} \biggl[ \frac{ \vert \frac {\partial ^{2}{}}{\partial t \partial s}f(a,c) \vert + \vert \frac {\partial ^{2}{}}{\partial t \partial s}f(a,b) \vert + \vert \frac {\partial ^{2}{}}{\partial t \partial s}f(b,c) \vert + \vert \frac {\partial ^{2}{}}{\partial t \partial s}f(b,d) \vert }{4} \biggr], \end{aligned}$$

(1.4)

where

$$\begin{aligned} A=\frac{1}{2} \biggl[\frac{1}{b-a} \int _{a}^{b} f(x,c)+f(x,d) \,dx + \frac{1}{d-c} \int _{c}^{d} f(a,y)+f(b,y) \,dx \biggr]. \end{aligned}$$

On the other hand, the concept of quantum calculus (sometimes called q-calculus) is known as the study of calculus with no limits. Note that q-calculus can be reduced to ordinary calculus if we take $\lim_{q\rightarrow 1}$. In 1910, Jackson [5] introduced the definite q-integral known as the q-Jackson integral. Quantum calculus has many applications in several mathematical areas such as combinatorics, number theory, orthogonal polynomials, basic hypergeometric functions, mechanics, quantum theory, and the theory of relativity, see for instance [6–11] and the references therein. The book by Kac and Cheung [12] covers the fundamental knowledge and also the basic theoretical concepts of quantum calculus.

2 Preliminaries

Throughout this paper, we let $\bigtriangleup :=[a,b]\times {}[ c,d]\subseteq \mathbb{R}$, $0< q<1$ and $0< q_{i}<1$ for $i=1,2$. Also, here and below, we use the following notation:

$$\begin{aligned}{} [n]_{q}:= \textstyle\begin{cases} \frac {1-q^{n}}{1-q}=1+q+\cdots +q^{n-2}+q^{n-1},& \text{if } n\in \mathbb{R}\mathbbm{^{+}}, \\ 0,& \text{if } n=0. \end{cases}\displaystyle \end{aligned}$$

In 2013, Tariboon and Ntouyas [13] defined the _aq-derivative and _aq-integral of a function on finite intervals and proved some of its properties.

Definition 2.1

([13])

Let $f:[a,b]\rightarrow \mathbb{R}$ be a continuous function. Then, the _aq-derivative of f at $x\in (a,b]$ is defined by

$$\begin{aligned} {}_{a}D_{q}f(x)= \frac{f ( x ) -f ( qx+(1-q)a ) }{(1-q)(x-a)}. \end{aligned}$$

The _aq-integral is defined by

$$\begin{aligned} \int _{a}^{x}{f(t)} \,{}_{a}d_{q}t=(1-q) (x-a)\sum_{n=0}^{\infty }{q^{n}f \bigl( q^{n}x+ \bigl( 1-q^{n} \bigr) a \bigr) }. \end{aligned}$$

In [14], Alp et al. proved the following quantum Hermite–Hadamard inequality for convex functions using the quantum integrals:

Theorem 2.1

If $f:[a,b]\rightarrow \mathbb{R}$ is a convex function, then we have

$$\begin{aligned} f \biggl( \frac{qa+b}{1+q} \biggr) \leq \frac{1}{b-a} \int _{a}^{b}f ( x ) \,{}_{a}d_{q}x \leq \frac{qf ( a ) +f ( b ) }{1+q}. \end{aligned}$$

(2.1)

In 2020, Bermudo et al. [15] defined the q-derivative and q-integral of a function on finite intervals that are called ^bq-calculus

Definition 2.2

([15])

Let $f:[a,b]\rightarrow \mathbb{R}$ be a continuous function. Then, the ^bq-derivative of f at $x\in {}[ a,b)$ is defined by

$$ {}^{b}D_{q}f(x)= \frac{f ( qx+(1-q)b ) -f ( x ) }{(1-q)(b-x)}. $$

The ^bq-integral is defined by

$$ \int _{x}^{b}{f(t)} \,{}^{b}d_{q}t=(1-q) (b-x)\sum_{n=0}^{\infty }{q^{n}f \bigl( q^{n}x+ \bigl( 1-q^{n} \bigr) b \bigr) }. $$

Bermudo et al. also proved the corresponding Hermite–Hadamard inequality for ^bq-integrals, as follows:

Theorem 2.2

If $f:[a,b]\rightarrow \mathbb{R}$ is a convex function, then we have

$$\begin{aligned} f \biggl( \frac{a+qb}{1+q} \biggr) \leq \frac{1}{b-a} \int _{a}^{b}f ( x ) \,{}^{b}d_{q}x \leq \frac{f ( a ) +qf ( b ) }{1+q}. \end{aligned}$$

(2.2)

In [16] and [17], the authors provide q-integrations by parts as follows:

Lemma 2.1

For continuous functions $h,f:[a,b]\to \mathbb{R}$, the following equality holds:

$$\begin{aligned} &\int _{0}^{c} h(t) \,{}_{a}D_{q}f \bigl(tb+(1-t)a \bigr) \,{}_{0}d_{q}t \\ &\quad =\frac{h(t)f(tb+(1-t)a)}{b-a}\bigg\vert _{0}^{c}- \int _{0}^{c} f \bigl(tb+(1-t)a \bigr)\, {}_{0}D_{q}h(t) \,{}_{0}d_{q}t. \end{aligned}$$

(2.3)

Lemma 2.2

For continuous functions $h,f:[a,b]\to \mathbb{R}$, the following equality holds:

$$\begin{aligned} &\int _{0}^{c} h(t) \,{}^{b}D_{q}f \bigl(ta+(1-t)b \bigr) \,{}_{0}d_{q}t \\ &\quad = \int _{0}^{c} f \bigl(ta+(1-t)b \bigr)\, {}_{0}D_{q}h(t) \,{}_{0}d_{q}t- \frac{h(t)f(ta+(1-t)b)}{b-a}\bigg\vert _{0}^{c}. \end{aligned}$$

(2.4)

Several papers were devoted to generalizations and estimations of the left and right sides of the inequalities (2.1) and (2.2). In [18], Noor et al. proved some bounds for the right-hand side of the inequality (2.1), whereas Alp et al. proved some bounds for the left-hand side of the inequality (2.1) in [14]. In [19], by using convex functions, Budak proved some bounds for the left- and right-hand sides of the inequality (2.2).

In [20], Ali et al. proved the following new version of the quantum Hermite–Hadamard inequality involving the _aq-integral and ^bq-integral. They also proved some inequalities for estimations of the left- and right-hand sides of this inequality.

Theorem 2.3

If $f:[a,b]\rightarrow \mathbb{R}$ is a convex function, then we have

$$\begin{aligned} f \biggl( \frac{a+b}{2} \biggr) \leq \frac{1}{b-a} \biggl[ \int _{a}^{ \frac{a+b}{2}}{f(x)} \,{}_{a}d_{q}x+ \int _{\frac{a+b}{2}}^{b}{f(x)} \,{}^{b}d_{q}x \biggr] \leq \frac{f(a)+f(b)}{2}. \end{aligned}$$

Recently, Sitthiwirattham et al. [21] proved some new quantum Hermite–Hadamard inequalities for convex functions by using their new techniques.

Theorem 2.4

If $f:[a,b]\to \mathbb{R}$ is a convex function, then we have

$$\begin{aligned} f \biggl(\frac{a+b}{2} \biggr)\leq \frac{1}{b-a} \biggl[ \int _{a}^{ \frac{a+b}{2}}{f(x)} \,{}^{\frac{a+b}{2}}d_{q}x+ \int _{\frac{a+b}{2}}^{b}{f(x)} \,{}_{\frac{a+b}{2}}d_{q}x \biggr]\leq \frac{f(a)+f(b)}{2}. \end{aligned}$$

(2.5)

On the other hand, in [22] Latif et al. and in [23] Budak et al. introduced the quantum integrals for functions of two variables

Definition 2.3

([22])

Suppose that $f:\bigtriangleup \rightarrow \mathbb{R}$ is a function of two variables. Then, the definite integral is given by

$$\begin{aligned} \int _{a}^{x} \int _{c}^{y}f(t,s) {\,{}_{c}d_{q_{2}}s} {\,{}_{a}d_{q_{1}}}t& =(1-q_{1}) (1-q_{2}) (x-a) (y-c) \\ &\quad{} \times \sum_{n=0}^{\infty }\sum _{m=0}^{\infty }{q_{1}^{n}q_{2}^{m}f \bigl(q_{1}^{n}x+ \bigl(1-q_{1}^{n} \bigr)a,q_{2}^{m}y+ \bigl(1-q_{2}^{m} \bigr)c \bigr)}. \end{aligned}$$

Definition 2.4

([23])

Suppose that $f:\bigtriangleup \rightarrow \mathbb{R}$ is a function of two variables. Then, the definite integrals are given by

$$\begin{aligned}& \begin{aligned} \int _{a}^{x} \int _{y}^{d}f(t,s) {\,{}^{d}d_{q_{2}}s} {\,{}_{a}d_{q_{1}}}t& =(1-q_{1}) (1-q_{2}) (x-a) (d-y) \\ &\quad{} \times \sum_{n=0}^{\infty }\sum _{m=0}^{\infty }{q_{1}^{n}q_{2}^{m}f \bigl(q_{1}^{n}x+ \bigl(1-q_{1}^{n} \bigr)a,q_{2}^{m}y+ \bigl(1-q_{2}^{m} \bigr)d \bigr)}, \end{aligned} \\& \begin{aligned} \int _{x}^{b} \int _{c}^{y}f(t,s) {\,{}_{c}d_{q_{2}}s} \ {}^{b}d_{q_{1}}t& =(1-q_{1}) (1-q_{2}) (b-x) (y-c) \\ &\quad{} \times \sum_{n=0}^{\infty }\sum _{m=0}^{\infty }{q_{1}^{n}q_{2}^{m}f \bigl(q_{1}^{n}x+ \bigl(1-q_{1}^{n} \bigr)b,q_{2}^{m}y+ \bigl(1-q_{2}^{m} \bigr)c \bigr)} \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \int _{x}^{b} \int _{y}^{d}f(t,s) {\,{}^{d}d_{q_{2}}s} {\ {}^{b}d_{q_{1}}t}& =(1-q_{1}) (1-q_{2}) (b-x) (d-y) \\ &\quad{} \times \sum_{n=0}^{\infty }\sum _{m=0}^{\infty }{q_{1}^{n}q_{2}^{m}f \bigl(q_{1}^{n}x+ \bigl(1-q_{1}^{n} \bigr)b,q_{2}^{m}y+ \bigl(1-q_{2}^{m} \bigr)d \bigr)}. \end{aligned}$$

In the papers [22] and [23], by using these integrals, the authors also proved the Hermite–Hadamard inequalities for coordinated convex functions. By using the concepts given in Definition 2.3 and Definition 2.4, many authors proved several important inequalities [24–28].

In [22] Latif et al. and in [29] Budak et al. introduced the following $q_{1},q_{2}$-derivatives for functions of two variables

Definition 2.5

([22])

Let $f:\bigtriangleup \subseteq \mathbb{R} ^{2}\rightarrow \mathbb{R}$ be a continuous function of two variables. Then, the partial _aq-, _cq-, and ${}_{a,c}q$-derivatives at $( x,y ) \in \bigtriangleup$ can be given as follows:

$$\begin{aligned}& \frac{{}_{a}\partial _{q_{1}}f ( x,y ) }{{}_{a}\partial _{q_{1}}x} = \frac{f ( q_{1}x+ ( 1-q_{1} ) a,y ) -f ( x,y ) }{ ( 1-q_{1} ) ( x-a ) }, \quad x\neq b, \\& \frac{{}_{c}\partial _{q_{2}}f ( x,y ) }{{}_{c}\partial _{q_{2}}y} = \frac{f ( x,q_{2}y+ ( 1-q_{2} ) c ) -f ( x,y ) }{ ( 1-q_{2} ) ( y-c ) }, \quad y\neq c, \\& \frac{{}_{a,c}\partial _{q_{1},q_{2}}^{2}f ( x,y ) }{{}_{a} \partial _{q_{1}}x\,{}_{c}\partial _{q_{2}}y} = \frac{1}{ ( x-a ) ( y-c ) ( 1-q_{1} ) ( 1-q_{2} ) } \\& \hphantom{\frac{{}_{a,c}\partial _{q_{1},q_{2}}^{2}f ( x,y ) }{{}_{a} \partial _{q_{1}}x\,{}_{c}\partial _{q_{2}}y} = {}}{} \times \bigl[ f \bigl(q_{1}x+ ( 1-q_{1} ) a,q_{2}y+ ( 1-q_{2} ) c \bigr) \\& \hphantom{\frac{{}_{a,c}\partial _{q_{1},q_{2}}^{2}f ( x,y ) }{{}_{a} \partial _{q_{1}}x\,{}_{c}\partial _{q_{2}}y} = {}}{} - f \bigl( q_{1}x+ ( 1-q_{1} ) a,y \bigr) -f \bigl( x,q_{2}y+ ( 1-q_{2} ) c \bigr) +f ( x,y ) \bigr], \\& \hphantom{\frac{{}_{a,c}\partial _{q_{1},q_{2}}^{2}f ( x,y ) }{{}_{a} \partial _{q_{1}}x\,{}_{c}\partial _{q_{2}}y} = {}}x\neq a, y\neq c. \end{aligned}$$

Definition 2.6

([29])

Let $f:\bigtriangleup \subseteq \mathbb{R} ^{2}\rightarrow \mathbb{R}$ be a continuous function of two variables. Then, the partial ^bq, ^dq, ${}_{a}^{d}q$, ${}_{c}^{b}q$, and ${}^{b,d}q$-derivatives at $( x,y ) \in \bigtriangleup$ can be given as follows:

$$\begin{aligned}& \frac{^{b}\partial _{q_{1}}f ( x,y ) }{^{b}\partial _{q_{1}}x} = \frac{f ( q_{1}x+ ( 1-q_{1} ) b,y ) -f ( x,y ) }{ ( 1-q_{1} ) ( b-x ) }, \quad x\neq b, \\& \frac{^{d}\partial _{q_{2}}f ( x,y ) }{^{b}\partial _{q_{2}}y} = \frac{f ( x,q_{2}y+ ( 1-q_{2} ) d ) -f ( x,y ) }{ ( 1-q_{2} ) ( d-y ) },\quad y\neq d, \\& \begin{aligned} \frac{{}_{a}^{d}\partial _{q_{1},q_{2}}^{2}f ( x,y ) }{{}_{a}\partial _{q_{1}}x\,{}^{d}\partial _{q_{2}}y} & = \frac{1}{ ( x-a ) ( d-y ) ( 1-q_{1} ) ( 1-q_{2} ) } \\ &\quad{} \times \bigl[ f \bigl(q_{1}x+ ( 1-q_{1} ) a,q_{2}y+ ( 1-q_{2} ) d \bigr) \\ &\quad{} - f \bigl( q_{1}x+ ( 1-q_{1} ) a,y \bigr) -f \bigl( x,q_{2}y+ ( 1-q_{2} ) d \bigr) +f ( x,y ) \bigr] , \\ &\quad x\neq a, y\neq d, \end{aligned} \\& \begin{aligned} \frac{{}_{c}^{b}\partial _{q_{1},q_{2}}^{2}f ( x,y ) }{^{b} \partial _{q_{1}}x_{ c}\partial _{q_{2}}y} & = \frac{1}{ ( b-x ) ( y-c ) ( 1-q_{1} ) ( 1-q_{2} ) } \\ &\quad{} \times \bigl[ f \bigl( q_{1}x+ ( 1-q_{1} ) b,q_{2}y+ ( 1-q_{2} ) c \bigr) \\ &\quad{} - f \bigl( q_{1}x+ ( 1-q_{1} ) b,y \bigr) -f \bigl( x,q_{2}y+ ( 1-q_{2} ) c \bigr) +f ( x,y ) \bigr] , \\ &\quad x\neq b, y\neq c, \end{aligned} \\& \begin{aligned} \frac{^{b,d}\partial _{q_{1},q_{2}}^{2}f ( x,y ) }{^{b}\partial _{q_{1}}x\,{}^{d}\partial _{q_{2}}y} & = \frac{1}{ ( b-x ) ( d-y ) ( 1-q_{1} ) ( 1-q_{2} ) } \\ &\quad{} \times \bigl[ f \bigl( q_{1}x+ ( 1-q_{1} ) b,q_{2}y+ ( 1-q_{2} ) d \bigr) \\ &\quad{} - f \bigl( q_{1}x+ ( 1-q_{1} ) b,y \bigr) -f \bigl( x,q_{2}y+ ( 1-q_{2} ) d \bigr) +f ( x,y ) \bigr] , \\ &\quad x\neq b, y\neq d. \end{aligned} \end{aligned}$$

Recently, in [30], new versions of q-Hermite–Hadamard-type inequalities for coordinated convex functions have been established.

Theorem 2.5

Let $f:\bigtriangleup \rightarrow \mathbb{R}$ be a coordinated convex function. Then, we have

$$\begin{aligned} &f \biggl( \frac{a+b}{2},\frac{c+d}{2} \biggr) \\ &\quad \leq \frac{1}{2(b-a)} \biggl[ \int _{a}^{\frac{a+b}{2}}{f \biggl( x, \frac{c+d}{2} \biggr) } \,{}^{\frac{a+b}{2}}d_{q_{1}}x+ \int _{\frac{a+b}{2}}^{b}{f \biggl( x,\frac{c+d}{2} \biggr) } \, {}_{\frac{a+b}{2}}d_{q_{1}}x \biggr] \\ &\quad \quad{} + \frac{1}{2(d-c)} \biggl[ \int _{c}^{\frac{c+d}{2}}{f \biggl( \frac{a+b}{2},y \biggr) } \,{}^{\frac{c+d}{2}}d_{q_{2}}y+ \int _{\frac{c+d}{2}}^{d}{f \biggl( \frac{a+b}{2},y \biggr) } \,{}_{\frac{c+d}{2}}d_{q_{2}}y \biggr] \\ &\quad \leq \frac{1}{(b-a)(d-c)} \biggl[ \int _{a}^{\frac{a+b}{2}} \int _{c}^{ \frac{c+d}{2}}{f(x,y}) \,{}^{\frac{c+d}{2}}d_{q_{2}}y \,{}^{\frac{a+b}{2}}d_{q_{1}}x \\ &\quad \quad{} + \int _{a}^{\frac{a+b}{2}} \int _{\frac{c+d}{2}}^{d}{f(x,y}) \,{}_{\frac{c+d}{2}}d_{q_{2}}y \,{}^{\frac{a+b}{2}}d_{q_{1}}x + \int _{\frac{a+b}{2}}^{b} \int _{c}^{\frac{c+d}{2}}{f(x,y}) \,{}^{\frac{c+d}{2}}d_{q_{2}}y \,{}_{\frac{a+b}{2}}d_{q_{1}}x \\ &\quad \quad{} + \int _{\frac{a+b}{2}}^{b} \int _{\frac{c+d}{2}}^{d}{f(x,y}) \,{}_{\frac{c+d}{2}}d_{q_{2}}y \,{}_{\frac{a+b}{2}}d_{q_{1}}x \biggr] \\ &\quad \leq \frac{1}{4(b-a)} \biggl[ \int _{a}^{\frac{a+b}{2}}{f(x,c)+f(x,d)} {}^{\frac{a+b}{2}}d_{q_{1}}x+ \int _{\frac{a+b}{2}}^{b}{f(x,c)+f(x,d)} \,{}_{\frac{a+b}{2}}d_{q_{1}}x \biggr] \\ &\quad \quad{} + \frac{1}{4(d-c)} \biggl[ \int _{c}^{\frac{c+d}{2}}{f(a,y)+f(b,y)} \,{}^{\frac{c+d}{2}}d_{q_{2}}y+ \int _{\frac{c+d}{2}}^{d}{f(a,y)+f(b,y)} \,{}_{\frac{c+d}{2}}d_{q_{2}}y \biggr] \\ &\quad \leq \frac{f(a,c)+f(a,d)+f(b,c)+f(b,d)}{4}. \end{aligned}$$

In this paper, we study some estimations between the middle and the rightmost terms of the above inequalities.

3 Main results

In this section, we prove several new inequalities for q-differentiable coordinated convex functions that are related to the right side of Hermite–Hadamard inequalities for coordinated convex functions. We may start with some lemmas, which are useful in further considerations.

Lemma 3.1

Let $f:[a,b]\to \mathbb{R}$. Then, we have

$$\begin{aligned} &\int _{0}^{1} qt \,{}_{a}D_{q}f \biggl(tb+(1-t)\frac{a+b}{2} \biggr) \,{}_{0}d_{q}t \\ &\quad =\frac{2}{b-a}f (b )-\frac{4}{(b-a)^{2}} \int _{ \frac{a+b}{2}}^{b} f(x) \,{}_{\frac{a+b}{2}}d_{q}x. \end{aligned}$$

(3.1)

Proof

By (2.3) and the definition of a q-integral, we have

$$\begin{aligned} & \int _{0}^{1}qt \,{}_{a}D_{q}f \biggl( tb+(1-t)\frac{a+b}{2} \biggr) \,{}_{0}d_{q}t \\ &\quad = \frac{2qt f ( tb+(1-t)\frac{a+b}{2} ) }{b-a}\bigg\vert _{0}^{1}- \frac{2q}{b-a} \int _{0}^{1}f \biggl( qtb+(1-qt) \frac{a+b}{2} \biggr) \,{}_{0}d_{q}t \\ &\quad =\frac{2qf(b)}{b-a}-\frac{2q}{b-a}(1-q)\sum _{n=0}^{\infty }q^{n}f \biggl( q^{n+1}b+ \bigl(1-q^{n+1} \bigr)\frac{a+b}{2} \biggr) \\ &\quad =\frac{2qf(b)}{b-a}-\frac{2}{b-a}(1-q)\sum _{n=0}^{\infty }q^{n+1}f \biggl( q^{n+1}b+ \bigl(1-q^{n+1} \bigr)\frac{a+b}{2} \biggr) \\ &\quad =\frac{2qf(b)}{b-a}-\frac{2}{b-a}(1-q)\sum _{n=1}^{\infty }q^{n}f \biggl( q^{n}b+ \bigl(1-q^{n} \bigr) \frac{a+b}{2} \biggr) \\ &\quad =\frac{2qf(b)}{b-a}-\frac{2}{b-a}(1-q) \Biggl\{ \sum _{n=0}^{\infty }q^{n}f \biggl( q^{n}b+ \bigl(1-q^{n} \bigr)\frac{a+b}{2} \biggr) -f(b) \Biggr\} \\ &\quad =\frac{2}{b-a}f (b ) -\frac{4}{(b-a)^{2}} \int _{\frac{a+b}{2}}^{b}f(x) \,{}_{\frac{a+b}{2}}d_{q}x, \end{aligned}$$

which completes the proof. □

Lemma 3.2

Let $f:[a,b]\to \mathbb{R}$. Then, we have

$$\begin{aligned} &\int _{0}^{1} qt \,{}^{b}D_{q}f \biggl(ta+(1-t)\frac{a+b}{2} \biggr) \,{}_{0}d_{q}t \\ &\quad =-\frac{2}{b-a}f (a )+\frac{4}{(b-a)^{2}} \int ^{ \frac{a+b}{2}}_{a} f(x) \,{}^{\frac{a+b}{2}}d_{q}x. \end{aligned}$$

(3.2)

Proof

By (2.4) and the definition of a q-integral, we have

$$\begin{aligned} & \int _{0}^{1}qt \,{}^{b}D_{q}f \biggl( ta+(1-t)\frac{a+b}{2} \biggr) \,{}_{0}d_{q}t \\ &\quad =\frac{2q}{b-a} \int _{0}^{1}f \biggl( qta+(1-qt) \frac{a+b}{2} \biggr) \,{}_{0}d_{q}t- \frac{2qt f ( ta+(1-t)\frac{a+b}{2} ) }{b-a}\bigg\vert _{0}^{1} \\ &\quad =\frac{2q}{b-a}(1-q)\sum_{n=0}^{\infty }q^{n}f \biggl( q^{n+1}a+ \bigl(1-q^{n+1} \bigr)\frac{a+b}{2} \biggr) -\frac{2qf(a)}{b-a} \\ &\quad =-\frac{2qf(a)}{b-a}+\frac{2}{b-a}(1-q)\sum _{n=0}^{\infty }q^{n+1}f \biggl( q^{n+1}a+ \bigl(1-q^{n+1} \bigr)\frac{a+b}{2} \biggr) \\ &\quad =-\frac{2qf(a)}{b-a}+\frac{2}{b-a}(1-q)\sum _{n=1}^{\infty }q^{n}f \biggl( q^{n}a+ \bigl(1-q^{n} \bigr) \frac{a+b}{2} \biggr) \\ &\quad =-\frac{2qf(a)}{b-a}+\frac{2}{b-a}(1-q) \Biggl\{ \sum _{n=0}^{\infty }q^{n}f \biggl( q^{n}a+ \bigl(1-q^{n} \bigr)\frac{a+b}{2} \biggr) -f(a) \Biggr\} \\ &\quad =-\frac{2}{b-a}f ( a ) +\frac{4}{(b-a)^{2}} \int _{a}^{\frac{a+b}{2}}f(x) \,{}^{\frac{a+b}{2}}d_{q}x. \end{aligned}$$

The proof is completed. □

For convenience, we will use the following notations:

$$\begin{aligned}& \Phi (t,s):= \frac {{}_{a,c}\partial ^{2}{}_{q_{1},q_{2}}f(t,s)}{{}_{a}\partial _{q_{1}}t{}_{c}\partial _{q_{2}}s},\qquad \Theta (t,s):= \frac {{}_{a}^{d}\partial ^{2}{}_{q_{1},q_{2}}f(t,s)}{{}_{a}\partial _{q_{1}}t{}^{d}\partial _{q_{2}}s}, \\& \Psi (t,s):= \frac {{}_{c}^{b}\partial ^{2}{}_{q_{1},q_{2}}f(t,s)}{{}^{b}\partial _{q_{1}}t{}_{c}\partial _{q_{2}}s}\quad \text{and} \quad \Omega (t,s):=\frac {{}^{b,d}\partial ^{2}{}_{q_{1},q_{2}}f(t,s)}{{}^{b}\partial _{q_{1}}t{}^{d}\partial _{q_{2}}s}. \end{aligned}$$

Lemma 3.3

Let $f:\triangle \rightarrow \mathbb{R}$ be a q-partially differential function on $\triangle ^{\circ }$. If $\Phi (t,s)$, $\Theta (t,s)$, $\Psi (t,s)$, and $\Omega (t,s)$ are q-integrable on △, then the following identity holds:

$$\begin{aligned} & \frac{f(a,c)+f(a,d)+f(b,c)+f(b,d)}{4} \\ &\quad \quad{} + \frac{1}{(b-a)(d-c)} \biggl[ \int _{a}^{\frac{a+b}{2}} \int _{c}^{ \frac{c+d}{2}}{f(x,y}) \,{}^{\frac{c+d}{2}}d_{q_{2}}y \,{}^{\frac{a+b}{2}}d_{q_{1}}x \\ &\quad \quad{} + \int _{a}^{\frac{a+b}{2}} \int _{\frac{c+d}{2}}^{d}{f(x,y}) \,{}_{\frac{c+d}{2}}d_{q_{2}}y \,{}^{\frac{a+b}{2}}d_{q_{1}}x+ \int _{ \frac{a+b}{2}}^{b} \int _{c}^{\frac{c+d}{2}}{f(x,y}) \,{}^{\frac{c+d}{2}}d_{q_{2}}y \, {}_{\frac{a+b}{2}}d_{q_{1}}x \\ &\quad \quad{} + \int _{\frac{a+b}{2}}^{b} \int _{\frac{c+d}{2}}^{d}{f(x,y}) \,{}_{\frac{c+d}{2}}d_{q_{2}}y \,{}_{\frac{a+b}{2}}d_{q_{1}}x \biggr] \\ &\quad \quad{} - \frac{1}{2(b-a)} \biggl[ \int _{a}^{\frac{a+b}{2}}{f(x,c)+f(x,d) } {}^{\frac{a+b}{2}}d_{q_{1}}x+ \int _{\frac{a+b}{2}}^{b}{f(x,c)+f(x,d)} \,{}_{\frac{a+b}{2}}d_{q_{1}}x \biggr] \\ &\quad \quad{} - \frac{1}{2(d-c)} \biggl[ \int _{c}^{\frac{c+d}{2}}{f(a,y)+f(b,y) } \,{}^{\frac{c+d}{2}}d_{q_{2}}y+ \int _{\frac{c+d}{2}}^{d}{f(a,y)+f(b,y) } \, {}_{\frac{c+d}{2}}d_{q_{2}}y \biggr] \\ &\quad =\frac{(b-a)(d-c)}{16} \int _{0}^{1} \int _{0}^{1}(q_{1}t) (q_{2}s) \biggl[ \Phi \biggl( \frac{1-t}{2}a+ \frac{1+t}{2}b,\frac{1-s}{2}c+\frac{1+s}{2}d \biggr) \\ &\quad \quad{}-\Theta \biggl( \frac{1-t}{2}a+\frac{1+t}{2}b,\frac{1+s}{2}c+\frac{1-s}{2}d \biggr) \\ &\quad \quad{}-\Psi \biggl( \frac{1+t}{2}a+\frac{1-t}{2}b, \frac{1-s}{2}c+ \frac{1+s}{2}d \biggr) \\ &\quad \quad{}+\Omega \biggl( \frac{1+t}{2}a+\frac{1-t}{2}b,\frac{1+s}{2}c+\frac{1-s}{2}d \biggr) \biggr] \,{}_{0}d_{q_{2}}s \ {}_{0}d_{q_{1}}t. \end{aligned}$$

Proof

We use Lemma 3.1 for variables s and t, and we obtain

$$\begin{aligned} I_{1}&: = \int _{0}^{1} \int _{0}^{1}(q_{1}t) (q_{2}s)\Phi \biggl( \frac{1-t}{2}a+\frac{1+t}{2}b, \frac{1-s}{2}c+\frac{1+s}{2}d \biggr) \,{}_{0}d_{q_{2}}s \ {}_{0}d_{q_{1}}t \\ & = \int _{0}^{1} \int _{0}^{1}(q_{1}t) (q_{2}s)\Phi \biggl( tb+(1-t) \frac{a+b}{2},sd+(1-s) \frac{c+d}{2} \biggr) \,{}_{0}d_{q_{2}}s \ {}_{0}d_{q_{1}}t \\ & = \int _{0}^{1}q_{1}t \biggl[ \int _{0}^{1}q_{2}s \Phi \biggl( tb+(1-t) \frac{a+b}{2},sd+(1-s)\frac{c+d}{2} \biggr) \,{}_{0}d_{q_{2}}s \biggr] \,{}_{0}d_{q_{1}}t \\ & = \int _{0}^{1}q_{1}t \biggl[ \frac{2}{d-c} \frac{_{a}\partial _{q_{1}}}{{}_{a}\partial _{q_{1}}t}f \biggl( tb+(1-t)\frac{a+b}{2},d \biggr) \\ & \quad{}-\frac{4}{(d-c)^{2}} \int _{\frac{c+d}{2}}^{d} \frac{_{a}\partial _{q_{1}}}{{}_{a}\partial _{q_{1}}t}f \biggl( tb+(1-t)\frac{a+b}{2},y \biggr) \,{}_{\frac{c+d}{2}}d_{q_{2}}y \biggr] \,{}_{0}d_{q_{1}}t \\ & =\frac{2}{d-c} \int _{0}^{1}q_{1}t \frac{_{a}\partial _{q_{1}}}{{}_{a}\partial _{q_{1}}t}f \biggl( tb+(1-t) \frac{a+b}{2},d \biggr) \,{}_{0}d_{q_{1}}t \\ & \quad{}-\frac{4}{(d-c)^{2}} \int _{\frac{c+d}{2}}^{d} \biggl[ \int _{0}^{1}q_{1}t \frac{_{a}\partial _{q_{1}}}{{}_{a}\partial _{q_{1}}t}f \biggl( tb+(1-t) \frac{a+b}{2},y \biggr) \,{}_{0}d_{q_{1}}t \biggr] \,{}_{\frac{c+d}{2}}d_{q_{2}}y \\ & =\frac{2}{d-c} \biggl[ \frac{2}{b-a}f ( b,d ) - \frac{4}{(b-a)^{2}} \int _{\frac{a+b}{2}}^{b}f ( x,d ) \,{}_{\frac{a+b}{2}}d_{q_{1}}x \biggr] \\ & \quad{}-\frac{4}{(d-c)^{2}} \int _{\frac{c+d}{2}}^{d} \biggl[ \frac{2}{b-a}f ( b,y ) -\frac{4}{(b-a)^{2}} \int _{\frac{a+b}{2}}^{b}f ( x,y ) \,{}_{\frac{a+b}{2}}d_{q_{1}}x \biggr] \,{}_{\frac{c+d}{2}}d_{q_{2}}y \\ & =\frac{4}{(b-a)(d-c)}f ( b,d ) - \frac{8}{(b-a)^{2}(d-c)} \int _{\frac{a+b}{2}}^{b}f ( x,d )\, {}_{\frac{a+b}{2}}d_{q_{1}}x \\ & \quad{}-\frac{8}{(b-a)(d-c)^{2}} \int _{\frac{c+d}{2}}^{d}f ( b,y ) \,{}_{\frac{c+d}{2}}d_{q_{2}}y \\ &\quad{}+\frac{16}{(b-a)^{2}(d-c)^{2}} \int _{\frac{a+b}{2}}^{b} \int _{ \frac{c+d}{2}}^{d}f ( x,y ) \,{}_{\frac{c+d}{2}}d_{q_{2}}y \,{}_{\frac{a+b}{2}}d_{q_{1}}x. \end{aligned}$$

Using Lemma 3.2 and Lemma 3.1 for variables s and t, respectively, we obtain

$$\begin{aligned} I_{2}& := \int _{0}^{1} \int _{0}^{1}(q_{1}t) (q_{2}s)\Theta \biggl( \frac{1-t}{2}a+\frac{1+t}{2}b,\frac{1+s}{2}c+\frac{1-s}{2}d \biggr) \,{}_{0}d_{q_{2}}s \ {}_{0}d_{q_{1}}t \\ & = \int _{0}^{1} \int _{0}^{1}(q_{1}t) (q_{2}s)\Theta \biggl( tb+(1-t) \frac{a+b}{2},sc+(1-s) \frac{c+d}{2} \biggr) \,{}_{0}d_{q_{2}}s \ {}_{0}d_{q_{1}}t \\ & = \int _{0}^{1}q_{1}t \biggl[ \int _{0}^{1}q_{2}s \Theta \biggl( tb+(1-t) \frac{a+b}{2},sc+(1-s)\frac{c+d}{2} \biggr) \,{}_{0}d_{q_{2}}s \biggr] \,{}_{0}d_{q_{1}}t \\ & = \int _{0}^{1}q_{1}t \biggl[ - \frac{2}{d-c}{} \frac{{}_{a}\partial _{q_{1}}}{_{a}\partial _{q_{1}}t}f \biggl( tb+(1-t) \frac{a+b}{2},c \biggr) \\ & \quad{}+\frac{4}{(d-c)^{2}} \int _{c}^{\frac{c+d}{2}}{} \frac{_{a}\partial _{q_{1}}}{{}_{a}\partial _{q_{1}}t}f \biggl( tb+(1-t) \frac{a+b}{2},y \biggr) \,{}^{\frac{c+d}{2}}d_{q_{2}}y \biggr] \,{}_{0}d_{q_{1}}t \\ & =-\frac{2}{d-c} \int _{0}^{1}q_{1}t {} \frac{{}_{a}\partial _{q_{1}}}{_{a}\partial _{q_{1}}t}f \biggl( tb+(1-t)\frac{a+b}{2},c \biggr) \,{}_{0}d_{q_{1}}t \\ & \quad{}+\frac{4}{(d-c)^{2}} \int _{c}^{\frac{c+d}{2}} \biggl[ \int _{0}^{1}q_{1}t {} \frac{{}_{a}\partial _{q_{1}}}{{}_{a}\partial _{q_{1}}t}f \biggl( tb+(1-t)\frac{a+b}{2},y \biggr) \,{}_{0}d_{q_{1}}t \biggr] \,{}^{\frac{c+d}{2}}d_{q_{2}}y \\ & =-\frac{2}{d-c} \biggl[ \frac{2}{b-a}f ( b,c ) - \frac{4}{(b-a)^{2}} \int _{\frac{a+b}{2}}^{b}f ( x,c ) \,{}_{\frac{a+b}{2}}d_{q_{1}}x \biggr] \\ & \quad{}+\frac{4}{(d-c)^{2}} \int _{c}^{\frac{c+d}{2}} \biggl[ \frac{2}{b-a}f ( b,y ) -\frac{4}{(b-a)^{2}} \int _{\frac{a+b}{2}}^{b}f ( x,y ) \,{}_{\frac{a+b}{2}}d_{q_{1}}x \biggr] \,{}^{\frac{c+d}{2}}d_{q_{2}}y \\ & =-\frac{4}{(b-a)(d-c)}f ( b,c ) + \frac{8}{(b-a)^{2}(d-c)} \int _{\frac{a+b}{2}}^{b}f ( x,c ) \, {}_{\frac{a+b}{2}}d_{q_{1}}x \\ & \quad{}+\frac{8}{(b-a)(d-c)^{2}} \int _{c}^{\frac{c+d}{2}}f ( b,y ) \,{}^{\frac{c+d}{2}}d_{q_{2}}y \\ &\quad{}-\frac{16}{(b-a)^{2}(d-c)^{2}}\int _{\frac{a+b}{2}}^{b} \int _{c}^{\frac{c+d}{2}}f ( x,y ) {}^{\frac{c+d}{2}}d_{q_{2}}y \,{}_{\frac{a+b}{2}}d_{q_{1}}x. \end{aligned}$$

Similarly, using Lemma 3.1 and Lemma 3.2 for variables s and t, respectively, we obtain

$$\begin{aligned} I_{3}&: = \int _{0}^{1} \int _{0}^{1}(q_{1}t) (q_{2}s)\Psi \biggl( \frac{1+t}{2}a+\frac{1-t}{2}b,\frac{1-s}{2}c+\frac{1+s}{2}d \biggr) \,{}_{0}d_{q_{2}}s \ {}_{0}d_{q_{1}}t \\ & =-\frac{4}{(b-a)(d-c)}f ( a,d ) + \frac{8}{(b-a)^{2}(d-c)} \int _{a}^{\frac{a+b}{2}}f ( x,d ) {}^{\frac{a+b}{2}}d_{q_{1}}x \\ & \quad{}+\frac{8}{(b-a)(d-c)^{2}} \int _{\frac{c+d}{2}}^{d}f ( a,y ) \,{}_{\frac{c+d}{2}}d_{q_{2}}y \\ &\quad{}-\frac{16}{(b-a)^{2}(d-c)^{2}} \int _{a}^{\frac{a+b}{2}} \int _{ \frac{c+d}{2}}^{d}f ( x,y ) \,{}_{\frac{c+d}{2}}d_{q_{2}}y \,{}^{\frac{a+b}{2}}d_{q_{1}}x. \end{aligned}$$

Using Lemma 3.2 for variables s and t, we obtain

$$\begin{aligned} I_{4}&: = \int _{0}^{1} \int _{0}^{1}(q_{1}t) (q_{2}s)\Omega \biggl( \frac{1+t}{2}a+\frac{1-t}{2}b, \frac{1+s}{2}c+\frac{1-s}{2}d \biggr) \,{}_{0}d_{q_{2}}s \ {}_{0}d_{q_{1}}t \\ & =\frac{4}{(b-a)(d-c)}f ( a,c ) - \frac{8}{(b-a)^{2}(d-c)} \int _{a}^{\frac{a+b}{2}}f ( x,c ) {}^{\frac{a+b}{2}}d_{q_{1}}x \\ & \quad{}-\frac{8}{(b-a)(d-c)^{2}} \int _{c}^{\frac{c+d}{2}}f ( a,y ) \,{}^{\frac{c+d}{2}}d_{q_{2}}y \\ &\quad{}+\frac{16}{(b-a)^{2}(d-c)^{2}}\int _{a}^{\frac{a+b}{2}} \int _{c}^{\frac{c+d}{2}}f ( x,y ) {}^{\frac{c+d}{2}}d_{q_{2}}y \,{}^{\frac{a+b}{2}}d_{q_{1}}x. \end{aligned}$$

Multiplying $(I_{1}-I_{2}-I_{3}+I_{4})$ by $\frac{(b-a)(d-c)}{16}$, the proof is completed. □

Theorem 3.1

Under the assumptions of Lemma 3.3. If $|\Phi (t,s)|$, $|\Theta (t,s)|$, $|\Psi (t,s)|$, and $|\Omega (t,s)|$ are coordinated convex on △, then the following inequality holds:

$$\begin{aligned} & \biggl\vert \frac{f(a,c)+f(a,d)+f(b,c)+f(b,d)}{4} \\ &\quad \quad{} + \frac{1}{(b-a)(d-c)} \biggl[ \int _{a}^{\frac{a+b}{2}} \int _{c}^{ \frac{c+d}{2}}{f(x,y}) \,{}^{\frac{c+d}{2}}d_{q_{2}}y \,{}^{\frac{a+b}{2}}d_{q_{1}}x \\ &\quad \quad{} + \int _{a}^{\frac{a+b}{2}} \int _{\frac{c+d}{2}}^{d}{f(x,y}) \,{}_{\frac{c+d}{2}}d_{q_{2}}y \,{}^{\frac{a+b}{2}}d_{q_{1}}x+ \int _{ \frac{a+b}{2}}^{b} \int _{c}^{\frac{c+d}{2}}{f(x,y}) \,{}^{\frac{c+d}{2}}d_{q_{2}}y \, {}_{\frac{a+b}{2}}d_{q_{1}}x \\ &\quad \quad{} + \int _{\frac{a+b}{2}}^{b} \int _{\frac{c+d}{2}}^{d}{f(x,y}) \,{}_{\frac{c+d}{2}}d_{q_{2}}y \,{}_{\frac{a+b}{2}}d_{q_{1}}x \biggr] \\ &\quad \quad{} - \frac{1}{2(b-a)} \biggl[ \int _{a}^{\frac{a+b}{2}}{f(x,c)+f(x,d) } {}^{\frac{a+b}{2}}d_{q_{1}}x+ \int _{\frac{a+b}{2}}^{b}{f(x,c)+f(x,d)} \,{}_{\frac{a+b}{2}}d_{q_{1}}x \biggr] \\ &\quad \quad{} - \frac{1}{2(d-c)} \biggl[ \int _{c}^{\frac{c+d}{2}}{f(a,y)+f(b,y) } \,{}^{\frac{c+d}{2}}d_{q_{2}}y+ \int _{\frac{c+d}{2}}^{d}{f(a,y)+f(b,y) } \,{}_{\frac{c+d}{2}}d_{q_{2}}y \biggr] \biggr\vert \\ &\quad \leq \frac{(b-a)(d-c)}{64[2]_{q_{1}}[2]_{q_{2}}[3]_{q_{1}}[3]_{q_{2}}} \\ &\quad \quad{} \times \bigl[ q_{1}^{3}q_{2}^{3} \bigl( \bigl\vert \Phi (a,c) \bigr\vert + \bigl\vert \Theta (a,d) \bigr\vert + \bigl\vert \Psi (b,c) \bigr\vert + \bigl\vert \Omega (b,d) \bigr\vert \bigr) \\ &\quad \quad{} + q_{1}^{3} \bigl(2q_{2}+2q_{2}^{2}+q_{2}^{3} \bigr) \bigl( \bigl\vert \Phi (a,d) \bigr\vert + \bigl\vert \Theta (a,c) \bigr\vert + \bigl\vert \Psi (b,d) \bigr\vert + \bigl\vert \Omega (b,c) \bigr\vert \bigr) \\ &\quad \quad{} + q_{2}^{3} \bigl(2q_{1}+2q_{1}^{2}+q_{1}^{3} \bigr) \bigl( \bigl\vert \Phi (b,c) \bigr\vert + \bigl\vert \Theta (b,d) \bigr\vert + \bigl\vert \Psi (a,c) \bigr\vert + \bigl\vert \Omega (a,d) \bigr\vert \bigr) \\ &\quad \quad{} + \bigl(2q_{1}+2q_{1}^{2}+q_{1}^{3} \bigr) \bigl(2q_{2}+2q_{2}^{2}+q_{2}^{3} \bigr) \bigl( \bigl\vert \Phi (b,d) \bigr\vert + \bigl\vert \Theta (b,c) \bigr\vert + \bigl\vert \Psi (a,d) \bigr\vert + \bigl\vert \Omega (a,c) \bigr\vert \bigr) \bigr] . \end{aligned}$$

(3.3)

Proof

From Lemma 3.3, we have

$$\begin{aligned} & \biggl\vert \frac{f(a,c)+f(a,d)+f(b,c)+f(b,d)}{4} \\ &\quad \quad{} + \frac{1}{(b-a)(d-c)} \biggl[ \int _{a}^{\frac{a+b}{2}} \int _{c}^{ \frac{c+d}{2}}{f(x,y}) \,{}^{\frac{c+d}{2}}d_{q_{2}}y \,{}^{\frac{a+b}{2}}d_{q_{1}}x \\ &\quad \quad{} + \int _{a}^{\frac{a+b}{2}} \int _{\frac{c+d}{2}}^{d}{f(x,y}) \,{}_{\frac{c+d}{2}}d_{q_{2}}y \,{}^{\frac{a+b}{2}}d_{q_{1}}x+ \int _{ \frac{a+b}{2}}^{b} \int _{c}^{\frac{c+d}{2}}{f(x,y}) \,{}^{\frac{c+d}{2}}d_{q_{2}}y \, {}_{\frac{a+b}{2}}d_{q_{1}}x \\ &\quad \quad{} + \int _{\frac{a+b}{2}}^{b} \int _{\frac{c+d}{2}}^{d}{f(x,y}) \,{}_{\frac{c+d}{2}}d_{q_{2}}y \,{}_{\frac{a+b}{2}}d_{q_{1}}x \biggr] \\ &\quad \quad{} - \frac{1}{2(b-a)} \biggl[ \int _{a}^{\frac{a+b}{2}}{f(x,c)+f(x,d) } {}^{\frac{a+b}{2}}d_{q_{1}}x+ \int _{\frac{a+b}{2}}^{b}{f(x,c)+f(x,d)} \,{}_{\frac{a+b}{2}}d_{q_{1}}x \biggr] \\ &\quad \quad{} - \frac{1}{2(d-c)} \biggl[ \int _{c}^{\frac{c+d}{2}}{f(a,y)+f(b,y) } \,{}^{\frac{c+d}{2}}d_{q_{2}}y+ \int _{\frac{c+d}{2}}^{d}{f(a,y)+f(b,y) } \, {}_{\frac{c+d}{2}}d_{q_{2}}y \biggr] \biggr\vert \\ &\quad \leq \frac{(b-a)(d-c)}{16} \int _{0}^{1} \int _{0}^{1}(q_{1}t) (q_{2}s) \biggl[ \biggl\vert \Phi \biggl( \frac{1-t}{2}a+\frac{1+t}{2}b, \frac{1-s}{2}c+\frac{1+s}{2}d \biggr) \biggr\vert \\ &\quad \quad{}+ \biggl\vert \Theta \biggl( \frac{1-t}{2}a+ \frac{1+t}{2}b, \frac{1+s}{2}c+\frac{1-s}{2}d \biggr) \biggr\vert \\ &\quad \quad{}+ \biggl\vert \Psi \biggl( \frac{1+t}{2}a+\frac{1-t}{2}b,\frac{1-s}{2}c+\frac{1+s}{2}d \biggr) \biggr\vert \\ &\quad \quad{}+ \biggl\vert \Omega \biggl( \frac{1+t}{2}a+ \frac{1-t}{2}b,\frac{1+s}{2}c+\frac{1-s}{2}d \biggr) \biggr\vert \biggr] \,{}_{0}d_{q_{2}}s \ {}_{0}d_{q_{1}}t. \end{aligned}$$

(3.4)

By the coordinated convexity of $|\Phi (t,s)|$, $|\Theta (t,s)|$, $|\Psi (t,s)|$ and $|\Omega (t,s)|$, we obtain

$$\begin{aligned}& \begin{aligned}[b] & \biggl\vert \Phi \biggl( \frac{1-t}{2}a+\frac{1+t}{2}b, \frac{1-s}{2}c+\frac{1+s}{2}d \biggr) \biggr\vert \\ &\quad \leq \frac{(1-t)(1-s)}{4} \bigl\vert \Phi (a,c) \bigr\vert + \frac{(1-t)(1+s)}{4} \bigl\vert \Phi (a,d) \bigr\vert \\ &\quad \quad{}+\frac{(1+t)(1-s)}{4} \bigl\vert \Phi (b,c) \bigr\vert + \frac{(1+t)(1+s)}{4} \bigl\vert \Phi (b,d) \bigr\vert , \end{aligned} \end{aligned}$$

(3.5)

$$\begin{aligned}& \begin{aligned}[b] & \biggl\vert \Theta \biggl( \frac{1-t}{2}a+\frac{1+t}{2}b, \frac{1+s}{2}c+\frac{1-s}{2}d \biggr) \biggr\vert \\ &\quad \leq \frac{(1-t)(1+s)}{4} \bigl\vert \Theta (a,c) \bigr\vert + \frac{(1-t)(1-s)}{4} \bigl\vert \Theta (a,d) \bigr\vert \\ &\quad \quad{}+\frac{(1+t)(1+s)}{4} \bigl\vert \Theta (b,c) \bigr\vert + \frac{(1+t)(1-s)}{4} \bigl\vert \Theta (b,d) \bigr\vert , \end{aligned} \end{aligned}$$

(3.6)

$$\begin{aligned}& \begin{aligned}[b] & \biggl\vert \Psi \biggl( \frac{1+t}{2}a+\frac{1-t}{2}b, \frac{1-s}{2}c+\frac{1+s}{2}d \biggr) \biggr\vert \\ &\quad \leq \frac{(1+t)(1-s)}{4} \bigl\vert \Psi (a,c) \bigr\vert + \frac{(1+t)(1+s)}{4} \bigl\vert \Psi (a,d) \bigr\vert \\ &\quad \quad{}+\frac{(1-t)(1-s)}{4} \bigl\vert \Psi (b,c) \bigr\vert + \frac{(1-t)(1+s)}{4} \bigl\vert \Psi (b,d) \bigr\vert \end{aligned} \end{aligned}$$

(3.7)

and

$$\begin{aligned} & \biggl\vert \Omega \biggl( \frac{1+t}{2}a+\frac{1-t}{2}b, \frac{1+s}{2}c+\frac{1-s}{2}d \biggr) \biggr\vert \\ &\quad \leq \frac{(1+t)(1+s)}{4} \bigl\vert \Omega (a,c) \bigr\vert + \frac{(1+t)(1-s)}{4} \bigl\vert \Omega (a,d) \bigr\vert \\ &\quad \quad{}+\frac{(1-t)(1+s)}{4} \bigl\vert \Omega (b,c) \bigr\vert + \frac{(1-t)(1-s)}{4} \bigl\vert \Omega (b,d) \bigr\vert . \end{aligned}$$

(3.8)

Replacing (3.5)–(3.8) in (3.4), we obtain

$$\begin{aligned} & \biggl\vert \frac{f(a,c)+f(a,d)+f(b,c)+f(b,d)}{4} \\ &\quad \quad{} + \frac{1}{(b-a)(d-c)} \biggl[ \int _{a}^{\frac{a+b}{2}} \int _{c}^{ \frac{c+d}{2}}{f(x,y}) \,{}^{\frac{c+d}{2}}d_{q_{2}}y \,{}^{\frac{a+b}{2}}d_{q_{1}}x \\ &\quad \quad{} + \int _{a}^{\frac{a+b}{2}} \int _{\frac{c+d}{2}}^{d}{f(x,y}) \,{}_{\frac{c+d}{2}}d_{q_{2}}y \,{}^{\frac{a+b}{2}}d_{q_{1}}x+ \int _{ \frac{a+b}{2}}^{b} \int _{c}^{\frac{c+d}{2}}{f(x,y}) \,{}^{\frac{c+d}{2}}d_{q_{2}}y \, {}_{\frac{a+b}{2}}d_{q_{1}}x \\ &\quad \quad{} + \int _{\frac{a+b}{2}}^{b} \int _{\frac{c+d}{2}}^{d}{f(x,y}) \,{}_{\frac{c+d}{2}}d_{q_{2}}y \,{}_{\frac{a+b}{2}}d_{q_{1}}x \biggr] \\ &\quad \quad{} - \frac{1}{2(b-a)} \biggl[ \int _{a}^{\frac{a+b}{2}}{f(x,c)+f(x,d) } {}^{\frac{a+b}{2}}d_{q_{1}}x+ \int _{\frac{a+b}{2}}^{b}{f(x,c)+f(x,d)} \,{}_{\frac{a+b}{2}}d_{q_{1}}x \biggr] \\ &\quad \quad{} - \frac{1}{2(d-c)} \biggl[ \int _{c}^{\frac{c+d}{2}}{f(a,y)+f(b,y) } \,{}^{\frac{c+d}{2}}d_{q_{2}}y+ \int _{\frac{c+d}{2}}^{d}{f(a,y)+f(b,y) } \, {}_{\frac{c+d}{2}}d_{q_{2}}y \biggr] \biggr\vert \\ &\quad \leq \frac{(b-a)(d-c)}{64} \int _{0}^{1} \int _{0}^{1}(q_{1}t) (q_{2}s) \\ &\quad \quad{}\times \bigl[(1-t) (1-s) \bigl( \bigl\vert \Phi (a,c) \bigr\vert + \bigl\vert \Theta (a,d) \bigr\vert + \bigl\vert \Psi (b,c) \bigr\vert + \bigl\vert \Omega (b,d) \bigr\vert \bigr) \\ &\quad \quad{}+(1-t) (1+s) \bigl( \bigl\vert \Phi (a,d) \bigr\vert + \bigl\vert \Theta (a,c) \bigr\vert + \bigl\vert \Psi (b,d) \bigr\vert + \bigl\vert \Omega (b,c) \bigr\vert \bigr) \\ &\quad \quad{}+(1+t) (1-s) \bigl( \bigl\vert \Phi (b,c) \bigr\vert + \bigl\vert \Theta (b,d) \bigr\vert + \bigl\vert \Psi (a,c) \bigr\vert + \bigl\vert \Omega (a,d) \bigr\vert \bigr) \\ &\quad \quad{}+(1+t) (1+s) \bigl( \bigl\vert \Phi (b,d) \bigr\vert + \bigl\vert \Theta (b,c) \bigr\vert + \bigl\vert \Psi (a,d) \bigr\vert + \bigl\vert \Omega (a,c) \bigr\vert \bigr) \bigr] \,{}_{0}d_{q_{2}}s \ {}_{0}d_{q_{1}}t. \end{aligned}$$

(3.9)

Evaluating each integral in (3.9), we obtain (3.3). The proof is completed. □

Remark 3.1

If we take the limit $q_{1}, q_{2}\to 1$, then (3.3) reduces to (1.4).

Theorem 3.2

Under the assumptions of Lemma 3.3. If $|\Phi (t,s)|^{r}$, $|\Theta (t,s)|^{r}$, $|\Psi (t,s)|^{r}$, and $|\Omega (t,s)|^{r}$ are coordinated convex on △ and $p,r>1$, $\frac{1}{p}+\frac{1}{r}=1$, then the following inequality holds:

$$\begin{aligned} & \biggl\vert \frac{f(a,c)+f(a,d)+f(b,c)+f(b,d)}{4} \\ &\quad \quad{} + \frac{1}{(b-a)(d-c)} \biggl[ \int _{a}^{\frac{a+b}{2}} \int _{c}^{ \frac{c+d}{2}}{f(x,y}) \,{}^{\frac{c+d}{2}}d_{q_{2}}y \,{}^{\frac{a+b}{2}}d_{q_{1}}x \\ &\quad \quad{} + \int _{a}^{\frac{a+b}{2}} \int _{\frac{c+d}{2}}^{d}{f(x,y}) \,{}_{\frac{c+d}{2}}d_{q_{2}}y \,{}^{\frac{a+b}{2}}d_{q_{1}}x+ \int _{ \frac{a+b}{2}}^{b} \int _{c}^{\frac{c+d}{2}}{f(x,y}) \,{}^{\frac{c+d}{2}}d_{q_{2}}y \, {}_{\frac{a+b}{2}}d_{q_{1}}x \\ &\quad \quad{} + \int _{\frac{a+b}{2}}^{b} \int _{\frac{c+d}{2}}^{d}{f(x,y}) \,{}_{\frac{c+d}{2}}d_{q_{2}}y \,{}_{\frac{a+b}{2}}d_{q_{1}}x \biggr] \\ &\quad \quad{} - \frac{1}{2(b-a)} \biggl[ \int _{a}^{\frac{a+b}{2}}{f(x,c)+f(x,d) } {}^{\frac{a+b}{2}}d_{q_{1}}x+ \int _{\frac{a+b}{2}}^{b}{f(x,c)+f(x,d)} \,{}_{\frac{a+b}{2}}d_{q_{1}}x \biggr] \\ &\quad \quad{} - \frac{1}{2(d-c)} \biggl[ \int _{c}^{\frac{c+d}{2}}{f(a,y)+f(b,y) } \,{}^{\frac{c+d}{2}}d_{q_{2}}y+ \int _{\frac{c+d}{2}}^{d}{f(a,y)+f(b,y) } \, {}_{\frac{c+d}{2}}d_{q_{2}}y \biggr] \biggr\vert \\ &\quad \leq \frac{(b-a)(d-c)}{16} \biggl[ \frac{q_{1}^{p}q_{2}^{p}}{[p+1]_{q_{1}}[p+1]_{q_{2}}} \biggr] ^{1/p} \biggl[\frac{1}{4[2]_{q_{1}}[2]_{q_{2}}} \biggr]^{1/r} \\ &\quad \quad{}\times \bigl[ \bigl(q_{1}q_{2} \bigl\vert \Phi (a,c) \bigr\vert ^{r}+q_{1}(2+q_{2}) \bigl\vert \Phi (a,d) \bigr\vert ^{r}+q_{2}(2+q_{1}) \bigl\vert \Phi (b,c) \bigr\vert ^{r} \\ &\quad \quad{}+(2+q_{1}) (2+q_{2}) \bigl\vert \Phi (b,d) \bigr\vert ^{r} \bigr)^{1/r} \\ &\quad \quad{}+ \bigl(q_{1}(2+q_{2}) \bigl\vert \Theta (a,c) \bigr\vert ^{r}+q_{1}q_{2} \bigl\vert \Theta (a,d) \bigr\vert ^{r}+(2+q_{1}) (2+q_{2}) \bigl\vert \Theta (b,c) \bigr\vert ^{r} \\ &\quad \quad{}+q_{2}(2+q_{1}) \bigl\vert \Theta (b,d) \bigr\vert ^{r} \bigr)^{1/r} \\ &\quad \quad{}+ \bigl(q_{2}(2+q_{1}) \bigl\vert \Psi (a,c) \bigr\vert ^{r}+(2+q_{1}) (2+q_{2}) \bigl\vert \Psi (a,d) \bigr\vert ^{r}+q_{1}q_{2} \bigl\vert \Psi (b,c) \bigr\vert ^{r} \\ &\quad \quad{}+q_{1}(2+q_{2}) \bigl\vert \Psi (b,d) \bigr\vert ^{r} \bigr)^{1/r} \\ &\quad \quad{}+ \bigl((2+q_{1}) (2+q_{2}) \bigl\vert \Omega (a,c) \bigr\vert ^{r}+q_{2}(2+q_{1}) \bigl\vert \Omega (a,d) \bigr\vert ^{r}+q_{1}(2+q_{2}) \bigl\vert \Omega (b,c) \bigr\vert ^{r} \\ &\quad \quad{}+q_{1}q_{2} \bigl\vert \Omega (b,d) \bigr\vert ^{r} \bigr)^{1/r} \bigr] . \end{aligned}$$

(3.10)

Proof

From Lemma 3.3, we have

$$\begin{aligned} & \biggl\vert \frac{f(a,c)+f(a,d)+f(b,c)+f(b,d)}{4} \\ &\quad \quad{} + \frac{1}{(b-a)(d-c)} \biggl[ \int _{a}^{\frac{a+b}{2}} \int _{c}^{ \frac{c+d}{2}}{f(x,y}) \,{}^{\frac{c+d}{2}}d_{q_{2}}y \,{}^{\frac{a+b}{2}}d_{q_{1}}x \\ &\quad \quad{} + \int _{a}^{\frac{a+b}{2}} \int _{\frac{c+d}{2}}^{d}{f(x,y}) \,{}_{\frac{c+d}{2}}d_{q_{2}}y \,{}^{\frac{a+b}{2}}d_{q_{1}}x+ \int _{ \frac{a+b}{2}}^{b} \int _{c}^{\frac{c+d}{2}}{f(x,y}) \,{}^{\frac{c+d}{2}}d_{q_{2}}y \, {}_{\frac{a+b}{2}}d_{q_{1}}x \\ &\quad \quad{} + \int _{\frac{a+b}{2}}^{b} \int _{\frac{c+d}{2}}^{d}{f(x,y}) \,{}_{\frac{c+d}{2}}d_{q_{2}}y \,{}_{\frac{a+b}{2}}d_{q_{1}}x \biggr] \\ &\quad \quad{} - \frac{1}{2(b-a)} \biggl[ \int _{a}^{\frac{a+b}{2}}{f(x,c)+f(x,d) } {}^{\frac{a+b}{2}}d_{q_{1}}x+ \int _{\frac{a+b}{2}}^{b}{f(x,c)+f(x,d)} \,{}_{\frac{a+b}{2}}d_{q_{1}}x \biggr] \\ &\quad \quad{} - \frac{1}{2(d-c)} \biggl[ \int _{c}^{\frac{c+d}{2}}{f(a,y)+f(b,y) } \,{}^{\frac{c+d}{2}}d_{q_{2}}y+ \int _{\frac{c+d}{2}}^{d}{f(a,y)+f(b,y) } \, {}_{\frac{c+d}{2}}d_{q_{2}}y \biggr] \biggr\vert \\ &\quad \leq \frac{(b-a)(d-c)}{16} \int _{0}^{1} \int _{0}^{1}(q_{1}t) (q_{2}s) \biggl[ \biggl\vert \Phi \biggl( \frac{1-t}{2}a+ \frac{1+t}{2}b, \frac{1-s}{2}c+\frac{1+s}{2}d \biggr) \biggr\vert \\ &\quad \quad{}+ \biggl\vert \Theta \biggl( \frac{1-t}{2}a+ \frac{1+t}{2}b, \frac{1+s}{2}c+\frac{1-s}{2}d \biggr) \biggr\vert \\ &\quad \quad{}+ \biggl\vert \Psi \biggl( \frac{1+t}{2}a+\frac{1-t}{2}b,\frac{1-s}{2}c+\frac{1+s}{2}d \biggr) \biggr\vert \\ &\quad \quad{}+ \biggl\vert \Omega \biggl( \frac{1+t}{2}a+ \frac{1-t}{2}b,\frac{1+s}{2}c+\frac{1-s}{2}d \biggr) \biggr\vert \biggr] \,{}_{0}d_{q_{2}}s \ {}_{0}d_{q_{1}}t. \end{aligned}$$

(3.11)

Now, using the q-Hölder inequality for double integrals, since $|\Phi (t,s)|^{r}$ is coordinated convex, we obtain

$$\begin{aligned} & \int _{0}^{1} \int _{0}^{1}(q_{1}t) (q_{2}s) \biggl\vert \Phi \biggl( \frac{1-t}{2}a+ \frac{1+t}{2}b,\frac{1-s}{2}c+\frac{1+s}{2}d \biggr) \biggr\vert \,{}_{0}d_{q_{2}}s \ {}_{0}d_{q_{1}}t \\ &\quad \leq \biggl[ \int _{0}^{1} \int _{0}^{1}(q_{1}t)^{p}(q_{2}s)^{p} \,{}_{0}d_{q_{2}}s \ {}_{0}d_{q_{1}}t \biggr] ^{1/p} \\ &\quad \quad{}\times \biggl[ \int _{0}^{1} \int _{0}^{1} \biggl\vert \Phi \biggl( \frac{1-t}{2}a+\frac{1+t}{2}b,\frac{1-s}{2}c+ \frac{1+s}{2}d \biggr) \biggr\vert ^{r} \,{}_{0}d_{q_{2}}s \ {}_{0}d_{q_{1}}t \biggr] ^{1/r} \\ &\quad = \biggl[ \frac{q_{1}^{p}q_{2}^{p}}{[p+1]_{q_{1}}[p+1]_{q_{2}}} \biggr] ^{1/p} \\ &\quad \quad{}\times \biggl[ \int _{0}^{1} \int _{0}^{1} \biggl\vert \Phi \biggl( \frac{1-t}{2}a+\frac{1+t}{2}b,\frac{1-s}{2}c+ \frac{1+s}{2}d \biggr) \biggr\vert ^{r} \,{}_{0}d_{q_{2}}s \ {}_{0}d_{q_{1}}t \biggr] ^{1/r} \\ &\quad \leq \biggl[ \frac{q_{1}^{p}q_{2}^{p}}{[p+1]_{q_{1}}[p+1]_{q_{2}}} \biggr] ^{1/p} \\ &\quad \quad{}\times \biggl[ \int _{0}^{1} \int _{0}^{1}\frac{(1-t)(1-s)}{4} \bigl\vert \Phi (a,c) \bigr\vert ^{r}+\frac{(1-t)(1+s)}{4} \bigl\vert \Phi (a,d) \bigr\vert ^{r} \\ &\quad \quad{}+\frac{(1+t)(1-s)}{4} \bigl\vert \Phi (b,c) \bigr\vert ^{r}+ \frac{(1+t)(1+s)}{4} \bigl\vert \Phi (b,d) \bigr\vert ^{r} \,{}_{0}d_{q_{2}}s \ {}_{0}d_{q_{1}}t \biggr] ^{1/r}. \end{aligned}$$

(3.12)

Similarly, we have

$$\begin{aligned}& \begin{aligned}[b] & \int _{0}^{1} \int _{0}^{1}(q_{1}t) (q_{2}s) \biggl\vert \Theta \biggl( \frac{1-t}{2}a+ \frac{1+t}{2}b,\frac{1+s}{2}c+\frac{1-s}{2}d \biggr) \biggr\vert \,{}_{0}d_{q_{2}}s \ {}_{0}d_{q_{1}}t \\ &\quad \leq \biggl[ \frac{q_{1}^{p}q_{2}^{p}}{[p+1]_{q_{1}}[p+1]_{q_{2}}} \biggr] ^{1/p} \\ &\quad \quad{}\times \biggl[ \int _{0}^{1} \int _{0}^{1}\frac{(1-t)(1+s)}{4} \bigl\vert \Theta (a,c) \bigr\vert ^{r}+\frac{(1-t)(1-s)}{4} \bigl\vert \Theta (a,d) \bigr\vert ^{r} \\ &\quad \quad{}+\frac{(1+t)(1+s)}{4} \bigl\vert \Theta (b,c) \bigr\vert ^{r}+ \frac{(1+t)(1-s)}{4} \bigl\vert \Theta (b,d) \bigr\vert ^{r} \,{}_{0}d_{q_{2}}s \ {}_{0}d_{q_{1}}t \biggr] ^{1/r}, \end{aligned} \end{aligned}$$

(3.13)

$$\begin{aligned}& \begin{aligned}[b] & \int _{0}^{1} \int _{0}^{1}(q_{1}t) (q_{2}s) \biggl\vert \Psi \biggl( \frac{1+t}{2}a+ \frac{1-t}{2}b,\frac{1-s}{2}c+\frac{1+s}{2}d \biggr) \biggr\vert \,{}_{0}d_{q_{2}}s \ {}_{0}d_{q_{1}}t \\ &\quad \leq \biggl[ \frac{q_{1}^{p}q_{2}^{p}}{[p+1]_{q_{1}}[p+1]_{q_{2}}} \biggr] ^{1/p} \\ &\quad \quad{}\times \biggl[ \int _{0}^{1} \int _{0}^{1}\frac{(1+t)(1-s)}{4} \bigl\vert \Psi (a,c) \bigr\vert ^{r}+\frac{(1+t)(1+s)}{4} \bigl\vert \Psi (a,d) \bigr\vert ^{r} \\ &\quad \quad{}+\frac{(1-t)(1-s)}{4} \bigl\vert \Psi (b,c) \bigr\vert ^{r}+ \frac{(1-t)(1+s)}{4} \bigl\vert \Psi (b,d) \bigr\vert ^{r} \,{}_{0}d_{q_{2}}s \ {}_{0}d_{q_{1}}t \biggr] ^{1/r} \end{aligned} \end{aligned}$$

(3.14)

and

$$\begin{aligned} & \int _{0}^{1} \int _{0}^{1}(q_{1}t) (q_{2}s) \biggl\vert \Omega \biggl( \frac{1+t}{2}a+ \frac{1-t}{2}b,\frac{1+s}{2}c+\frac{1-s}{2}d \biggr) \biggr\vert \,{}_{0}d_{q_{2}}s \ {}_{0}d_{q_{1}}t \\ &\quad \leq \biggl[ \frac{q_{1}^{p}q_{2}^{p}}{[p+1]_{q_{1}}[p+1]_{q_{2}}} \biggr] ^{1/p} \\ &\quad \quad{}\times \biggl[ \int _{0}^{1} \int _{0}^{1}\frac{(1+t)(1+s)}{4} \bigl\vert \Omega (a,c) \bigr\vert ^{r}+\frac{(1+t)(1-s)}{4} \bigl\vert \Omega (a,d) \bigr\vert ^{r} \\ &\quad \quad{}+\frac{(1-t)(1+s)}{4} \bigl\vert \Omega (b,c) \bigr\vert ^{r}+ \frac{(1-t)(1-s)}{4} \bigl\vert \Omega (b,d) \bigr\vert ^{r} \,{}_{0}d_{q_{2}}s \ {}_{0}d_{q_{1}}t \biggr] ^{1/r}. \end{aligned}$$

(3.15)

Replacing (3.12)–(3.15) in (3.11) and calculating each integral, we obtain

$$\begin{aligned} & \biggl\vert \frac{f(a,c)+f(a,d)+f(b,c)+f(b,d)}{4} \\ &\quad \quad{} + \frac{1}{(b-a)(d-c)} \biggl[ \int _{a}^{\frac{a+b}{2}} \int _{c}^{ \frac{c+d}{2}}{f(x,y}) \,{}^{\frac{c+d}{2}}d_{q_{2}}y \,{}^{\frac{a+b}{2}}d_{q_{1}}x \\ &\quad \quad{} + \int _{a}^{\frac{a+b}{2}} \int _{\frac{c+d}{2}}^{d}{f(x,y}) \,{}_{\frac{c+d}{2}}d_{q_{2}}y \,{}^{\frac{a+b}{2}}d_{q_{1}}x+ \int _{ \frac{a+b}{2}}^{b} \int _{c}^{\frac{c+d}{2}}{f(x,y}) \,{}^{\frac{c+d}{2}}d_{q_{2}}y \, {}_{\frac{a+b}{2}}d_{q_{1}}x \\ &\quad \quad{} + \int _{\frac{a+b}{2}}^{b} \int _{\frac{c+d}{2}}^{d}{f(x,y}) \,{}_{\frac{c+d}{2}}d_{q_{2}}y \,{}_{\frac{a+b}{2}}d_{q_{1}}x \biggr] \\ &\quad \quad{} - \frac{1}{2(b-a)} \biggl[ \int _{a}^{\frac{a+b}{2}}{f(x,c)+f(x,d) } {}^{\frac{a+b}{2}}d_{q_{1}}x+ \int _{\frac{a+b}{2}}^{b}{f(x,c)+f(x,d)} \,{}_{\frac{a+b}{2}}d_{q_{1}}x \biggr] \\ &\quad \quad{} - \frac{1}{2(d-c)} \biggl[ \int _{c}^{\frac{c+d}{2}}{f(a,y)+f(b,y) } \,{}^{\frac{c+d}{2}}d_{q_{2}}y+ \int _{\frac{c+d}{2}}^{d}{f(a,y)+f(b,y) } \, {}_{\frac{c+d}{2}}d_{q_{2}}y \biggr] \biggr\vert \\ &\quad \leq \frac{(b-a)(d-c)}{16} \biggl[ \frac{q_{1}^{p}q_{2}^{p}}{[p+1]_{q_{1}}[p+1]_{q_{2}}} \biggr] ^{1/p} \biggl[\frac{1}{4[2]_{q_{1}}[2]_{q_{2}}} \biggr]^{1/r} \\ &\quad \quad{}\times \bigl[ \bigl(q_{1}q_{2} \bigl\vert \Phi (a,c) \bigr\vert ^{r}+q_{1}(2+q_{2}) \bigl\vert \Phi (a,d) \bigr\vert ^{r}+q_{2}(2+q_{1}) \bigl\vert \Phi (b,c) \bigr\vert ^{r} \\ &\quad \quad{}+ (2+q_{1}) (2+q_{2}) \bigl\vert \Phi (b,d) \bigr\vert ^{r} \bigr)^{1/r} \\ &\quad \quad{}+ \bigl(q_{1}(2+q_{2}) \bigl\vert \Theta (a,c) \bigr\vert ^{r}+q_{1}q_{2} \bigl\vert \Theta (a,d) \bigr\vert ^{r}+(2+q_{1}) (2+q_{2}) \bigl\vert \Theta (b,c) \bigr\vert ^{r} \\ &\quad \quad{}+q_{2}(2+q_{1}) \bigl\vert \Theta (b,d) \bigr\vert ^{r} \bigr)^{1/r} \\ &\quad \quad{}+ \bigl(q_{2}(2+q_{1}) \bigl\vert \Psi (a,c) \bigr\vert ^{r}+(2+q_{1}) (2+q_{2}) \bigl\vert \Psi (a,d) \bigr\vert ^{r}+q_{1}q_{2} \bigl\vert \Psi (b,c) \bigr\vert ^{r} \\ &\quad \quad{}+q_{1}(2+q_{2}) \bigl\vert \Psi (b,d) \bigr\vert ^{r} \bigr)^{1/r} \\ &\quad \quad{}+ \bigl((2+q_{1}) (2+q_{2}) \bigl\vert \Omega (a,c) \bigr\vert ^{r}+q_{2}(2+q_{1}) \bigl\vert \Omega (a,d) \bigr\vert ^{r}+q_{1}(2+q_{2}) \bigl\vert \Omega (b,c) \bigr\vert ^{r} \\ &\quad \quad{}+ q_{1}q_{2} \bigl\vert \Omega (b,d) \bigr\vert ^{r} \bigr)^{1/r} \bigr] . \end{aligned}$$

The proof is completed. □

Theorem 3.3

Under the assumptions of Lemma 3.3. If $|\Phi (t,s)|^{r}$, $|\Theta (t,s)|^{r}$, $|\Psi (t,s)|^{r}$, and $|\Omega (t,s)|^{r}$ are coordinated convex on △ and $r\geq 1$, then the following inequality holds:

$$\begin{aligned} & \biggl\vert \frac{f(a,c)+f(a,d)+f(b,c)+f(b,d)}{4} \\ &\quad \quad{} + \frac{1}{(b-a)(d-c)} \biggl[ \int _{a}^{\frac{a+b}{2}} \int _{c}^{ \frac{c+d}{2}}{f(x,y}) \,{}^{\frac{c+d}{2}}d_{q_{2}}y \,{}^{\frac{a+b}{2}}d_{q_{1}}x \\ &\quad \quad{} + \int _{a}^{\frac{a+b}{2}} \int _{\frac{c+d}{2}}^{d}{f(x,y}) \,{}_{\frac{c+d}{2}}d_{q_{2}}y \,{}^{\frac{a+b}{2}}d_{q_{1}}x+ \int _{ \frac{a+b}{2}}^{b} \int _{c}^{\frac{c+d}{2}}{f(x,y}) \,{}^{\frac{c+d}{2}}d_{q_{2}}y \, {}_{\frac{a+b}{2}}d_{q_{1}}x \\ &\quad \quad{} + \int _{\frac{a+b}{2}}^{b} \int _{\frac{c+d}{2}}^{d}{f(x,y}) \,{}_{\frac{c+d}{2}}d_{q_{2}}y \,{}_{\frac{a+b}{2}}d_{q_{1}}x \biggr] \\ &\quad \quad{} - \frac{1}{2(b-a)} \biggl[ \int _{a}^{\frac{a+b}{2}}{f(x,c)+f(x,d) } {}^{\frac{a+b}{2}}d_{q_{1}}x+ \int _{\frac{a+b}{2}}^{b}{f(x,c)+f(x,d)} \,{}_{\frac{a+b}{2}}d_{q_{1}}x \biggr] \\ &\quad \quad{} - \frac{1}{2(d-c)} \biggl[ \int _{c}^{\frac{c+d}{2}}{f(a,y)+f(b,y) } \,{}^{\frac{c+d}{2}}d_{q_{2}}y+ \int _{\frac{c+d}{2}}^{d}{f(a,y)+f(b,y) } \, {}_{\frac{c+d}{2}}d_{q_{2}}y \biggr] \biggr\vert \\ &\quad \leq \frac{(b-a)(d-c)}{16} \biggl[ \frac{q_{1}q_{2}}{[2]_{q_{1}}[2]_{q_{2}}} \biggr] ^{1-1/r} \biggl[ \frac{1}{4[2]_{q_{1}}[2]_{q_{2}}[3]_{q_{1}}[3]_{q_{2}}} \biggr]^{1/r} \\ &\quad \quad{} \times \bigl[ \bigl( q_{1}^{3}q_{2}^{3} \bigl\vert \Phi (a,c) \bigr\vert ^{r}+q_{1}^{3} \bigl(2q_{2}+2q_{2}^{2}+q_{2}^{3} \bigr) \bigl\vert \Phi (a,d) \bigr\vert ^{r} \\ &\quad \quad{} + q_{2}^{3} \bigl(2q_{1}+2q_{1}^{2}+q_{1}^{3} \bigr) \bigl\vert \Phi (b,c) \bigr\vert ^{r}+ \bigl(2q_{1}+2q_{1}^{2}+q_{1}^{3} \bigr) \bigl(2q_{2}+2q_{2}^{2}+q_{2}^{3} \bigr) \bigl\vert \Phi (b,d) \bigr\vert ^{r} \bigr) ^{1/r} \\ &\quad \quad{} + \bigl( q_{1}^{3} \bigl(2q_{2}+2q_{2}^{2}+q_{2}^{3} \bigr) \bigl\vert \Theta (a,c) \bigr\vert ^{r}+q_{1}^{3}q_{2}^{3} \bigl\vert \Theta (a,d) \bigr\vert ^{r} \\ &\quad \quad{} + \bigl(2q_{1}+2q_{1}^{2}+q_{1}^{3} \bigr) \bigl(2q_{2}+2q_{2}^{2}+q_{2}^{3} \bigr) \bigl\vert \Theta (b,c) \bigr\vert ^{r}+q_{2}^{3} \bigl(2q_{1}+2q_{1}^{2}+q_{1}^{3} \bigr) \bigl\vert \Theta (b,d) \bigr\vert ^{r} \bigr) ^{1/r} \\ &\quad \quad{} + \bigl( q_{2}^{3} \bigl(2q_{1}+2q_{1}^{2}+q_{1}^{3} \bigr) \bigl\vert \Psi (a,c) \bigr\vert ^{r}+ \bigl(2q_{1}+2q_{1}^{2}+q_{1}^{3} \bigr) \bigl(2q_{2}+2q_{2}^{2}+q_{2}^{3} \bigr) \bigl\vert \Psi (a,d) \bigr\vert ^{r} \\ &\quad \quad{} + q_{1}^{3}q_{2}^{3} \bigl\vert \Psi (b,c) \bigr\vert ^{r}+q_{1}^{3} \bigl(2q_{2}+2q_{2}^{2}+q_{2}^{3} \bigr) \bigl\vert \Psi (b,d) \bigr\vert ^{r} \bigr) ^{1/r} \\ &\quad \quad{} + \bigl( \bigl(2q_{1}+2q_{1}^{2}+q_{1}^{3} \bigr) \bigl(2q_{2}+2q_{2}^{2}+q_{2}^{3} \bigr) \bigl\vert \Omega (a,c) \bigr\vert ^{r}+q_{2}^{3} \bigl(2q_{1}+2q_{1}^{2}+q_{1}^{3} \bigr) \bigl\vert \Omega (a,d) \bigr\vert ^{r} \\ &\quad \quad{} + q_{1}^{3} \bigl(2q_{2}+2q_{2}^{2}+q_{2}^{3} \bigr) \bigl\vert \Omega (b,c) \bigr\vert ^{r}+q_{1}^{3}q_{2}^{3} \bigl\vert \Omega (b,d) \bigr\vert ^{r} \bigr) ^{1/r} \bigr]. \end{aligned}$$

(3.16)

Proof

From Lemma 3.3, we have

$$\begin{aligned} & \biggl\vert \frac{f(a,c)+f(a,d)+f(b,c)+f(b,d)}{4} \\ &\quad \quad{} + \frac{1}{(b-a)(d-c)} \biggl[ \int _{a}^{\frac{a+b}{2}} \int _{c}^{ \frac{c+d}{2}}{f(x,y}) \,{}^{\frac{c+d}{2}}d_{q_{2}}y \,{}^{\frac{a+b}{2}}d_{q_{1}}x \\ &\quad \quad{} + \int _{a}^{\frac{a+b}{2}} \int _{\frac{c+d}{2}}^{d}{f(x,y}) \,{}_{\frac{c+d}{2}}d_{q_{2}}y \,{}^{\frac{a+b}{2}}d_{q_{1}}x+ \int _{ \frac{a+b}{2}}^{b} \int _{c}^{\frac{c+d}{2}}{f(x,y}) \,{}^{\frac{c+d}{2}}d_{q_{2}}y \, {}_{\frac{a+b}{2}}d_{q_{1}}x \\ &\quad \quad{} + \int _{\frac{a+b}{2}}^{b} \int _{\frac{c+d}{2}}^{d}{f(x,y}) \,{}_{\frac{c+d}{2}}d_{q_{2}}y \,{}_{\frac{a+b}{2}}d_{q_{1}}x \biggr] \\ &\quad \quad{} - \frac{1}{2(b-a)} \biggl[ \int _{a}^{\frac{a+b}{2}}{f(x,c)+f(x,d) } {}^{\frac{a+b}{2}}d_{q_{1}}x+ \int _{\frac{a+b}{2}}^{b}{f(x,c)+f(x,d)} \,{}_{\frac{a+b}{2}}d_{q_{1}}x \biggr] \\ &\quad \quad{} - \frac{1}{2(d-c)} \biggl[ \int _{c}^{\frac{c+d}{2}}{f(a,y)+f(b,y) } \,{}^{\frac{c+d}{2}}d_{q_{2}}y+ \int _{\frac{c+d}{2}}^{d}{f(a,y)+f(b,y) } \, {}_{\frac{c+d}{2}}d_{q_{2}}y \biggr] \biggr\vert \\ &\quad \leq \frac{(b-a)(d-c)}{16} \int _{0}^{1} \int _{0}^{1}(q_{1}t) (q_{2}s) \biggl[ \biggl\vert \Phi \biggl( \frac{1-t}{2}a+\frac{1+t}{2}b, \frac{1-s}{2}c+\frac{1+s}{2}d \biggr) \biggr\vert \\ &\quad \quad{}+ \biggl\vert \Theta \biggl( \frac{1-t}{2}a+ \frac{1+t}{2}b, \frac{1+s}{2}c+\frac{1-s}{2}d \biggr) \biggr\vert \\ &\quad \quad{}+ \biggl\vert \Psi \biggl( \frac{1+t}{2}a+\frac{1-t}{2}b,\frac{1-s}{2}c+\frac{1+s}{2}d \biggr) \biggr\vert \\ &\quad \quad{}+ \biggl\vert \Omega \biggl( \frac{1+t}{2}a+ \frac{1-t}{2}b,\frac{1+s}{2}c+\frac{1-s}{2}d \biggr) \biggr\vert \biggr] \,{}_{0}d_{q_{2}}s \ {}_{0}d_{q_{1}}t. \end{aligned}$$

(3.17)

Now, using the q-power mean inequality for double integrals, since $|\Phi (t,s)|^{r}$ is coordinated convex, we obtain

$$\begin{aligned} & \int _{0}^{1} \int _{0}^{1}(q_{1}t) (q_{2}s) \biggl\vert \Phi \biggl( \frac{1-t}{2}a+ \frac{1+t}{2}b,\frac{1-s}{2}c+\frac{1+s}{2}d \biggr) \biggr\vert \,{}_{0}d_{q_{2}}s \ {}_{0}d_{q_{1}}t \\ &\quad \leq \biggl[ \int _{0}^{1} \int _{0}^{1}(q_{1}t) (q_{2}s) \,{}_{0}d_{q_{2}}s \ {}_{0}d_{q_{1}}t \biggr] ^{1-1/r} \\ &\quad \quad{} \times \biggl[ \int _{0}^{1} \int _{0}^{1}(q_{1}t) (q_{2}s) \biggl\vert \Phi \biggl( \frac{1-t}{2}a+ \frac{1+t}{2}b,\frac{1-s}{2}c+ \frac{1+s}{2}d \biggr) \biggr\vert ^{r} \,{}_{0}d_{q_{2}}s \ {}_{0}d_{q_{1}}t \biggr] ^{1/r} \\ &\quad = \biggl[ \frac{q_{1}q_{2}}{[2]_{q_{1}}[2]_{q_{2}}} \biggr] ^{1-1/r} \\ &\quad \quad{} \times \biggl[ \int _{0}^{1} \int _{0}^{1}(q_{1}t) (q_{2}s) \biggl\vert \Phi \biggl( \frac{1-t}{2}a+ \frac{1+t}{2}b,\frac{1-s}{2}c+ \frac{1+s}{2}d \biggr) \biggr\vert ^{r} \,{}_{0}d_{q_{2}}s \ {}_{0}d_{q_{1}}t \biggr] ^{1/r} \\ &\quad \leq \biggl[ \frac{q_{1}q_{2}}{[2]_{q_{1}}[2]_{q_{2}}} \biggr] ^{1-1/r} \\ &\quad \quad{} \times \biggl[ \int _{0}^{1} \int _{0}^{1}(q_{1}t) (q_{2}s) \biggl\{ \frac{(1-t)(1-s)}{4} \bigl\vert \Phi (a,c) \bigr\vert ^{r}+ \frac{(1-t)(1+s)}{4} \bigl\vert \Phi (a,d) \bigr\vert ^{r} \\ &\quad \quad{} + \frac{(1+t)(1-s)}{4} \bigl\vert \Phi (b,c) \bigr\vert ^{r}+ \frac{(1+t)(1+s)}{4} \bigl\vert \Phi (b,d) \bigr\vert ^{r} \biggr\} \,{}_{0}d_{q_{2}}s \ {}_{0}d_{q_{1}}t \biggr] ^{1/r} \\ &\quad = \biggl[ \frac{q_{1}q_{2}}{[2]_{q_{1}}[2]_{q_{2}}} \biggr] ^{1-1/r} \biggl[ \frac{1}{4[2]_{q_{1}}[2]_{q_{2}}[3]_{q_{1}}[3]_{q_{2}}} \biggr]^{1/r} \\ &\quad \quad{} \times \bigl[ q_{1}^{3}q_{2}^{3} \bigl\vert \Phi (a,c) \bigr\vert ^{r}+q_{1}^{3} \bigl(2q_{2}+2q_{2}^{2}+q_{2}^{3} \bigr) \bigl\vert \Phi (a,d) \bigr\vert ^{r} \\ &\quad \quad{} + q_{2}^{3} \bigl(2q_{1}+2q_{1}^{2}+q_{1}^{3} \bigr) \bigl\vert \Phi (b,c) \bigr\vert ^{r} \\ &\quad \quad{} + \bigl(2q_{1}+2q_{1}^{2}+q_{1}^{3} \bigr) \bigl(2q_{2}+2q_{2}^{2}+q_{2}^{3} \bigr) \bigl\vert \Phi (b,d) \bigr\vert ^{r} \bigr] ^{1/r}. \end{aligned}$$

(3.18)

Similarly, we have

$$\begin{aligned}& \begin{aligned}[b] & \int _{0}^{1} \int _{0}^{1}(q_{1}t) (q_{2}s) \biggl\vert \Theta \biggl( \frac{1-t}{2}a+ \frac{1+t}{2}b,\frac{1+s}{2}c+\frac{1-s}{2}d \biggr) \biggr\vert \,{}_{0}d_{q_{2}}s \ {}_{0}d_{q_{1}}t \\ &\quad \leq \biggl[ \frac{q_{1}q_{2}}{[2]_{q_{1}}[2]_{q_{2}}} \biggr] ^{1-1/r} \biggl[ \frac{1}{4[2]_{q_{1}}[2]_{q_{2}}[3]_{q_{1}}[3]_{q_{2}}} \biggr]^{1/r} \\ &\quad \quad{}\times \bigl[ q_{1}^{3} \bigl(2q_{2}+2q_{2}^{2}+q_{2}^{3} \bigr) \bigl\vert \Theta (a,c) \bigr\vert ^{r}+q_{1}^{3}q_{2}^{3} \bigl\vert \Theta (a,d) \bigr\vert ^{r} \\ &\quad \quad{}+ \bigl(2q_{1}+2q_{1}^{2}+q_{1}^{3} \bigr) \bigl(2q_{2}+2q_{2}^{2}+q_{2}^{3} \bigr) \bigl\vert \Theta (b,c) \bigr\vert ^{r} \\ &\quad \quad{}+ q_{2}^{3} \bigl(2q_{1}+2q_{1}^{2}+q_{1}^{3} \bigr) \bigl\vert \Theta (b,d) \bigr\vert ^{r} \bigr] ^{1/r}, \end{aligned} \end{aligned}$$

(3.19)

$$\begin{aligned}& \begin{aligned}[b] & \int _{0}^{1} \int _{0}^{1}(q_{1}t) (q_{2}s) \biggl\vert \Psi \biggl( \frac{1+t}{2}a+ \frac{1-t}{2}b,\frac{1-s}{2}c+\frac{1+s}{2}d \biggr) \biggr\vert \,{}_{0}d_{q_{2}}s \ {}_{0}d_{q_{1}}t \\ &\quad \leq \biggl[ \frac{q_{1}q_{2}}{[2]_{q_{1}}[2]_{q_{2}}} \biggr] ^{1-1/r} \biggl[ \frac{1}{4[2]_{q_{1}}[2]_{q_{2}}[3]_{q_{1}}[3]_{q_{2}}} \biggr]^{1/r} \\ &\quad \quad{}\times \bigl[ q_{2}^{3} \bigl(2q_{1}+2q_{1}^{2}+q_{1}^{3} \bigr) \bigl\vert \Psi (a,c) \bigr\vert ^{r}+ \bigl(2q_{1}+2q_{1}^{2}+q_{1}^{3} \bigr) \bigl(2q_{2}+2q_{2}^{2}+q_{2}^{3} \bigr) \bigl\vert \Psi (a,d) \bigr\vert ^{r} \\ &\quad \quad{}+ q_{1}^{3}q_{2}^{3} \bigl\vert \Psi (b,c) \bigr\vert ^{r}+q_{1}^{3} \bigl(2q_{2}+2q_{2}^{2}+q_{2}^{3} \bigr) \bigl\vert \Psi (b,d) \bigr\vert ^{r} \bigr] ^{1/r} \end{aligned} \end{aligned}$$

(3.20)

and

$$\begin{aligned} & \int _{0}^{1} \int _{0}^{1}(q_{1}t) (q_{2}s) \biggl\vert \Omega \biggl( \frac{1+t}{2}a+ \frac{1-t}{2}b,\frac{1+s}{2}c+\frac{1-s}{2}d \biggr) \biggr\vert \,{}_{0}d_{q_{2}}s \ {}_{0}d_{q_{1}}t \\ &\quad \leq \biggl[ \frac{q_{1}q_{2}}{[2]_{q_{1}}[2]_{q_{2}}} \biggr] ^{1-1/r} \biggl[ \frac{1}{4[2]_{q_{1}}[2]_{q_{2}}[3]_{q_{1}}[3]_{q_{2}}} \biggr]^{1/r} \\ &\quad \quad{}\times \bigl[ \bigl(2q_{1}+2q_{1}^{2}+q_{1}^{3} \bigr) \bigl(2q_{2}+2q_{2}^{2}+q_{2}^{3} \bigr) \bigl\vert \Omega (a,c) \bigr\vert ^{r}+q_{2}^{3} \bigl(2q_{1}+2q_{1}^{2}+q_{1}^{3} \bigr) \bigl\vert \Omega (a,d) \bigr\vert ^{r} \\ &\quad \quad{}+ q_{1}^{3} \bigl(2q_{2}+2q_{2}^{2}+q_{2}^{3} \bigr) \bigl\vert \Omega (b,c) \bigr\vert ^{r}+q_{1}^{3}q_{2}^{3} \bigl\vert \Omega (b,d) \bigr\vert ^{r} \bigr] ^{1/r}. \end{aligned}$$

(3.21)

Replacing (3.18)–(3.21) in (3.17), we obtain

$$\begin{aligned} & \biggl\vert \frac{f(a,c)+f(a,d)+f(b,c)+f(b,d)}{4} \\ &\quad \quad{} + \frac{1}{(b-a)(d-c)} \biggl[ \int _{a}^{\frac{a+b}{2}} \int _{c}^{ \frac{c+d}{2}}{f(x,y}) \,{}^{\frac{c+d}{2}}d_{q_{2}}y \,{}^{\frac{a+b}{2}}d_{q_{1}}x \\ &\quad \quad{} + \int _{a}^{\frac{a+b}{2}} \int _{\frac{c+d}{2}}^{d}{f(x,y}) \,{}_{\frac{c+d}{2}}d_{q_{2}}y \,{}^{\frac{a+b}{2}}d_{q_{1}}x+ \int _{ \frac{a+b}{2}}^{b} \int _{c}^{\frac{c+d}{2}}{f(x,y}) \,{}^{\frac{c+d}{2}}d_{q_{2}}y \, {}_{\frac{a+b}{2}}d_{q_{1}}x \\ &\quad \quad{} + \int _{\frac{a+b}{2}}^{b} \int _{\frac{c+d}{2}}^{d}{f(x,y}) \,{}_{\frac{c+d}{2}}d_{q_{2}}y \,{}_{\frac{a+b}{2}}d_{q_{1}}x \biggr] \\ &\quad \quad{} - \frac{1}{2(b-a)} \biggl[ \int _{a}^{\frac{a+b}{2}}{f(x,c)+f(x,d) } {}^{\frac{a+b}{2}}d_{q_{1}}x+ \int _{\frac{a+b}{2}}^{b}{f(x,c)+f(x,d)} \,{}_{\frac{a+b}{2}}d_{q_{1}}x \biggr] \\ &\quad \quad{} - \frac{1}{2(d-c)} \biggl[ \int _{c}^{\frac{c+d}{2}}{f(a,y)+f(b,y) } \,{}^{\frac{c+d}{2}}d_{q_{2}}y+ \int _{\frac{c+d}{2}}^{d}{f(a,y)+f(b,y) } \, {}_{\frac{c+d}{2}}d_{q_{2}}y \biggr] \biggr\vert \\ &\quad \leq \frac{(b-a)(d-c)}{16} \biggl[ \frac{q_{1}q_{2}}{[2]_{q_{1}}[2]_{q_{2}}} \biggr] ^{1-1/r} \biggl[ \frac{1}{4[2]_{q_{1}}[2]_{q_{2}}[3]_{q_{1}}[3]_{q_{2}}} \biggr]^{1/r} \\ &\quad \quad{} \times \bigl[ \bigl( q_{1}^{3}q_{2}^{3} \bigl\vert \Phi (a,c) \bigr\vert ^{r}+q_{1}^{3} \bigl(2q_{2}+2q_{2}^{2}+q_{2}^{3} \bigr) \bigl\vert \Phi (a,d) \bigr\vert ^{r} \\ &\quad \quad{} + q_{2}^{3} \bigl(2q_{1}+2q_{1}^{2}+q_{1}^{3} \bigr) \bigl\vert \Phi (b,c) \bigr\vert ^{r}+ \bigl(2q_{1}+2q_{1}^{2}+q_{1}^{3} \bigr) \bigl(2q_{2}+2q_{2}^{2}+q_{2}^{3} \bigr) \bigl\vert \Phi (b,d) \bigr\vert ^{r} \bigr) ^{1/r} \\ &\quad \quad{} + \bigl( q_{1}^{3} \bigl(2q_{2}+2q_{2}^{2}+q_{2}^{3} \bigr) \bigl\vert \Theta (a,c) \bigr\vert ^{r}+q_{1}^{3}q_{2}^{3} \bigl\vert \Theta (a,d) \bigr\vert ^{r} \\ &\quad \quad{} + \bigl(2q_{1}+2q_{1}^{2}+q_{1}^{3} \bigr) \bigl(2q_{2}+2q_{2}^{2}+q_{2}^{3} \bigr) \bigl\vert \Theta (b,c) \bigr\vert ^{r}+q_{2}^{3} \bigl(2q_{1}+2q_{1}^{2}+q_{1}^{3} \bigr) \bigl\vert \Theta (b,d) \bigr\vert ^{r} \bigr) ^{1/r} \\ &\quad \quad{} + \bigl( q_{2}^{3} \bigl(2q_{1}+2q_{1}^{2}+q_{1}^{3} \bigr) \bigl\vert \Psi (a,c) \bigr\vert ^{r}+ \bigl(2q_{1}+2q_{1}^{2}+q_{1}^{3} \bigr) \bigl(2q_{2}+2q_{2}^{2}+q_{2}^{3} \bigr) \bigl\vert \Psi (a,d) \bigr\vert ^{r} \\ &\quad \quad{} + q_{1}^{3}q_{2}^{3} \bigl\vert \Psi (b,c) \bigr\vert ^{r}+q_{1}^{3} \bigl(2q_{2}+2q_{2}^{2}+q_{2}^{3} \bigr) \bigl\vert \Psi (b,d) \bigr\vert ^{r} \bigr) ^{1/r} \\ &\quad \quad{} + \bigl( \bigl(2q_{1}+2q_{1}^{2}+q_{1}^{3} \bigr) \bigl(2q_{2}+2q_{2}^{2}+q_{2}^{3} \bigr) \bigl\vert \Omega (a,c) \bigr\vert ^{r}+q_{2}^{3} \bigl(2q_{1}+2q_{1}^{2}+q_{1}^{3} \bigr) \bigl\vert \Omega (a,d) \bigr\vert ^{r} \\ &\quad \quad{} + q_{1}^{3} \bigl(2q_{2}+2q_{2}^{2}+q_{2}^{3} \bigr) \bigl\vert \Omega (b,c) \bigr\vert ^{r}+q_{1}^{3}q_{2}^{3} \bigl\vert \Omega (b,d) \bigr\vert ^{r} \bigr) ^{1/r} \bigr]. \end{aligned}$$

The proof is completed. □

4 Examples

Now, we give some examples of our main results to demonstrate our theorems.

Example 4.1

Let $f:[0,1]\times [0,1]\to \mathbb{R}$ be a function defined by $f(x,y)=x^{2}y^{2}$. Then, f is q-partially differentiable. Moreover, $\vert \Phi (t,s) \vert$, $\vert \Theta (t,s) \vert$, $\vert \Psi (t,s) \vert$, and $\vert \Omega (t,s) \vert$ are q-integrable coordinated convex on $[0,1]\times [0,1]$. By applying Theorem 3.1 with $q_{1}=\frac{1}{4}$ and $q_{2}=\frac{3}{4}$, we have

$$\begin{aligned} &\frac{f(0,0)+f(0,1)+f(1,0)+f(1,1)}{4}=\frac{1}{4}, \\ &\frac{1}{(1-0)(1-0)} \biggl[ \int _{0}^{\frac{0+1}{2}} \int _{0}^{ \frac{0+1}{2}}{x^{2}y^{2}} \,{}^{\frac{0+1}{2}}d_{\frac{3}{4}}y \,{}^{\frac{0+1}{2}}d_{ \frac{1}{4}}x \\ &\quad \quad{}+ \int _{0}^{\frac{0+1}{2}} \int _{\frac{0+1}{2}}^{1}{x^{2}y^{2}} \, {}_{\frac{0+1}{2}}d_{\frac{3}{4}}y \,{}^{\frac{0+1}{2}}d_{\frac{1}{4}}x + \int _{\frac{0+1}{2}}^{1} \int _{0}^{\frac{0+1}{2}}{x^{2}y^{2}} \,{}^{\frac{0+1}{2}}d_{\frac{3}{4}}y \,{}_{\frac{0+1}{2}}d_{\frac{1}{4}}x \\ &\quad \quad{}+ \int _{\frac{0+1}{2}}^{1} \int _{\frac{0+1}{2}}^{1}{x^{2}y^{2}} \, {}_{\frac{0+1}{2}}d_{\frac{3}{4}}y \,{}_{\frac{0+1}{2}}d_{\frac{1}{4}}x \biggr] \\ &\quad = \int _{0}^{\frac{1}{2}} \int _{0}^{\frac{1}{2}}{x^{2}y^{2}} \,{}^{\frac{1}{2}}d_{\frac{3}{4}}y \,{}^{\frac{1}{2}}d_{\frac{1}{4}}x+ \int _{0}^{\frac{1}{2}} \int _{\frac{1}{2}}^{1}{x^{2}y^{2}} \,{}_{\frac{1}{2}}d_{\frac{3}{4}}y \,{}^{\frac{1}{2}}d_{\frac{1}{4}}x \\ &\quad \quad{}+ \int _{\frac{1}{2}}^{1} \int _{0}^{\frac{1}{2}}{x^{2}y^{2}} \,{}^{\frac{1}{2}}d_{\frac{3}{4}}y \,{}_{\frac{1}{2}}d_{\frac{1}{4}}x+ \int _{\frac{1}{2}}^{1} \int _{\frac{1}{2}}^{1}{x^{2}y^{2}} \,{}_{\frac{1}{2}}d_{\frac{3}{4}}y \,{}_{\frac{1}{2}}d_{\frac{1}{4}}x \\ &\quad =\frac{1}{4}\cdot \frac{3}{4}\cdot \frac{1}{2} \cdot \frac{1}{2}\sum_{n=0}^{\infty} \sum_{m=0}^{\infty}{ \biggl( \frac{1}{4} \biggr)^{n} \biggl(\frac{3}{4} \biggr)^{m}}\frac{1}{4}\cdot \frac{1}{4} \biggl(1- \biggl( \frac{1}{4} \biggr)^{n} \biggr)^{2} \biggl(1- \biggl(\frac{3}{4} \biggr)^{m} \biggr)^{2} \\ &\quad \quad{}+\frac{1}{4}\cdot \frac{3}{4}\cdot \frac{1}{2}\cdot \frac{1}{2}\sum _{n=0}^{\infty}\sum_{m=0}^{\infty}{ \biggl(\frac{1}{4} \biggr)^{n} \biggl(\frac{3}{4} \biggr)^{m}}\frac{1}{4}\cdot \frac{1}{4} \biggl(1- \biggl( \frac{1}{4} \biggr)^{n} \biggr)^{2} \biggl(1+ \biggl(\frac{3}{4} \biggr)^{m} \biggr)^{2} \\ &\quad \quad{}+\frac{1}{4}\cdot \frac{3}{4}\cdot \frac{1}{2}\cdot \frac{1}{2}\sum _{n=0}^{\infty}\sum_{m=0}^{\infty}{ \biggl(\frac{1}{4} \biggr)^{n} \biggl(\frac{3}{4} \biggr)^{m}}\frac{1}{4}\cdot \frac{1}{4} \biggl(1+ \biggl( \frac{1}{4} \biggr)^{n} \biggr)^{2} \biggl(1- \biggl(\frac{3}{4} \biggr)^{m} \biggr)^{2} \\ &\quad \quad{}+\frac{1}{4}\cdot \frac{3}{4}\cdot \frac{1}{2}\cdot \frac{1}{2}\sum _{n=0}^{\infty}\sum_{m=0}^{\infty}{ \biggl(\frac{1}{4} \biggr)^{n} \biggl(\frac{3}{4} \biggr)^{m}}\frac{1}{4}\cdot \frac{1}{4} \biggl(1+ \biggl( \frac{1}{4} \biggr)^{n} \biggr)^{2} \biggl(1+ \biggl(\frac{3}{4} \biggr)^{m} \biggr)^{2} \\ &\quad = \frac{85}{116{,}032}+\frac{11{,}339}{1{,}740{,}480}+\frac{1765}{116{,}032}+ \frac{235{,}451}{1{,}740{,}480}=\frac{53}{336} \end{aligned}$$

and

$$\begin{aligned} &\frac{1}{2(1-0)} \biggl[ \int _{0}^{\frac{0+1}{2}}{f(x,0)+f(x,1)} \,{}^{\frac{0+1}{2}}d_{\frac{1}{4}}x+ \int _{\frac{0+1}{2}}^{1}{f(x,0)+f(x,1)} \,{}_{\frac{0+1}{2}}d_{\frac{1}{4}}x \biggr] \\ &\quad \quad{} + \frac{1}{2(1-0)} \biggl[ \int _{0}^{\frac{0+1}{2}}{f(0,y)+f(1,y)} \,{}^{\frac{0+1}{2}}d_{\frac{3}{4}}y + \int _{\frac{0+1}{2}}^{1}{f(0,y)+f(1,y)} \,{}_{\frac{0+1}{2}}d_{\frac{3}{4}}y \biggr] \\ &\quad =\frac{1}{2} \biggl[ \int _{0}^{\frac{1}{2}}{x^{2}} \,{}^{\frac{1}{2}}d_{ \frac{1}{4}}x+ \int _{\frac{1}{2}}^{1}{x^{2}} \,{}_{\frac{1}{2}}d_{ \frac{1}{4}}x+ \int _{0}^{\frac{1}{2}}{y^{2}} \,{}^{\frac{1}{2}}d_{ \frac{3}{4}}y + \int _{\frac{1}{2}}^{1}{y^{2}} \,{}_{\frac{1}{2}}d_{ \frac{3}{4}}y \biggr] \\ &\quad =\frac{1}{2} \Biggl[ \frac{3}{4}\cdot \frac{1}{2} \sum_{n=0}^{\infty}{ \biggl( \frac{1}{4} \biggr)^{n}\frac{1}{4} \biggl(1- \biggl(\frac{1}{4} \biggr)^{n} \biggr)^{2}}+ \frac{3}{4}\cdot \frac{1}{2}\sum_{n=0}^{ \infty}{ \biggl(\frac{1}{4} \biggr)^{n}\frac{1}{4} \biggl(1+ \biggl( \frac{1}{4} \biggr)^{n} \biggr)^{2}} \\ &\quad \quad{}+ \frac{1}{2}\cdot \frac{1}{2}\sum _{n=0}^{\infty}{ \biggl(\frac{3}{4} \biggr)^{n}\frac{1}{4} \biggl(1- \biggl( \frac{3}{4} \biggr)^{n} \biggr)^{2}}+ \frac{1}{4}\cdot \frac{1}{2}\sum_{n=0}^{ \infty}{ \biggl(\frac{3}{4} \biggr)^{n}\frac{1}{4} \biggl(1+ \biggl( \frac{3}{4} \biggr)^{n} \biggr)^{2}} \Biggr] \\ &\quad =\frac{1}{2} \biggl[\frac{68}{3360}+\frac{1412}{3360}+ \frac{300}{8288}+\frac{2668}{8288} \biggr]=\frac{1241}{3108}. \end{aligned}$$

Thus, the left-hand side of (3.3) is

$$ \biggl\vert \frac{1}{4}+\frac{53}{336}-\frac{1241}{3108} \biggr\vert = \frac{5}{592}. $$

Next, we consider

$$\begin{aligned}& \begin{aligned} \Phi (t,s)&= \frac {{}_{0,0}\partial ^{2}\,{}_{\frac{1}{4},\frac{3}{4}} t^{2}s^{2}}{{}_{0}\partial _{\frac{1}{4}}t{}_{0}\partial _{\frac{3}{4}}s}= \frac{4\cdot 4}{3\cdot 1\cdot ts} \biggl[ \frac{9t^{2}s^{2}}{16\cdot 16}- \frac{t^{2}s^{2}}{16}-\frac{9t^{2}s^{2}}{16}+t^{2}s^{2} \biggr] \\ &=\frac{35ts}{16}, \end{aligned} \\& \begin{aligned} \Theta (t,s)&= \frac {{}_{0}^{1}\partial ^{2}\,{}_{\frac{1}{4},\frac{3}{4}} t^{2}s^{2}}{{}_{0}\partial _{\frac{1}{4}}t{}^{1}\partial _{\frac{3}{4}}s} \\ &=\frac{4\cdot 4}{3\cdot 1\cdot t(1-s)} \biggl[\frac{t^{2}(3s+1)^{2}}{16\cdot 16}- \frac{t^{2}s^{2}}{16}-\frac{t^{2}(3s+1)^{2}}{16}+t^{2}s^{2} \biggr] \\ &=\frac{-5t(7s+1)}{16}, \end{aligned} \\& \begin{aligned} \Psi (t,s)&= \frac {{}_{0}^{1}\partial ^{2}\,{}_{\frac{1}{4},\frac{3}{4}} t^{2}s^{2}}{{}^{1}\partial _{\frac{1}{4}}t{}_{0}\partial _{\frac{3}{4}}s} \\ &=\frac{4\cdot 4}{3\cdot 1\cdot (1-t)s} \biggl[\frac{9(t+3)^{2}s^{2}}{16\cdot 16}- \frac{(t+3)^{2}s^{2}}{16}- \frac{9t^{2}s^{2}}{16}+t^{2}s^{2} \biggr] \\ &=\frac{-7(5t+3)s}{16} \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \Omega (t,s)&= \frac {{}^{1,1}\partial ^{2}\,{}_{\frac{1}{4},\frac{3}{4}} t^{2}s^{2}}{{}^{1}\partial _{\frac{1}{4}}t{}^{1}\partial _{\frac{3}{4}}s} \\ &=\frac{4\cdot 4}{3\cdot 1\cdot (1-t)(1-s)} \biggl[ \frac{(t+3)^{2}(3s+1)^{2}}{16\cdot 16}-\frac{(t+3)^{2}s^{2}}{16}- \frac{t^{2}(3s+1)^{2}}{16} + t^{2}s^{2} \biggr] \\ &=\frac{(5t+3)(7s+1)}{16}. \end{aligned}$$

Hence, the right-hand side of (3.3) is

$$\begin{aligned} & \frac{(1-0)(1-0)}{64 (1+\frac{1}{4} ) (1+\frac{3}{4} ) (1+\frac{1}{4}+\frac{1}{16} ) (1+\frac{3}{4}+\frac{9}{16} )} \\ &\quad \quad{}\times \biggl[ \biggl(\frac{1}{4} \biggr)^{3} \biggl( \frac{3}{4} \biggr)^{3} \bigl( \bigl\vert \Phi (0,0) \bigr\vert + \bigl\vert \Theta (0,1) \bigr\vert + \bigl\vert \Psi (1,0) \bigr\vert + \bigl\vert \Omega (1,1) \bigr\vert \bigr) \\ &\quad \quad{}+ \biggl(\frac{1}{4} \biggr)^{3} \biggl( \frac{2\cdot 3}{4}+ \frac{2\cdot 9}{16}+\frac{27}{64} \biggr) \bigl( \bigl\vert \Phi (0,1) \bigr\vert + \bigl\vert \Theta (0,0) \bigr\vert + \bigl\vert \Psi (1,1) \bigr\vert + \bigl\vert \Omega (1,0) \bigr\vert \bigr) \\ &\quad \quad{}+ \biggl(\frac{3}{4} \biggr)^{3} \biggl( \frac{2\cdot 1}{4}+ \frac{2\cdot 1}{16}+\frac{1}{64} \biggr) \bigl( \bigl\vert \Phi (1,0) \bigr\vert + \bigl\vert \Theta (1,1) \bigr\vert + \bigl\vert \Psi (0,0) \bigr\vert + \bigl\vert \Omega (0,1) \bigr\vert \bigr) \\ &\quad \quad{}+ \biggl(\frac{2\cdot 1}{4}+\frac{2\cdot 1}{16}+ \frac{1}{64} \biggr) \biggl(\frac{2\cdot 3}{4}+\frac{2\cdot 9}{16}+ \frac{27}{64} \biggr) \bigl( \bigl\vert \Phi (1,1) \bigr\vert + \bigl\vert \Theta (1,0) \bigr\vert \\ &\quad \quad{}+ \bigl\vert \Psi (0,1) \bigr\vert + \bigl\vert \Omega (0,0) \bigr\vert \bigr) \biggr]. \\ &\quad =\frac{4\cdot 4 \cdot 16 \cdot 16}{64\cdot 5\cdot 7\cdot 21\cdot 37} \biggl[\frac{1\cdot 27\cdot 4}{64\cdot 64}+\frac{1\cdot 195\cdot 4}{64\cdot 64}+ \frac{27\cdot 41\cdot 4}{64\cdot 64}+ \frac{41\cdot 195\cdot 4}{64\cdot 64} \biggr] \\ &\quad =\frac{3}{140}. \end{aligned}$$

It is clear that

$$ \frac{5}{592}\leq \frac{3}{140}, $$

which demonstrates the result described in Theorem 3.1.

Example 4.2

Let $f:[0,1]\times {}[ 0,1]\rightarrow \mathbb{R}$ be a function defined by $f(x,y)=x^{2}y^{2}$ and $p=r=2$. Then, f is q-partially differentiable. Moreover, $|\Phi (t,s)|^{2}$, $|\Theta (t,s)|^{2}$, $|\Psi (t,s)|^{2}$, and $|\Omega (t,s)|^{2}$ are q-integrable coordinated convex on $[0,1]\times [0,1]$.

By applying Theorem 3.2 with $q_{1}=\frac{1}{4}$ and $q_{2}=\frac{3}{4}$, the left-hand side of (3.10) is similar to the left-hand side of (3.3), which is $\frac{5}{592}$.

Since

$$\begin{aligned} \Phi (t,s)& =\frac{35ts}{16},\qquad \Theta (t,s)=\frac{-5t(7s+1)}{16}, \\ \Psi (t,s)& =\frac{-7(5t+3)s}{16},\quad \text{and}\quad \Omega (t,s)= \frac{(5t+3)(7s+1)}{16}, \end{aligned}$$

the right-hand side of (3.10) is

$$\begin{aligned} & \frac{(1-0)(1-0)}{16} \biggl[ \frac{1\cdot 9}{16\cdot 16 (1+\frac{1}{4}+\frac{1}{16} ) (1+\frac{3}{4}+\frac{9}{16} )} \biggr] ^{1/2} \biggl[ \frac{1}{4 (1+\frac{1}{4} ) (1+\frac{3}{4} )} \biggr] ^{1/2} \\ &\quad \quad{}\times \biggl[ \biggl( \biggl( 2+\frac{1}{4} \biggr) \biggl( 2+\frac{3}{4} \biggr) \bigl\vert \Phi (1,1) \bigr\vert ^{2} \biggr) ^{1/2} \\ &\quad \quad{}+ \biggl( \biggl( 2+\frac{1}{4} \biggr) \biggl( 2+ \frac{3}{4} \biggr) \bigl\vert \Theta (1,0) \bigr\vert ^{2}+\frac{3}{4} \biggl( 2+\frac{1}{4} \biggr) \bigl\vert \Theta (1,1) \bigr\vert ^{2} \biggr) ^{1/2} \\ &\quad \quad{}+ \biggl( \biggl( 2+\frac{1}{4} \biggr) \biggl( 2+ \frac{3}{4} \biggr) \bigl\vert \Psi (0,1) \bigr\vert ^{2}+\frac{1}{4} \biggl( 2+\frac{3}{4} \biggr) \bigl\vert \Psi (1,1) \bigr\vert ^{2} \biggr) ^{1/2} \\ &\quad \quad{}+ \biggl( \biggl( 2+\frac{1}{4} \biggr) \biggl( 2+ \frac{3}{4} \biggr) \bigl\vert \Omega (0,0) \bigr\vert ^{2}+ \frac{3}{4} \biggl( 2+\frac{1}{4} \biggr) \bigl\vert \Omega (0,1) \bigr\vert ^{2} \\ &\quad \quad{}+ \frac{1}{4} \biggl( 2+\frac{3}{4} \biggr) \bigl\vert \Omega (1,0) \bigr\vert ^{2}+\frac{1}{4}\cdot \frac{3}{4} \bigl\vert \Omega (1,1) \bigr\vert ^{2} \biggr) ^{1/2} \biggr] \\ &\quad =\frac{1}{16} \biggl[ \frac{3}{7\cdot 37} \biggr] ^{1/2} \biggl[ \frac{4}{5\cdot 7} \biggr] ^{1/2} \\ &\quad \quad{}\times \biggl[ \biggl( \frac{99}{4096} \biggr) ^{1/2}+ \biggl( \frac{43{,}475}{4096} \biggr) ^{1/2}+ \biggl( \frac{78{,}155}{4096} \biggr) ^{1/2}+ \biggl( \frac{38{,}435}{4096} \biggr) ^{1/2} \biggr] \\ &\quad =0.02466\ldots . \end{aligned}$$

It is clear that

$$ \frac{5}{592}\leq 0.02466\ldots , $$

which demonstrates the result described in Theorem 3.2.

Example 4.3

Let $f:[0,1]\times [0,1]\to \mathbb{R}$ be a function defined by $f(x,y)=x^{2}y^{2}$ and $r=2$. Then, f is q-partially differentiable. Moreover, $\vert \Phi (t,s) \vert ^{2}$, $\vert \Theta (t,s) \vert ^{2}$, $\vert \Psi (t,s) \vert ^{2}$, and $\vert \Omega (t,s) \vert ^{2}$ are q-integrable coordinated convex on $[0,1]\times [0,1]$.

By applying Theorem 3.3 with $q_{1}=\frac{1}{4}$ and $q_{2}=\frac{3}{4}$, the left-hand side of (3.16) is similar to the left-hand side of (3.3), which is $\frac{5}{592}$.

Since

$$\begin{aligned} \Phi (t,s)&=\frac{35ts}{16}, \qquad \Theta (t,s)=\frac{-5t(7s+1)}{16}, \\ \Psi (t,s)&=\frac{-7(5t+3)s}{16}, \quad \text{and}\quad \Omega (t,s)= \frac{(5t+3)(7s+1)}{16}, \end{aligned}$$

the right-hand side of (3.16) is

$$\begin{aligned} &\frac{(1-0)(1-0)}{16} \biggl[ \frac{1\cdot 3}{4\cdot 4\cdot (1+\frac{1}{4} ) (1+\frac{3}{4} )} \biggr]^{1/2} \\ &\quad \quad{}\times \biggl[ \frac{1}{4 (1+\frac{1}{4} ) (1+\frac{3}{4} ) (1+\frac{1}{4}+\frac{1}{16} ) (1+\frac{3}{4}+\frac{9}{16} )} \biggr]^{1/2} \\ &\quad \quad{}\times \biggl\lbrace \biggl[ \biggl(\frac{2\cdot 1}{4}+ \frac{2\cdot 1}{16}+\frac{1}{64} \biggr) \biggl(\frac{2\cdot 3}{4}+ \frac{2\cdot 9}{16}+\frac{27}{64} \biggr) \bigl\vert \Phi (1,1) \bigr\vert ^{2} \biggr]^{1/2} \\ &\quad \quad{}+ \biggl[ \biggl(\frac{2\cdot 1}{4}+\frac{2\cdot 1}{16}+ \frac{1}{64} \biggr) \biggl(\frac{2\cdot 3}{4}+\frac{2\cdot 9}{16}+ \frac{27}{64} \biggr) \bigl\vert \Theta (1,0) \bigr\vert ^{2} \\ &\quad \quad{}+ \biggl(\frac{3}{4} \biggr)^{3} \biggl(\frac{2\cdot 1}{4}+\frac{2\cdot 1}{16}+ \frac{1}{64} \biggr) \bigl\vert \Theta (1,1) \bigr\vert ^{2} \biggr]^{1/2} \\ &\quad \quad{}+ \biggl( \biggl(\frac{2\cdot 1}{4}+\frac{2\cdot 1}{16}+ \frac{1}{64} \biggr) \biggl(\frac{2\cdot 3}{4}+\frac{2\cdot 9}{16}+ \frac{27}{64} \biggr) \bigl\vert \Psi (0,1) \bigr\vert ^{2} \\ &\quad \quad{}+ \biggl(\frac{1}{4} \biggr)^{3} \biggl(\frac{2\cdot 3}{4}+\frac{2\cdot 9}{16}+ \frac{27}{64} \biggr) \bigl\vert \Psi (1,1) \bigr\vert ^{2} \biggr)^{1/2} \\ &\quad \quad{}+ \biggl( \biggl(\frac{2\cdot 1}{4}+\frac{2\cdot 1}{16}+ \frac{1}{64} \biggr) \biggl(\frac{2\cdot 3}{4}+\frac{2\cdot 9}{16}+ \frac{27}{64} \biggr) \bigl\vert \Omega (0,0) \bigr\vert ^{2} \\ &\quad \quad{}+ \biggl(\frac{3}{4} \biggr)^{3} \biggl( \frac{2\cdot 1}{4}+\frac{2\cdot 1}{16}+\frac{1}{64} \biggr) \bigl\vert \Omega (0,1) \bigr\vert ^{2} \\ &\quad \quad{}+ \biggl(\frac{1}{4} \biggr)^{3} \biggl(\frac{2\cdot 3}{4}+\frac{2\cdot 9}{16}+ \frac{27}{64} \biggr) \bigl\vert \Omega (1,0) \bigr\vert ^{2}+ \biggl( \frac{1}{4} \biggr)^{3} \biggl(\frac{3}{4} \biggr)^{3} \bigl\vert \Omega (1,1) \bigr\vert ^{2} \biggr)^{1/2} \biggr\rbrace \\ &\quad =\frac{1}{16} \biggl[\frac{1\cdot 3}{5\cdot 7} \biggr]^{1/2} \biggl[ \frac{4\cdot 16\cdot 16}{5\cdot 7\cdot 21\cdot 37} \biggr]^{1/2} \\ &\quad \quad{}\times \biggl[ \biggl(\frac{9{,}793{,}875}{1{,}048{,}576} \biggr)^{1/2}+ \biggl(\frac{1{,}971{,}075}{1{,}048{,}576} \biggr)^{1/2}+ \biggl( \frac{4{,}137{,}315}{1{,}048{,}576} \biggr)^{1/2}+ \biggl(\frac{832{,}659}{1{,}048{,}576} \biggr)^{1/2} \biggr] \\ &\quad = 0.02593\ldots . \end{aligned}$$

It is clear that

$$ \frac{5}{592}\leq 0.02593\ldots , $$

which demonstrates the result described in Theorem 3.3.

5 Conclusion

We established several new inequalities for q-differentiable coordinated convex functions that are related to the right side of Hermite–Hadamard inequalities for coordinated convex functions. We also showed that the inequalities proved in this paper generalize the results given in earlier works. Moreover, we gave some examples in order to demonstrate our main results.