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Quantum variant of Montgomery identity and Ostrowski-type inequalities for the mappings of two variables

Abstract

In this investigation, we demonstrate the quantum version of Montgomery identity for the functions of two variables. Then we use the result to derive some new Ostrowski-type inequalities for the functions of two variables via quantum integrals. We also consider the particular cases of the key results and offer some new integral inequalities.

Introduction

In the field of q-analysis, many studies have recently been carried out, starting with Euler owing to a vast requirement for mathematics that models quantum computing q-calculus occurred for the relationship between physics and mathematics. In different areas of mathematics, it has numerous applications such as combinatorics, number theory, basic hypergeometric functions, orthogonal polynomials, mechanics, the theory of relativity, and quantum theory [1, 2]. Apparently, Euler invented this important mathematics branch. He used the q parameter in Newton’s work on infinite series. Later, in a methodical manner, the q-calculus without limit calculus was firstly given by Jackson [3]. In 1908–1909 the general form of the q-integral and q-difference operator was defined by Jackson [4]. In 1969, for the first time, Agarwal [5] defined the q-fractional derivative. In 1966–1967, Al-Salam [6] introduced a q-analog of the q-fractional integral and q-Riemann–Liouville fractional. In 2004, Rajkovic gave a definition of the Riemann-type q-integral, which was generalized to Jackson q-integral. In 2013, Tariboon [7] introduced the \({}_{a}D_{q}\)-difference operator. Recently, in 2020, Bermudo et al. [8] introduced the notions of the \({}^{b}D_{q}\)-derivative and integral.

Many well-known integral inequalities, such as the Hölder, Hermite–Hadamard, Simpson, Newton, Ostrowski, Cauchy–Bunyakovsky–Schwarz, Gruss, Gruss–Chebyshev, and other integral inequalities, have been studied in the setup of q-calculus using the concept of classical convexity. For more results in this direction, we refer to [920].

In 1938, Ostrowski [21] established the following interesting integral inequality.

Theorem 1

Let \(F:[a,b]\rightarrow \mathbb{R} \) be a differentiable function on \((a,b)\) with bounded derivative, that is, \(\Vert F' \Vert _{\infty }:=\sup_{x \in (a,b)} \vert F'(x ) \vert <\infty \). Then we have the following integral inequality:

$$ \biggl\vert F(\tau )-\frac{1}{b-a} \int _{a}^{b}F(\tau )\,d\tau \biggr\vert \leq \biggl[ \frac{1}{4}+ \frac{(\tau -\frac{a+b}{2})}{(b-a)^{2}} \biggr] (b-a) \bigl\Vert F' \bigr\Vert _{\infty } $$
(1.1)

for all \(\tau \in {}[ a,b]\). The constant \(\frac{1}{4}\) is the best possible.

Inequality (1.1) can be rewritten in the equivalent form

$$ \biggl\vert F(\tau )-\frac{1}{b-a} \int _{a}^{b}F(\tau )\,d\tau \biggr\vert \leq \biggl[ \frac{(\tau -a)^{2}+(b-\tau )^{2}}{2(b-a)} \biggr] \bigl\Vert F' \bigr\Vert _{\infty }. $$
(1.2)

Since 1938, when Ostrowski proved his famous inequality (see [21]), this inequality has been studied by many mathematicians in various fields, such as numerical analysis and probability.

Various generalizations and extensions of the Ostrowski integral inequality for bounded-variation, monotonic, Lipschitzian, convex, absolutely continuous, and n times differentiable mappings with error estimates for some special means and some numerical quadrature rules were considered by many scientists. For more recent results, we refer to [2232] and the references therein.

A formal definition of coordinated convex (concave) functions may be expressed as follows.

Definition 1

A function \(F:\Delta \rightarrow \mathbb{R} \) is said to be coordinated convex on Δ if it satisfies the following inequality for all \((x ,y),(z,w)\in \Delta \) and \(\lambda ,\mu \in {}[ 0,1]\):

$$\begin{aligned}& F \bigl(\lambda x +(1-\lambda )z,\mu y+(1-\mu )w \bigr) \\& \quad \leq \lambda \mu F(x ,y)+\lambda (1-\mu )F(x ,w)+\mu (1-\lambda )F(z,y)+(1- \lambda ) (1-\mu )F(z,w). \end{aligned}$$
(1.3)

The mapping F is coordinated concave on Δ if inequality (1.3) holds in the reversed direction for all \((x ,y),(z,w)\in \Delta \) and \(\lambda ,\mu \in {}[ 0,1]\).

Latif et al. [33] established the following Ostrowski-type inequalities for coordinated convex functions.

Theorem 2

Let \(F:\Delta :=[a,b]\times {}[ c,d]\rightarrow \mathbb{R} \) be a twice differentiable mapping on \(\Delta ^{\circ }\) with \(a< b\), \(c< d\), \(a,c\geq 0\) such that \(\frac{\partial ^{2}F}{\partial s\,\partial t}\in L(\Delta )\). If \(\vert \frac{\partial ^{2}F}{\partial s\,\partial t} \vert \) is coordinated convex on Δ and \(\vert \frac{\partial ^{2}F}{\partial s\,\partial t} \vert \leq M\), \((x,y)\in \Delta \), then we have the following inequality:

$$\begin{aligned}& \biggl\vert F(x,y)+\frac{1}{(b-a)(d-c)} \int _{a}^{b} \int _{c}^{d}F(t,s)\,dt \,ds-A_{1} \biggr\vert \\& \quad \leq M \biggl[ \frac{(x-a)^{2}+(b-x)^{2}}{2(b-a)} \biggr] \biggl[ \frac{(y-c)^{2}+(d-y)^{2}}{2(d-c)} \biggr], \end{aligned}$$
(1.4)

where

$$ A_{1}=\frac{1}{d-c} \int _{c}^{d}F(x,s)\,ds+\frac{1}{b-a} \int _{a}^{b}F(t,y)\,dt. $$

Theorem 3

Let \(F:\Delta :=[a,b]\times {}[ c,d]\rightarrow \mathbb{R} \) be a twice differentiable mapping on \(\Delta ^{\circ }\) with \(a< b\), \(c< d\), \(a,c\geq 0\) such that \(\frac{\partial ^{2}F}{\partial s\,\partial t}\in L(\Delta )\). If \(\vert \frac{\partial ^{2}F}{\partial s\,\partial t} \vert ^{p}\) is coordinated convex on Δ, \(p>1\), \(\frac{1}{p}+\frac{1}{r}=1\), and \(\vert \frac{\partial ^{2}F}{\partial s\,\partial t}(x,y) \vert \leq M\), \((x,y)\in \Delta \), then we have the following inequality:

$$\begin{aligned}& \biggl\vert F(x,y)+\frac{1}{(b-a)(d-c)} \int _{a}^{b} \int _{c}^{d}F(t,s)\,dt \,ds-A_{1} \biggr\vert \\& \quad \leq \frac{M}{(1+r)^{\frac{2}{r}}} \biggl[ \frac{(x-a)^{2}+(b-x)^{2}}{2(b-a)} \biggr] \biggl[ \frac{(y-c)^{2}+(d-y)^{2}}{2(d-c)} \biggr] , \end{aligned}$$
(1.5)

where \(A_{1}\) is defined in Theorem 2.

Theorem 4

Let \(F:\Delta :=[a,b]\times {}[ c,d]\rightarrow \mathbb{R} \) be a twice differentiable mapping on \(\Delta ^{\circ }\) with \(a< b\), \(c< d\), \(a,c\geq 0\) such that \(\frac{\partial ^{2}F}{\partial s\,\partial t}\in L(\Delta )\). If \(\vert \frac{\partial ^{2}F}{\partial s\,\partial t} \vert ^{p}\) is coordinated convex on Δ, \(p> 1\), and \(\vert \frac{\partial ^{2}F}{\partial s\,\partial t}(x,y) \vert \leq M\), \((x,y)\in \Delta \), then we have the following inequality:

$$\begin{aligned}& \biggl\vert F(x,y)+\frac{1}{(b-a)(d-c)} \int _{a}^{b} \int _{c}^{d}F(t,s)\,dt \,ds-A_{1} \biggr\vert \\& \quad \leq \frac{M}{4} \biggl[ \frac{(x-a)^{2}+(b-x)^{2}}{2(b-a)} \biggr] \biggl[ \frac{(y-c)^{2}+(d-y)^{2}}{2(d-c)} \biggr] , \end{aligned}$$
(1.6)

where \(A_{1}\) is defined in Theorem 2.

Inspired by this ongoing study, we establish some new quantum Ostrowski inequalities for \(q_{1}q_{2}\)-differentiable coordinated convex functions. This is the primary motivation of this paper. The ideas and strategies of the paper may open new venues for further research in this field.

Preliminaries of q-calculus and some inequalities

In this section, we review the basic notions and findings needed to prove our crucial results. Moreover, we use the following notation (see [34]):

$$ [ n ] _{q}=\frac{1-q^{n}}{1-q}=1+q+q^{2}+ \cdots+q^{n-1}, \quad q\in ( 0,1 ) . $$

Jackson [4] has defined the q-Jackson integral from 0 to b for \(0< q<1\) as follows:

$$ \int _{0}^{b}F ( x ) \,d_{q}x= ( 1-q ) b \sum_{n=0}^{\infty }q^{n}F \bigl( bq^{n} \bigr), $$
(2.1)

provided that the series converges absolutely.

Moreover, he defined the q-Jackson integral in a general interval \([a,b]\) as

$$ \int _{a}^{b}F ( x ) \,d_{q}x= \int _{0}^{b}F ( x ) \,d_{q}x- \int _{0}^{a}F ( x ) \,d_{q}x. $$

Definition 2

([35])

For a continuous function \(F: [ a,b ] \rightarrow \mathbb{R} \), the \(q_{a}\)-derivative of F at \(x\in [ a,b ] \) is defined by the expression

$$ {} _{a}D_{q}F ( x ) = \frac{F ( x ) -F ( qx+ ( 1-q ) a ) }{ ( 1-q ) ( x-a ) }, \quad x\neq a. $$
(2.2)

The function F is said to be \(q_{a}\)-differentiable on \([ a,b ] \) if \({} _{a}D_{q}F ( x ) \) exists for all \(x\in [ a,b ] \). If \(a=0\) in (2.2), then \({} _{0}D_{q}F ( x ) =D_{q}F ( x ) \), where \(D_{q}F ( x ) \) is the familiar q-derivative of F at \(x\in [ a,b ] \) defined by the expression (see [34])

$$ D_{q}F ( x ) = \frac{F ( x ) -F ( qx ) }{ ( 1-q ) x}, \quad x\neq 0. $$

Definition 3

([8])

For a continuous function \(F: [ a,b ] \rightarrow \mathbb{R} \), the \(q^{b}\)-derivative of F at \(x\in [ a,b ] \) is characterized by the expression

$$ {}^{b}D_{q}F ( x ) = \frac{F ( qx+ ( 1-q ) b ) -F ( x ) }{ ( 1-q ) ( b-x ) }, \quad x\neq b. $$

The function F is said to be \(q^{b}\)-differentiable on \([ a,b ] \) if \({} ^{b}D_{q}F ( x ) \) exists for all \(x\in [ a,b ] \). If \(b=0\) in (2.2), then \({} ^{0}D_{q}F ( x ) =D_{q}F ( x )\), where \(D_{q}F ( x ) \) is the familiar q-derivative of F at \(x\in [ a,b ] \) defined by the expression (see [34])

$$ D_{q}F ( x ) = \frac{F ( x ) -F ( qx ) }{ ( 1-q ) x}, \quad x\neq 0. $$

Definition 4

([35])

Let \(F: [ a,b ] \rightarrow \mathbb{R} \) be a continuous function. Then the \(q_{a}\)-definite integral on \([ a,b ] \) is defined as

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

On the other hand, Bermudo et al. [8] gave the following new definition of the quantum integral.

Definition 5

Let \(F: [ a,b ] \rightarrow \mathbb{R} \) be a continuous function. Then the \(q^{b}\)-definite integral on \([ a,b ] \) is defined as

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

For more detail about \(q^{b}\)-integrals and corresponding inequalities, we refer to [8].

We have to give the following notation, which will be used many times in the next sections (see [34]):

$$ [ n ] _{q}=\frac{q^{n}-1}{q-1}. $$

Lemma 1

([36])

We have the equality

$$ \int _{a}^{b} ( x -a ) ^{\alpha } \,{}_{a}d_{q}x= \frac{ ( b-a ) ^{\alpha +1}}{ [ \alpha +1 ] _{q}} $$

for \(\alpha \in \mathbb{R} \backslash \{ -1 \} \).

Latif et al. [37] defined the \(q_{ac}\)-integral and partial q-derivatives for two-variable functions as follows.

Definition 6

Suppose that \(F: [ a,b ] \times [ c,d ] \subset \mathbb{R} ^{2}\rightarrow \mathbb{R} \) is a continuous function. Then the definite \(q_{ac}\)-integral on \([ a,b ] \times [ c,d ] \) is defined as

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

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

Lemma 2

If the assumptions of Definition 6hold, then

$$\begin{aligned} \int _{y_{1}}^{y} \int _{x _{1}}^{x }F ( t,s )\, {}_{a}d_{q_{1}}t \,{}_{c}d_{q_{2}}s =& \int _{y_{1}}^{y} \int _{a}^{x }F ( t,s ) \,{}_{a}d_{q_{1}}t \,{}_{c}d_{q_{2}}s- \int _{y_{1}}^{y} \int _{a}^{x _{1}}F ( t,s ) \,{}_{a}d_{q_{1}}t \,{}_{c}d_{q_{2}}s \\ =& \int _{c}^{y} \int _{a}^{x }F ( t,s ) \,{}_{a}d_{q_{1}}t \,{}_{c}d_{q_{2}}s- \int _{c}^{y_{1}} \int _{a}^{x }F ( t,s ) \,{}_{a}d_{q_{1}}t \,{}_{c}d_{q_{2}}s \\ &{}- \int _{c}^{y} \int _{a}^{x _{1}}F ( t,s ) \,{}_{a}d_{q_{1}}t\,{}_{c} d_{q_{2}}s+ \int _{c}^{y_{1}} \int _{a}^{x _{1}}F ( t,s ) \,{}_{a}d_{q_{1}}t \,{}_{c}d_{q_{2}}s. \end{aligned}$$

Definition 7

([37])

Let \(F: [ a,b ] \times [ c,d ] \subseteq \mathbb{R} ^{2}\rightarrow \mathbb{R} \) be a continuous function of two variables. Then the partial \(q_{1}\)-derivatives, \(q_{2}\)-derivatives, and \(q_{1}q_{2}\)-derivatives at \(( x,y ) \in [ a,b ] \times [ c,d ] \) can be given as follows:

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

For more detail on the related to q-integrals and derivatives for the functions of two variables, we refer to [37].

On the other hand, Budak et al. [38] gave the following definitions of \(q_{a}^{d}\)-, \(q_{b}^{c}\)-, and \(q^{bd}\)-integrals.

Definition 8

Suppose that \(F: [ a,b ] \times [ c,d ] \subset \mathbb{R} ^{2}\rightarrow \mathbb{R} \) is continuous function. Then the following \(q_{a}^{d}\)-, \(q_{c}^{b}\)-, and \(q^{bd}\)-integrals on \([ a,b ] \times [ c,d ] \) are defined by

$$\begin{aligned}& \begin{aligned}[t] \int _{a}^{x } \int _{y}^{d}F ( t,s ) \,{}^{d}d_{q_{2}}s{}_{a} \,d_{q_{1}}t&= ( 1-q_{1} ) ( 1-q_{2} ) ( x -a ) ( d-y ) \\ &\quad {}\times \sum_{n=0}^{\infty }\sum _{m=0}^{\infty }q_{1}^{n}q_{2}^{m}F \bigl( q_{1}^{n}x + \bigl( 1-q_{1}^{n} \bigr) a,q_{2}^{m}y+ \bigl( 1-q_{2}^{m} \bigr) d \bigr), \end{aligned} \end{aligned}$$
(2.3)
$$\begin{aligned}& \begin{aligned}[t] \int _{x }^{b} \int _{c}^{y}F ( t,s ) \,{}_{c}d_{q_{2}}s \,{}^{b}d_{q_{1}}t&= ( 1-q_{1} ) ( 1-q_{2} ) ( b-x ) ( y-c ) \\ &\quad {}\times \sum_{n=0}^{\infty }\sum _{m=0}^{\infty }q_{1}^{n}q_{2}^{m}F \bigl( q_{1}^{n}x + \bigl( 1-q_{1}^{n} \bigr) b,q_{2}^{m}y+ \bigl( 1-q_{2}^{m} \bigr) c \bigr), \end{aligned} \end{aligned}$$
(2.4)

and

$$\begin{aligned} \int _{x }^{b} \int _{y}^{d}F ( t,s ) \,{}^{d}d_{q_{2}}s\,{}^{b} d_{q_{1}}t =& ( 1-q_{1} ) ( 1-q_{2} ) ( b-x ) ( d-y ) \\ &{}\times \sum_{n=0}^{\infty }\sum _{m=0}^{\infty }q_{1}^{n}q_{2}^{m}F \bigl( q_{1}^{n}x + \bigl( 1-q_{1}^{n} \bigr) b,q_{2}^{m}y+ \bigl( 1-q_{2}^{m} \bigr) d \bigr), \end{aligned}$$
(2.5)

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

Definition 9

([39])

Let \(F: [ a,b ] \times [ c,d ] \subseteq \mathbb{R} ^{2}\rightarrow \mathbb{R} \) be a continuous function of two variables. Then the partial \(q_{1}\)-derivatives, \(q_{2}\)-derivatives, and \(q_{1}q_{2}\)-derivatives at \(( x,y ) \in [ a,b ] \times [ c,d ] \) can be given as follows:

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

Quantum Montgomery identity for the functions of two variables

In this section, we prove a quantum Montgomery identity via newly defined quantum integrals for functions of two variables.

Lemma 3

Let \(F:\Delta \subseteq \mathbb{R} ^{2}\rightarrow \mathbb{R} \) be a twice \(q_{1}q_{2}\)-differentiable function on \(\Delta ^{\circ }\). If the partial \(q_{1}q_{2}\)-derivatives \(\frac{{}^{b, d}\partial _{q_{1},q_{2}}^{2}F ( t,s ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s}\) are continuous and integrable on \([ a,b ] \times [ c,d ] \subseteq \Delta ^{\circ }\), then we have the following identity for \(q_{1}q_{2}\)-integrals:

$$\begin{aligned}& \frac{1}{ ( b-a ) ( d-c ) } \int _{a}^{b} \int _{c}^{d}F ( t,s ) \,{}^{b}d_{q_{1}}t \,{}^{d}d_{q_{2}}s-\frac{1}{b-a} \int _{a}^{b}F ( t,y ) \,{}^{b}d_{q_{1}}t \\& \qquad {}-\frac{1}{d-c} \int _{c}^{d}F ( x,s ) \,{}^{d}d_{q_{2}}s+F ( x,y ) \\& \quad = ( b-a ) ( d-c ) \int _{0}^{1} \int _{0}^{1} \Psi _{q_{1}} ( t ) \Psi _{q_{2}} ( s ) \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( ta+ ( 1-t ) b,sc+ ( 1-s ) d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s}\,d_{q_{1}}t\,d_{q_{2}}s, \end{aligned}$$
(3.1)

where

$$ \Psi _{q_{1}}(t)= \textstyle\begin{cases} q_{1}t, & t\in [ 0,\frac{b-x}{b-a} ), \\ q_{1}t-1, & t\in [ \frac{b-x}{b-a},1 ], \end{cases} $$

and

$$ \Psi _{q_{2}}(s)= \textstyle\begin{cases} q_{2}s, & s\in [ 0,\frac{d-y}{d-c} ), \\ q_{2}s-1, & s\in [ \frac{d-y}{d-c},1 ], \end{cases} $$

for \(q_{1},q_{2}\in ( 0,1 ) \).

Proof

By Lemma 2 and the definitions of \(\Psi _{q_{1}} ( t ) \) and \(\Psi _{q_{2}} ( s ) \) we obtain

$$\begin{aligned}& \int _{0}^{1} \int _{0}^{1}\Psi _{q_{1}} ( t ) \Psi _{q_{2}} ( s ) \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( ta+ ( 1-t ) b,sc+ ( 1-s ) d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s}\,d_{q_{1}}t\,d_{q_{2}}s \\& \quad = \int _{0}^{\frac{b-x}{b-a}} \int _{0}^{\frac{d-y}{d-c}} \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( ta+ ( 1-t ) b,sc+ ( 1-s ) d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s}\,d_{q_{1}}t \,d_{q_{2}}s \\& \qquad {}+ \int _{0}^{\frac{b-x}{b-a}} \int _{0}^{1} ( q_{2}s-1 ) \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( ta+ ( 1-t ) b,sc+ ( 1-s ) d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s}\,d_{q_{2}}s \\& \qquad {}+ \int _{0}^{1} \int _{0}^{\frac{d-y}{d-c}} ( q_{1}t-1 ) \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( ta+ ( 1-t ) b,sc+ ( 1-s ) d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s}\,d_{q_{1}}t \\& \qquad {}+ \int _{0}^{1} \int _{0}^{1} ( q_{2}s-1 ) ( q_{1}t-s ) \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( ta+ ( 1-t ) b,sc+ ( 1-s ) d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s}\,d_{q_{1}}t\,d_{q_{2}}s \\& \quad =I_{1}+I_{2}+I_{3}+I_{4}. \end{aligned}$$
(3.2)

From Definition 9 we have

$$\begin{aligned}& \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( ta+ ( 1-t ) b,sc+ ( 1-s ) d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \\& \quad =\frac{1}{ ( 1-q_{1} ) ( 1-q_{2} ) ( b-x ) ( d-y ) ts} \bigl[ F \bigl( tq_{1}a+ ( 1-tq_{1} ) b,sq_{2}c+ ( 1-sq_{2} ) d \bigr) \\& \qquad {}-F \bigl( tq_{1}a+ ( 1-tq_{1} ) b,sc+ ( 1-s ) d \bigr) -F \bigl( ta+ ( 1-t ) b,sq_{2}c+ ( 1-sq_{2} ) d \bigr) \\& \qquad {} +F \bigl( ta+ ( 1-t ) b,sc+ ( 1-s ) d \bigr) \bigr]. \end{aligned}$$
(3.3)

To conclude the proof, we need to calculate the integrals in the right side of (3.2). By the definition of \(q_{1}q_{2}\)-integrals we obtain that

$$\begin{aligned} & \int _{0}^{\frac{b-x}{b-a}} \int _{0}^{\frac{d-y}{d-c}} \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( ta+ ( 1-t ) b,sc+ ( 1-s ) d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s}\,d_{q_{1}}t \,d_{q_{2}}s \\ &\quad =\frac{1}{ ( b-a ) ( d-c ) } \\ &\qquad {} \times \Biggl[ \sum_{n=0}^{\infty } \sum_{m=0}^{\infty }F\biggl( q_{1}^{n+1} \biggl( \frac{b-x}{b-a} \biggr) a+ \biggl( 1-q_{1}^{n+1} \biggl( \frac{b-x}{b-a} \biggr) \biggr) b, \\ &\qquad q_{2}^{m+1} \biggl( \frac{d-y}{d-c} \biggr) c+ \biggl( 1-q_{2}^{m+1} \biggl( \frac{d-y}{d-c} \biggr) \biggr) d\biggr) \\ &\qquad {} -\sum_{n=0}^{\infty }\sum _{m=0}^{\infty }F\biggl( q_{1}^{n+1} \biggl( \frac{b-x}{b-a} \biggr) a+ \biggl( 1-q_{1}^{n+1} \biggl( \frac{b-x}{b-a} \biggr) \biggr) b, \\ &\qquad q_{2}^{m} \biggl( \frac{d-y}{d-c} \biggr) c+ \biggl( 1-q_{2}^{m} \biggl( \frac{d-y}{d-c} \biggr) \biggr) d\biggr) \\ &\qquad {} -\sum_{n=0}^{\infty }\sum _{m=0}^{\infty }F\biggl( q_{1}^{n} \biggl( \frac{b-x}{b-a} \biggr) a+ \biggl( 1-q_{1}^{n} \biggl( \frac{b-x}{b-a} \biggr) \biggr) b, \\ &\qquad q_{2}^{m+1} \biggl( \frac{d-y}{d-c} \biggr) c+ \biggl( 1-q_{2}^{m+1} \biggl( \frac{d-y}{d-c} \biggr) \biggr) d\biggr) \\ &\qquad {} +\sum_{n=0}^{\infty }\sum _{m=0}^{\infty }F\biggl( q_{1}^{n} \biggl( \frac{b-x}{b-a} \biggr) a+ \biggl( 1-q_{1}^{n} \biggl( \frac{b-x}{b-a} \biggr) \biggr) b, \\ &\qquad q_{2}^{m} \biggl( \frac{d-y}{d-c} \biggr) c+ \biggl( 1-q_{2}^{m} \biggl( \frac{d-y}{d-c} \biggr) \biggr) d\biggr) \Biggr] \\ &\quad =\frac{1}{ ( b-a ) ( d-c ) } \\ &\qquad {} \times \Biggl[ \sum_{n=0}^{\infty } \Biggl\{ \sum_{m=0}^{\infty }F \biggl( q_{1}^{n+1} \biggl( \frac{b-x}{b-a} \biggr) a+ \biggl( 1-q_{1}^{n+1} \biggl( \frac{b-x}{b-a} \biggr) \biggr) b, \\ &\qquad q_{2}^{m+1} \biggl( \frac{d-y}{d-c} \biggr) c+ \biggl( 1-q_{2}^{m+1} \biggl( \frac{d-y}{d-c} \biggr) \biggr) d\biggr) \\ &\qquad {} - \sum_{m=0}^{\infty }F \biggl( q_{1}^{n} \biggl( \frac{b-x}{b-a} \biggr) a+ \biggl( 1-q_{1}^{n} \biggl( \frac{b-x}{b-a} \biggr) \biggr) b, \\ &\qquad q_{2}^{m+1} \biggl( \frac{d-y}{d-c} \biggr) c+ \biggl( 1-q_{2}^{m+1} \biggl( \frac{d-y}{d-c} \biggr) \biggr) d\biggr)\Biggr\} \\ &\qquad {} +\sum_{n=0}^{\infty }\Biggl\{ \sum_{m=0}^{\infty }F\biggl( q_{1}^{n} \biggl( \frac{b-x}{b-a} \biggr) a+ \biggl( 1-q_{1}^{n} \biggl( \frac{b-x}{b-a} \biggr) \biggr) b, \\ &\qquad q_{2}^{m} \biggl( \frac{d-y}{d-c} \biggr) c+ \biggl( 1-q_{2}^{m} \biggl( \frac{d-y}{d-c} \biggr) \biggr) d\biggr) \\ &\qquad {} -\sum_{m=0}^{\infty }F \biggl( q_{1}^{n+1} \biggl( \frac{b-x}{b-a} \biggr) a+ \biggl( 1-q_{1}^{n+1} \biggl( \frac{b-x}{b-a} \biggr) \biggr) b, \\ &\qquad q_{2}^{m} \biggl( \frac{d-y}{d-c} \biggr) c+ \biggl( 1-q_{2}^{m} \biggl( \frac{d-y}{d-c} \biggr) \biggr) d\biggr) \Biggr\} \Biggr] \\ &\quad =\frac{1}{ ( b-a ) ( d-c ) } \Biggl[ \sum_{n=0}^{\infty }F \biggl( q_{1}^{n+1} \biggl( \frac{b-x}{b-a} \biggr) a+ \biggl( 1-q_{1}^{n+1} \biggl( \frac{b-x}{b-a} \biggr) \biggr) b,d \biggr) \\ &\qquad {} -\sum_{n=0}^{\infty }F \biggl( q_{1}^{n} \biggl( \frac{b-x}{b-a} \biggr) a+ \biggl( 1-q_{1}^{n} \biggl( \frac{b-x}{b-a} \biggr) \biggr) b,d \biggr) \\ &\qquad {} +\sum_{n=0}^{\infty }F \biggl( q_{1}^{n} \biggl( \frac{b-x}{b-a} \biggr) a+ \biggl( 1-q_{1}^{n} \biggl( \frac{b-x}{b-a} \biggr) \biggr) b,y \biggr) \\ &\qquad {} -\sum_{n=0}^{\infty }F \biggl( q_{1}^{n+1} \biggl( \frac{b-x}{b-a} \biggr) a+ \biggl( 1-q_{1}^{n+1} \biggl( \frac{b-x}{b-a} \biggr) \biggr) b,y \biggr) \Biggr] \\ &\quad =\frac{1}{ ( b-a ) ( d-c ) } \bigl[ F ( b,d ) -F ( x,d ) -F ( b,y ) +F ( x,y ) \bigr] . \end{aligned}$$
(3.4)

Similarly, we have

$$\begin{aligned}& \int _{0}^{\frac{b-x}{b-a}} \int _{0}^{1} \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( ta+ ( 1-t ) b,sc+ ( 1-s ) d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s}\,d_{q_{1}}t \,d_{q_{2}}s \end{aligned}$$
(3.5)
$$\begin{aligned}& \quad =\frac{1}{ ( b-a ) ( d-c ) } \bigl[ F ( b,d ) -F ( x,d ) -F ( b,c ) +F ( x,c ) \bigr] , \\& \int _{0}^{1} \int _{0}^{\frac{d-y}{d-c}} \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( ta+ ( 1-t ) b,sc+ ( 1-s ) d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s}\,d_{q_{1}}t \,d_{q_{2}}s \end{aligned}$$
(3.6)
$$\begin{aligned}& \quad =\frac{1}{ ( b-a ) ( d-c ) } \bigl[ F ( b,d ) -F ( b,y ) -F ( a,d ) +F ( a,y ) \bigr] , \\& \int _{0}^{1} \int _{0}^{1} \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( ta+ ( 1-t ) b,sc+ ( 1-s ) d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s}\,d_{q_{1}}t \,d_{q_{2}}s \\& \quad =\frac{1}{ ( b-a ) ( d-c ) } \bigl[ F ( b,d ) -F ( a,d ) -F ( b,c ) +F ( a,c ) \bigr] . \end{aligned}$$
(3.7)

Additionally, we have

$$\begin{aligned}& \int _{0}^{\frac{b-x}{b-a}} \int _{0}^{1}s \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( ta+ ( 1-t ) b,sc+ ( 1-s ) d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \,d_{q_{1}}t\,d_{q_{2}}s \\& \quad =\frac{1}{ ( b-a ) ( d-c ) } \\& \qquad {}\times \Biggl[ \sum_{n=0}^{\infty }\sum _{m=0}^{\infty }q_{2}^{m}F \biggl( q_{1}^{n+1} \biggl( \frac{b-x}{b-a} \biggr) a+ \biggl( 1-q_{1}^{n+1} \biggl( \frac{b-x}{b-a} \biggr) \biggr) b,q_{2}^{m+1}c+ \bigl( 1-q_{2}^{m+1} \bigr) d \biggr) \\& \qquad {}-\sum_{n=0}^{\infty }\sum _{m=0}^{\infty }q_{2}^{m}F \biggl( q_{1}^{n+1} \biggl( \frac{b-x}{b-a} \biggr) a+ \biggl( 1-q_{1}^{n+1} \biggl( \frac{b-x}{b-a} \biggr) \biggr) b,q_{2}^{m}c+ \bigl( 1-q_{2}^{m} \bigr) d \biggr) \\& \qquad {}-\sum_{n=0}^{\infty }\sum _{m=0}^{\infty }q_{2}^{m}F \biggl( q_{1}^{n} \biggl( \frac{b-x}{b-a} \biggr) a+ \biggl( 1-q_{1}^{n} \biggl( \frac{b-x}{b-a} \biggr) \biggr) b,q_{2}^{m+1}c+ \bigl( 1-q_{2}^{m+1} \bigr) d \biggr) \\& \qquad {} +\sum_{n=0}^{\infty }\sum _{m=0}^{\infty }q_{2}^{m}F \biggl( q_{1}^{n} \biggl( \frac{b-x}{b-a} \biggr) a+ \biggl( 1-q_{1}^{n} \biggl( \frac{b-x}{b-a} \biggr) \biggr) b,q_{2}^{m}c+ \bigl( 1-q_{2}^{m+1} \bigr) d \biggr) \Biggr] \\& \quad =\frac{1}{ ( b-a ) ( d-c ) } \\& \qquad {}\times \Biggl[ \sum_{m=0}^{\infty }q_{2}^{m}F \bigl( b,q_{2}^{m+1}c+ \bigl( 1-q_{2}^{m+1} \bigr) d \bigr) -\sum_{m=0}^{\infty }q_{2}^{m}F \bigl( b,q_{2}^{m}c+ \bigl( 1-q_{2}^{m} \bigr) d \bigr) \\& \qquad {}+ \sum_{m=0}^{\infty }q_{2}^{m}F \bigl( x,q_{2}^{m}c+ \bigl( 1-q_{2}^{m} \bigr) d \bigr) -\sum_{m=0}^{\infty }q_{2}^{m}F \bigl( x,q_{2}^{m+1}c+ \bigl( 1-q_{2}^{m+1} \bigr) d \bigr) \Biggr] \\& \quad =\frac{1}{ ( b-a ) ( d-c ) } \\& \qquad {}\times \Biggl[ \frac{1-q_{2}}{q_{2}}\sum_{m=0}^{\infty }q_{2}^{m}F \bigl( b,q_{2}^{m}c+ \bigl( 1-q_{2}^{m} \bigr) d \bigr) - \frac{1}{q_{2}}F ( b,c ) \\& \qquad {} -\frac{1-q_{2}}{q_{2}}\sum_{m=0}^{\infty }q_{2}^{m}F \bigl( x,q_{2}^{m}c+ \bigl( 1-q_{2}^{m} \bigr) d \bigr) +\frac{1}{q_{2}}F ( x,c ) \Biggr] \\& \quad =\frac{1}{ ( b-a ) ( d-c ) } \biggl[ \frac{1}{q_{2} ( d-c ) } \int _{c}^{d}F ( b,s ) \,{}^{d} d_{q_{2}}s- \frac{1}{q_{2} ( d-c ) } \int _{c}^{d}F ( x,s ) \,{}^{d} d_{q_{2}}s \\& \qquad {} -\frac{1}{q_{2}}F ( b,c ) +\frac{1}{q_{2}}F ( x,c ) \biggr] . \end{aligned}$$
(3.8)

By similar operations we have

$$\begin{aligned}& \int _{0}^{1} \int _{0}^{\frac{d-y}{d-c}}t \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( ta+ ( 1-t ) b,sc+ ( 1-s ) d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \,d_{q_{1}}t\,d_{q_{2}}s \end{aligned}$$
(3.9)
$$\begin{aligned}& \quad =\frac{1}{ ( b-a ) ( d-c ) } \biggl[ \frac{1}{q_{1} ( b-a ) } \int _{a}^{b}F ( t,d ) \,{}^{b} d_{q_{1}}t- \frac{1}{q_{1} ( b-a ) } \int _{a}^{b}F ( t,y ) \,{}^{b} d_{q_{1}}t \\& \qquad {} -\frac{1}{q_{1}}F ( a,d ) +\frac{1}{q_{2}}F ( a,y ) \biggr] , \\& \int _{0}^{1} \int _{0}^{1}s \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( ta+ ( 1-t ) b,sc+ ( 1-s ) d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \,d_{q_{1}}t \,d_{q_{2}}s \end{aligned}$$
(3.10)
$$\begin{aligned}& \quad =\frac{1}{ ( b-a ) ( d-c ) } \biggl\{ \frac{1}{q_{2} ( d-c ) } \int _{c}^{d}F ( b,s ) \,{}^{d}d_{q_{2}}s- \frac{1}{q_{2} ( d-c ) } \int _{c}^{d}F ( a,s ) \,{}^{d}d_{q_{2}}s \\& \qquad {} -\frac{1}{q_{2}}F ( b,c ) +\frac{1}{q_{2}}F ( a,c ) \biggr\} , \\& \int _{0}^{1} \int _{0}^{1}t \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( ta+ ( 1-t ) b,sc+ ( 1-s ) d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \,d_{q_{1}}t \,d_{q_{2}}s \\& \quad =\frac{1}{ ( b-a ) ( d-c ) } \biggl\{ \frac{1}{q_{1} ( b-a ) } \int _{a}^{b}F ( t,d ) \,{}^{b}d_{q_{1}}t- \frac{1}{q_{1} ( b-a ) } \int _{a}^{b}F ( t,c ) \,{}^{b}d_{q_{1}}t \\& \qquad {} -\frac{1}{q_{1}}F ( a,d ) +\frac{1}{q_{1}}F ( a,c ) \biggr\} , \end{aligned}$$
(3.11)

and

$$\begin{aligned}& \int _{0}^{1} \int _{0}^{1}ts \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( ta+ ( 1-t ) b,sc+ ( 1-s ) d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \,d_{q_{1}}t \,d_{q_{2}}s \\& \quad =\frac{1}{ ( b-a ) ( d-c ) } \Biggl\{ \sum_{m=0}^{\infty } \sum_{n=0}^{\infty }q_{1}^{n}q_{2}^{m}F \bigl( q_{1}^{n+1}a+ \bigl( 1-q_{1}^{n+1} \bigr) b,q_{2}^{m+1}c+ \bigl( 1-q_{2}^{m+1} \bigr) d \bigr) \\& \qquad {}-\sum_{m=0}^{\infty }\sum _{n=0}^{\infty }q_{1}^{n}q_{2}^{m}F \bigl( q_{1}^{n+1}a+ \bigl( 1-q_{1}^{n+1} \bigr) b,q_{2}^{m}c+ \bigl( 1-q_{2}^{m} \bigr) d \bigr) \\& \qquad {}-\sum_{m=0}^{\infty }\sum _{n=0}^{\infty }q_{1}^{n}q_{2}^{m}F \bigl( q_{1}^{n}a+ \bigl( 1-q_{1}^{n} \bigr) b,q_{2}^{m+1}c+ \bigl( 1-q_{2}^{m+1} \bigr) d \bigr) \\& \qquad {} +\sum_{m=0}^{\infty }\sum _{n=0}^{\infty }q_{1}^{n}q_{2}^{m}F \bigl( q_{1}^{n}a+ \bigl( 1-q_{1}^{n} \bigr) b,q_{2}^{m}c+ \bigl( 1-q_{2}^{m} \bigr) d \bigr) \Biggr\} \\& \quad =\frac{1}{ ( b-a ) ( d-c ) } \Biggl\{ \frac{1}{q_{1}q_{2}} \Biggl[ \sum _{m=0}^{\infty }\sum_{n=0}^{\infty }q_{1}^{n}q_{2}^{m}F \bigl( q_{1}^{n}a+ \bigl( 1-q_{1}^{n} \bigr) b,q_{2}^{m}c+ \bigl( 1-q_{2}^{m} \bigr) d \bigr) \\& \qquad {} -\sum_{m=0}^{\infty }q_{2}^{m}F \bigl( a,q_{2}^{m}c+ \bigl( 1-q_{2}^{m} \bigr) d \bigr) -\sum_{n=0}^{\infty }q_{1}^{n}F \bigl( q_{1}^{n}a+ \bigl( 1-q_{1}^{n} \bigr) b,c \bigr) +F ( a,c ) \Biggr] \\& \qquad {}-\frac{1}{q_{1}} \Biggl[ \sum_{m=0}^{\infty } \sum_{n=0}^{\infty }q_{1}^{n}q_{2}^{m}F \bigl( q_{1}^{n}a+ \bigl( 1-q_{1}^{n} \bigr) b,q_{2}^{m}c+ \bigl( 1-q_{2}^{m} \bigr) d \bigr) \\& \qquad {} -\sum_{m=0}^{\infty }q_{2}^{m}F \bigl( a,q_{2}^{m}c+ \bigl( 1-q_{2}^{m} \bigr) d \bigr) \Biggr] \\& \qquad {}-\frac{1}{q_{2}} \Biggl[ \sum_{m=0}^{\infty } \sum_{n=0}^{\infty }q_{1}^{n}q_{2}^{m}F \bigl( q_{1}^{n}a+ \bigl( 1-q_{1}^{n} \bigr) b,q_{2}^{m}c+ \bigl( 1-q_{2}^{m} \bigr) d \bigr) \\& \qquad {} -\sum_{n=0}^{\infty }q_{1}^{n}F \bigl( q_{1}^{n}a+ \bigl( 1-q_{1}^{n} \bigr) b,c \bigr) \Biggr] \\& \qquad {}+\sum_{m=0}^{\infty } \sum_{n=0}^{\infty }q_{1}^{n}q_{2}^{m}F \bigl( q_{1}^{n}a+ \bigl( 1-q_{1}^{n} \bigr) b,q_{2}^{m}c+ \bigl( 1-q_{2}^{m} \bigr) d \bigr) \Biggr\} \\& \quad =\frac{1}{ ( b-a ) ( d-c ) } \Biggl\{ \frac{ ( 1-q_{1} ) ( 1-q_{2} ) }{q_{1}q_{2}}\sum _{m=0}^{\infty }\sum_{n=0}^{\infty }q_{1}^{n}q_{2}^{m}F \bigl( q_{1}^{n}a+ \bigl( 1-q_{1}^{n} \bigr) b,q_{2}^{m}c+ \bigl( 1-q_{2}^{m} \bigr) d \bigr) \\& \qquad {}-\frac{1-q_{2}}{q_{1}q_{2}}\sum_{m=0}^{\infty }q_{2}^{m}F \bigl( a,q_{2}^{m}c+ \bigl( 1-q_{2}^{m} \bigr) d \bigr) \\& \qquad {} -\frac{1-q_{1}}{q_{1}q_{2}}\sum_{n=0}^{\infty }q_{1}^{n}F \bigl( q_{1}^{n}a+ \bigl( 1-q_{1}^{n} \bigr) b,c \bigr) + \frac{1}{q_{1}q_{2}}F ( a,c ) \Biggr\} \\& \quad =\frac{1}{ ( b-a ) ( d-c ) } \biggl\{ \frac{1}{q_{1}q_{2} ( b-a ) ( d-c ) } \int _{a}^{b} \int _{c}^{d}F ( t,s ) \,{}^{b}d_{q_{1}}t \,{}^{d}d_{q_{2}}s \\& \qquad {} -\frac{1}{q_{1}q_{2} ( b-a ) } \int _{a}^{b}F ( t,c )\, {}^{b}d_{q_{1}}t- \frac{1}{q_{1}q_{2} ( d-c ) } \int _{c}^{d}F ( a,s ) \,{}^{d}d_{q_{2}}s+ \frac{1}{q_{1}q_{2}}F ( a,c ) \biggr\} . \end{aligned}$$
(3.12)

Now from (3.4)–(3.12) we obtain the following relations:

$$\begin{aligned}& I_{1} =\frac{1}{ ( b-a ) ( d-c ) } \bigl[ F ( b,d ) -F ( x,d ) -F ( b,y ) +F ( x,y ) \bigr] , \\& \begin{aligned} I_{2} &=\frac{1}{ ( b-a ) ( d-c ) } \\ &\quad {}\times \biggl[ \frac{1}{d-c} \int _{c}^{d}F ( b,s ) \,{}^{d}d_{q_{2}}s- \frac{1}{d-c} \int _{c}^{d}F ( x,s ) \,{}^{d}d_{q_{2}}s-F ( b,d ) +F ( x,d ) \biggr] , \end{aligned} \\& \begin{aligned} I_{3} &=\frac{1}{ ( b-a ) ( d-c ) } \\ &\quad {}\times \biggl[ \frac{1}{b-a} \int _{a}^{b}F ( t,d ) \,{}^{b}d_{q_{1}}t- \frac{1}{b-a} \int _{a}^{b}F ( t,y ) \,{}^{b}d_{q_{1}}t-F ( b,d ) +F ( b,y ) \biggr] , \end{aligned} \\& \begin{aligned} I_{4} &=\frac{1}{ ( b-a ) ( d-c ) } \biggl[ \frac{1}{ ( b-a ) ( d-c ) } \int _{a}^{b} \int _{c}^{d}F ( t,s ) \,{}^{b}d_{q_{1}}t \,{}^{d}d_{q_{2}}s-\frac{1}{b-a} \int _{a}^{b}F ( t,d ) \,{}^{b}d_{q_{1}}t \\ &\quad {} -\frac{1}{d-c} \int _{c}^{d}F ( b,s ) \,{}^{d}d_{q_{2}}s+F ( b,d ) \biggr], \end{aligned} \end{aligned}$$

which finishes the proof. □

Some new quantum Ostrowski-type integral inequalities

In this section, we prove some new quantum Ostrowski-type inequalities for \(q_{1}q_{2}\)-differentiable coordinated convex functions using the lemma proved in the last section.

Theorem 5

Suppose that the assumptions of Lemma 3hold. If \(\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( t,s ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \vert ^{p_{1}}\), \(p_{1} > 1\), is coordinated convex on \([ a,b ] \times [ c,d ] \), then we have the inequality

$$\begin{aligned}& \biggl\vert \frac{1}{ ( b-a ) ( d-c ) } \int _{a}^{b} \int _{c}^{d}F ( t,s ) \,{}^{b}d_{q_{1}}t \,{}^{d}d_{q_{2}}s-\frac{1}{b-a} \int _{a}^{b}F ( t,y ) \,{}^{b}d_{q_{1}}t \\& \qquad {} -\frac{1}{d-c} \int _{c}^{d}F ( x ,s ) \,{}^{d}d_{q_{2}}s+F ( x ,y ) \biggr\vert \\& \quad \le ( b-a ) ( d-c ) \biggl[ A_{1}^{1- \frac{1}{p_{1}}} ( a,b,q_{1},x ) A_{1}^{1-\frac{1}{p_{1}}} ( c,d,q_{2},y ) \\& \qquad {}\times \biggl\{ A_{2} ( a,b,q_{1},x ) \biggl( A_{2} ( c,d,q_{2},y ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \\& \qquad {} +A_{3} ( c,d,q_{2},y ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \biggr) \\& \qquad {}+A_{3} ( a,b,q_{1},x ) \biggl( A_{2} ( c,d,q_{2},y ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \\& \qquad {} +A_{3} ( c,d,q_{2},y ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \biggr) \biggr\} ^{\frac{1}{p_{1}}} \\& \qquad {}+A_{1}^{1-\frac{1}{p_{1}}} ( a,b,q_{1},x ) A_{4}^{1- \frac{1}{p_{1}}} ( c,d,q_{2},y ) \\& \qquad {}\times \biggl\{ A_{2} ( a,b,q_{1},x ) \biggl( A_{5} ( c,d,q_{2},y ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \\& \qquad {} +A_{6} ( c,d,q_{2},y ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \biggr) \\& \qquad {}+A_{3} ( a,b,q_{1},x ) \biggl( A_{5} ( c,d,q_{2},y ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \\& \qquad {} +A_{6} ( c,d,q_{2},y ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \biggr) \biggr\} ^{\frac{1}{p_{1}}} \\& \qquad {}+A_{4}^{1-\frac{1}{p_{1}}} ( a,b,q_{1},x ) A_{1}^{1- \frac{1}{p_{1}}} ( c,d,q_{2},y ) \\& \qquad {}\times \biggl\{ A_{5} ( a,b,q_{1},x ) \biggl( A_{2} ( c,d,q_{2},y ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \\& \qquad {} +A_{3} ( c,d,q_{2},y ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \biggr) \\& \qquad {}+A_{6} ( a,b,q_{1},x ) \biggl( A_{2} ( c,d,q_{2},y ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \\& \qquad {} +A_{3} ( c,d,q_{2},y ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \biggr) \biggr\} ^{\frac{1}{p_{1}}} \\& \qquad {}+A_{4}^{1-\frac{1}{p_{1}}} ( a,b,q_{1},x ) A_{4}^{1- \frac{1}{p_{1}}} ( c,d,q_{2},y ) \\& \qquad {}\times \biggl\{ A_{5} ( a,b,q_{1},x ) \biggl( A_{5} ( c,d,q_{2},y ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \\& \qquad {} +A_{6} ( c,d,q_{2},y ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \biggr) \\& \qquad {}+A_{6} ( a,b,q_{1},x ) \biggl( A_{5} ( c,d,q_{2},y ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \\& \qquad {} +A_{6} ( c,d,q_{2},y ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \biggr) \biggr\} ^{\frac{1}{p_{1}}} \biggr], \end{aligned}$$
(4.1)

where

$$\begin{aligned}& A_{1} ( u,v,q,z ) = \int _{0}^{\frac{v-z}{v-u}}qt\,d_{q}t= \frac{q}{1+q} \biggl( \frac{v-z}{v-u} \biggr) ^{2}, \\& A_{2} ( u,v,q,z ) = \int _{0}^{\frac{v-z}{v-u}}qt^{2}\,d_{q}t= \frac{q}{1+q+q^{2}} \biggl( \frac{v-z}{v-u} \biggr) ^{3}, \\& A_{3} ( u,v,q,z ) = \int _{0}^{\frac{v-z}{v-u}}qt\,d_{q}t- \int _{0}^{\frac{v-z}{v-u}}qt^{2}\,d_{q}t=A_{1} ( u,v,q,z ) -A_{2} ( u,v,q,z ) , \\& \begin{aligned} A_{4} ( u,v,q,z ) &= \int _{\frac{v-z}{v-u}}^{1} ( 1-qt ) \,d_{q}t= \int _{0}^{1} ( 1-qt ) \,d_{q}t- \int _{0}^{\frac{v-z}{v-u}} ( 1-qt ) \,d_{q}t \\ &=\frac{1-q}{1+q} \biggl( \frac{z-u}{v-u} \biggr) +\frac{q}{1+q} \biggl( \frac{z-u}{v-u} \biggr) ^{2}, \end{aligned} \\& \begin{aligned} A_{5} ( u,v,q,z ) &= \int _{\frac{v-z}{v-u}}^{1} \bigl( t-qt^{2} \bigr) \,d_{q}t= \int _{0}^{1} \bigl( t-qt^{2} \bigr) \,d_{q}t- \int _{0}^{\frac{v-z}{v-u}} \bigl( t-qt^{2} \bigr) \,d_{q}t \\ &=\frac{1}{ ( 1+q ) ( 1+q+q^{2} ) }- \frac{1}{1+q} \biggl( \frac{v-z}{v-u} \biggr) ^{2}+\frac{q}{1+q+q^{2}} \biggl( \frac{v-z}{v-u} \biggr) ^{3}, \end{aligned} \\& \begin{aligned} A_{6} ( u,v,q,z ) &= \int _{\frac{v-z}{v-u}}^{1} ( 1-t ) ( 1-qt ) \,d_{q}t \\ &= \int _{0}^{1} ( 1-t ) ( 1-qt ) \,d_{q}t- \int _{0}^{\frac{v-z}{v-u}} ( 1-t ) ( 1-qt ) \,d_{q}t \\ &= \int _{0}^{1} ( 1-qt ) \,d_{q}t- \int _{0}^{1} \bigl( t-qt^{2} \bigr) \,d_{q}t\\ &\quad {}- \int _{0}^{\frac{v-z}{v-u}} ( 1-qt ) \,d_{q}t+ \int _{0}^{\frac{v-z}{v-u}} \bigl( t-qt^{2} \bigr) \,d_{q}t \\ &=A_{4} ( u,v,q,z ) -A_{5} ( u,v,q,z ) , \end{aligned} \end{aligned}$$

and \(q_{1},q_{2}\in ( 0,1 ) \).

Proof

By taking the modulus in Lemma 3 we have

$$\begin{aligned}& \biggl\vert \frac{1}{ ( b-a ) ( d-c ) } \int _{a}^{b} \int _{c}^{d}F ( t,s ) \,{}^{b}d_{q_{1}}t \,{}^{d}d_{q_{2}}s-\frac{1}{b-a} \int _{a}^{b}F ( t,y ) \,{}^{b}d_{q_{1}}t \\& \qquad {} -\frac{1}{d-c} \int _{c}^{d}F ( x,s ) \,{}^{d}d_{q_{2}}s+F ( x,y ) \biggr\vert \\& \quad \leq ( b-a ) ( d-c ) \\& \qquad {}\times \int _{0}^{1} \int _{0}^{1} \bigl\vert \Psi _{q_{1}} ( t ) \Psi _{q_{2}} ( s ) \bigr\vert \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( ta+ ( 1-t ) b,sc+ ( 1-s ) d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert \,d_{q_{1}}t\,d_{q_{2}}s \\& \quad = ( b-a ) ( d-c ) \\& \qquad {}\times \int _{0}^{\frac{b-x}{b-a}} \int _{0}^{\frac{d-y}{d-c}}q_{1}q_{2}ts \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( ta+ ( 1-t ) b,sc+ ( 1-s ) d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert \,d_{q_{1}}t \,d_{q_{2}}s \\& \qquad {}+ ( b-a ) ( d-c ) \\& \qquad {}\times \int _{0}^{\frac{b-x}{b-a}} \int _{\frac{d-y}{d-c}}^{1}q_{1}t ( 1-q_{2}s ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( ta+ ( 1-t ) b,sc+ ( 1-s ) d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert \,d_{q_{1}}t \,d_{q_{2}}s \\& \qquad {}+ ( b-a ) ( d-c ) \\& \qquad {}\times \int _{\frac{b-x}{b-a}}^{1} \int _{0}^{\frac{d-y}{d-c}} ( 1-q_{1}t ) q_{2}s \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( ta+ ( 1-t ) b,sc+ ( 1-s ) d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert \,d_{q_{1}}t\,d_{q_{2}}s \\& \qquad {}+ ( b-a ) ( d-c ) \\& \qquad {}\times \int _{\frac{b-x}{b-a}}^{1} \int _{\frac{d-y}{d-c}}^{1} ( 1-q_{1}t ) ( 1-q_{2}s ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( ta+ ( 1-t ) b,sc+ ( 1-s ) d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert \,d_{q_{1}}t\,d_{q_{2}}s . \end{aligned}$$
(4.2)

Applying the power mean inequality for quantum integrals, we obtain

$$\begin{aligned}& \biggl\vert \frac{1}{ ( b-a ) ( d-c ) } \int _{a}^{b} \int _{c}^{d}F ( t,s ) \,{}^{b}d_{q_{1}}t \,{}^{d}d_{q_{2}}s-\frac{1}{b-a} \int _{a}^{b}F ( t,y ) \,{}^{b}d_{q_{1}}t \\& \qquad {} -\frac{1}{d-c} \int _{c}^{d}F ( x,s ) \,{}^{d}d_{q_{2}}s+F ( x,y ) \biggr\vert \\& \quad \leq ( b-a ) ( d-c ) \biggl( \int _{0}^{\frac{b-x}{b-a}} \int _{0}^{\frac{d-y}{d-c}}q_{1}q_{2}ts \,d_{q_{1}}t\,d_{q_{2}}s \biggr) ^{1-\frac{1}{p_{1}}} \\& \qquad {}\times \biggl( \int _{0}^{\frac{b-x}{b-a}} \int _{0}^{\frac{d-y}{d-c}}q_{1}q_{2}ts \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( ta+ ( 1-t ) b,sc+ ( 1-s ) d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}}\,d_{q_{1}}t \,d_{q_{2}}s \biggr) ^{\frac{1}{p_{1}}} \\& \qquad {}+ ( b-a ) ( d-c ) \biggl( \int _{0}^{\frac{b-x}{b-a}} \int _{\frac{d-y}{d-c}}^{1}q_{1}t ( 1-q_{2}s ) \,d_{q_{1}}t\,d_{q_{2}}s \biggr) ^{1-\frac{1}{p_{1}}} \\& \qquad {}\times \biggl( \int _{0}^{\frac{b-x}{b-a}} \int _{\frac{d-y}{d-c}}^{1}q_{1}t ( 1-q_{2}s ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( ta+ ( 1-t ) b,sc+ ( 1-s ) d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \,d_{q_{1}}t\,d_{q_{2}}s \biggr) ^{\frac{1}{p_{1}}} \\& \qquad {}+ ( b-a ) ( d-c ) \biggl( \int _{\frac{b-x}{b-a}}^{1} \int _{0}^{\frac{d-y}{d-c}} ( 1-q_{1}t ) q_{2}s\,d_{q_{1}}t\,d_{q_{2}}s \biggr) ^{1-\frac{1}{p_{1}}} \\& \qquad {}\times \biggl( \int _{\frac{b-x}{b-a}}^{1} \int _{0}^{\frac{d-y}{d-c}} ( 1-q_{1}t ) q_{2}s \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( ta+ ( 1-t ) b,sc+ ( 1-s ) d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \,d_{q_{1}}t\,d_{q_{2}}s \biggr) ^{\frac{1}{p_{1}}} \\& \qquad {}+ ( b-a ) ( d-c ) \biggl( \int _{\frac{b-x}{b-a}}^{1} \int _{\frac{d-y}{d-c}}^{1} ( 1-q_{1}t ) ( 1-q_{2}s ) \,d_{q_{1}}t\,d_{q_{2}}s \biggr) ^{1-\frac{1}{p_{1}}} \\& \qquad {}\times \biggl( \int _{\frac{b-x}{b-a}}^{1} \int _{\frac{d-y}{d-c}}^{1} ( 1-q_{1}t ) ( 1-q_{2}s ) \\& \qquad {}\times \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( ta+ ( 1-t ) b,sc+ ( 1-s ) d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}}\,d_{q_{1}}t\,d_{q_{2}}s \biggr) ^{\frac{1}{p_{1}}} \end{aligned}$$
(4.3)

Now using the convexity of \(\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( t,s ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \vert ^{p_{1}}\), we obtain

$$\begin{aligned}& \biggl[ \int _{0}^{\frac{b-x}{b-a}} \int _{0}^{\frac{d-y}{d-c}}q_{1}q_{2}ts \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( ta+ ( 1-t ) b,sc+ ( 1-s ) d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}}\,d_{q_{1}}t \,d_{q_{2}}s \biggr] ^{\frac{1}{p_{1}}} \\& \quad \leq \biggl[ \int _{0}^{\frac{d-y}{d-c}}q_{2}s \biggl\{ \int _{0}^{\frac{b-x}{b-a}}q_{1}t \biggl( t \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,sc+ ( 1-s ) d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \\& \qquad {} + ( 1-t ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,sc+ ( 1-s ) d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \biggr) \,d_{q_{1}}t \biggr\} \,d_{q_{2}}s \biggr] ^{\frac{1}{p_{1}}} \\& \quad = \biggl[ \int _{0}^{\frac{d-y}{d-c}}q_{2}s \biggl\{ A_{2} ( a,b,q_{1},x ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,sc+ ( 1-s ) d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \\& \qquad {} +A_{3} ( a,b,q_{1},x ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,sc+ ( 1-s ) d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \biggr\} \,d_{q_{2}}s \biggr] ^{\frac{1}{p_{1}}} \\& \quad \leq \biggl[ A_{2} ( a,b,q_{1},x ) \int _{0}^{\frac{d-y}{d-c}}q_{2}s \biggl\{ s \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}}+ ( 1-s ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \biggr\} \\& \qquad {} +A_{3} ( a,b,q_{1},x ) \int _{0}^{\frac{d-y}{d-c}}q_{2}s \biggl\{ s \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}}+ ( 1-s ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \biggr\} \biggr] ^{\frac{1}{p_{1}}} \\& \quad = \biggl[ A_{2} ( a,b,q_{1},x ) \biggl\{ A_{2} ( c,d,q_{2},y ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}}+A_{3} ( c,d,q_{2},y ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \biggr\} \\& \qquad {} +A_{3} ( a,b,q_{1},x ) \biggl\{ A_{2} ( c,d,q_{2},y ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \\& \qquad {}+A_{3} ( c,d,q_{2},y ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \biggr\} \biggr] ^{\frac{1}{p_{1}}}. \end{aligned}$$
(4.4)

By using similar operations we find

$$\begin{aligned}& \biggl[ \int _{0}^{\frac{b-x}{b-a}} \int _{\frac{d-y}{d-c}}^{1}q_{1}t ( 1-q_{2}s ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( ta+ ( 1-t ) b,sc+ ( 1-s ) d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \,d_{q_{1}}t\,d_{q_{2}}s \biggr] ^{\frac{1}{p_{1}}} \end{aligned}$$
(4.5)
$$\begin{aligned}& \quad \leq \biggl[ A_{2} ( a,b,q_{1},x ) \biggl\{ A_{5} ( c,d,q_{2},y ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \\& \qquad {}+A_{6} ( c,d,q_{2},y ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \biggr\} \\& \qquad {} +A_{3} ( a,b,q_{1},x ) \biggl\{ A_{5} ( c,d,q_{2},y ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \\& \qquad {}+A_{6} ( c,d,q_{2},y ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \biggr\} \biggr] ^{\frac{1}{p_{1}}}, \\& \biggl[ \int _{\frac{b-x}{b-a}}^{1} \int _{0}^{\frac{d-y}{d-c}} ( 1-q_{1}t ) q_{2}s \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( ta+ ( 1-t ) b,sc+ ( 1-s ) d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \,d_{q_{1}}t\,d_{q_{2}}s \biggr] ^{\frac{1}{p_{1}}} \\& \quad \leq \biggl[ A_{5} ( a,b,q_{1},x ) \biggl\{ A_{2} ( c,d,q_{2},y ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \\& \qquad {}+A_{3} ( c,d,q_{2},y ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \biggr\} \\& \qquad {} +A_{6} ( a,b,q_{1},x ) \biggl\{ A_{2} ( c,d,q_{2},y ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \\& \qquad {}+A_{3} ( c,d,q_{2},y ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \biggr\} \biggr] ^{\frac{1}{p_{1}}} , \end{aligned}$$
(4.6)

and

$$\begin{aligned}& \biggl[ \int _{\frac{b-x}{b-a}}^{1} \int _{\frac{d-y}{d-c}}^{1} ( 1-q_{1}t ) ( 1-q_{2}s ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( ta+ ( 1-t ) b,sc+ ( 1-s ) d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}}\,d_{q_{1}}t\,d_{q_{2}}s \biggr] ^{\frac{1}{p_{1}}} \\& \quad \leq \biggl[ A_{5} ( a,b,q_{1},x ) \biggl\{ A_{5} ( c,d,q_{2},y ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \\& \qquad {}+A_{6} ( c,d,q_{2},y ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \biggr\} \\& \qquad {} +A_{6} ( a,b,q_{1},x ) \biggl\{ A_{5} ( c,d,q_{2},y ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \\& \qquad {}+A_{6} ( c,d,q_{2},y ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \biggr\} \biggr] ^{\frac{1}{p_{1}}}. \end{aligned}$$
(4.7)

We also observe that

$$\begin{aligned}& \begin{aligned}[t] \biggl( \int _{0}^{\frac{b-x}{b-a}} \int _{0}^{\frac{d-y}{d-c}}q_{1}q_{2}ts \,d_{q_{1}}t\,d_{q_{2}}s \biggr) ^{1-\frac{1}{p_{1}}}& = \biggl( \biggl( \int _{0}^{\frac{b-x}{b-a}}q_{1}t\,d_{q_{1}}t \biggr) \biggl( \int _{0}^{\frac{d-y}{d-c}}q_{2}s \,d_{q_{2}}s \biggr) \biggr) ^{1- \frac{1}{p_{1}}} \\ &=A_{1}^{1-\frac{1}{p_{1}}} ( a,b,q_{1},x ) A_{1}^{1- \frac{1}{p_{1}}} ( c,d,q_{2},y ) , \end{aligned} \end{aligned}$$
(4.8)
$$\begin{aligned}& \biggl( \int _{0}^{\frac{b-x}{b-a}} \int _{\frac{d-y}{d-c}}^{1}q_{1}t ( 1-q_{2}s ) \,d_{q_{1}}t\,d_{q_{2}}s \biggr) ^{1- \frac{1}{p_{1}}}=A_{1}^{1-\frac{1}{p_{1}}} ( a,b,q_{1},x ) A_{4}^{1-\frac{1}{p_{1}}} ( c,d,q_{2},y ) , \end{aligned}$$
(4.9)
$$\begin{aligned}& \biggl( \int _{\frac{b-x}{b-a}}^{1} \int _{0}^{\frac{d-y}{d-c}} ( 1-q_{1}t ) q_{2}s\,d_{q_{1}}t\,d_{q_{2}}s \biggr) ^{1-\frac{1}{p_{1}}}=A_{4}^{1-\frac{1}{p_{1}}} ( a,b,q_{1},x ) A_{1}^{1- \frac{1}{p_{1}}} ( c,d,q_{2},y ) , \end{aligned}$$
(4.10)
$$\begin{aligned}& \biggl( \int _{\frac{b-x}{b-a}}^{1} \int _{\frac{d-y}{d-c}}^{1} ( 1-q_{1}t ) ( 1-q_{2}s ) \,d_{q_{1}}t\,d_{q_{2}}s \biggr) ^{1- \frac{1}{p_{1}}}=A_{4}^{1-\frac{1}{p_{1}}} ( a,b,q_{1},x ) A_{4}^{1- \frac{1}{p_{1}}} ( c,d,q_{2},y ) . \end{aligned}$$
(4.11)

By (4.3)–(4.10) we obtain the desired inequality, which finishes the proof. □

Theorem 6

Suppose that the assumptions of Lemma 3hold. If \(\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( t,s ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \vert ^{p_{1}}\) is coordinated convex on \([ a,b ] \times [ c,d ] \), then we have the inequality

$$\begin{aligned}& \biggl\vert \frac{1}{ ( b-a ) ( d-c ) } \int _{a}^{b} \int _{c}^{d}F ( t,s ) \,{}^{b}d_{q_{1}}t \,{}^{d}d_{q_{2}}s-\frac{1}{b-a} \int _{a}^{b}F ( t,y ) \,{}^{b}d_{q_{1}}t \\& \qquad {} -\frac{1}{d-c} \int _{c}^{d}F ( x ,s ) \,{}^{d}d_{q_{2}}s+F ( x ,y ) \biggr\vert \\& \quad \leq ( b-a ) ( d-c ) \\& \qquad {}\times \biggl[ \biggl( \biggl( \frac{b-x }{b-a} \biggr) ^{1+ \frac{1}{r_{1}}} \biggl( \frac{d-y}{d-c} \biggr) ^{1+\frac{1}{r_{1}}} \biggl( \frac{q_{1}}{ [ r_{1}+1 ] _{q_{1}}} \biggr) ^{\frac{1}{r_{1}}} \biggl( \frac{q_{2}}{ [ r_{1}+1 ] _{q_{2}}} \biggr) ^{\frac{1}{r_{1}}} \biggr) \\& \qquad {}\times \biggl\{ \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \frac{1}{ [ 2 ] _{q_{1}} [ 2 ] _{q_{2}}} \biggl( \frac{b-x }{b-a} \biggr) ^{2} \biggl( \frac{d-y}{d-c} \biggr) ^{2} \\& \qquad {}+ \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \frac{1}{ [ 2 ] _{q_{1}}} \biggl( \frac{b-x }{b-a} \biggr) ^{2} \biggl( \frac{d-y}{d-c}-\frac{1}{ [ 2 ] _{q_{2}}} \biggl( \frac{d-y}{d-c} \biggr) ^{2} \biggr) \\& \qquad {}+ \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \frac{1}{ [ 2 ] _{q_{2}}} \biggl( \frac{d-y}{d-c} \biggr) ^{2} \biggl( \frac{b-x}{b-a}-\frac{1}{ [ 2 ] _{q_{1}}} \biggl( \frac{b-x }{b-a} \biggr) ^{2} \biggr) \\& \qquad {}+ \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \biggl( \frac{b-x }{b-a}-\frac{1}{ [ 2 ] _{q_{1}}} \biggl( \frac{b-x }{b-a} \biggr) ^{2} \biggr) \\& \qquad {} \times \biggl( \frac{d-y}{d-c}- \frac{1}{ [ 2 ] _{q_{2}}} \biggl( \frac{d-y}{d-c} \biggr) ^{2} \biggr) \biggr\} ^{\frac{1}{p_{1}}} \\& \qquad {}+ \biggl( \biggl( \frac{d-y}{d-c} \biggr) ^{1+\frac{1}{r_{1}}} \biggl( \frac{q_{2}}{ [ r_{1}+1 ] _{q_{2}}} \biggr) ^{\frac{1}{r_{1}}} \biggl( \int _{\frac{b-x }{b-a}}^{1} ( 1-q_{1}t ) ^{r_{1}}\,d_{q_{1}}t \biggr) ^{\frac{1}{r_{1}}} \biggr) \\& \qquad {}\times \biggl\{ \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \frac{1}{ [ 2 ] _{q_{1}} [ 2 ] _{q_{2}}} \biggl( \frac{d-y}{d-c} \biggr) ^{2} \biggl( 1- \biggl( \frac{b-x}{b-a} \biggr) ^{2} \biggr) \\& \qquad {}+ \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \frac{1}{ [ 2 ] _{q_{1}}} \biggl( 1- \biggl( \frac{b-x }{b-a} \biggr) ^{2} \biggr) \biggl( \frac{d-y}{d-c}-\frac{1}{ [ 2 ] _{q_{2}}} \biggl( \frac{d-y}{d-c} \biggr) ^{2} \biggr) \\& \qquad {}+ \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \biggl( \frac{q_{1}}{ [ 2 ] _{q_{1}}}-\frac{b-x }{b-a}+ \frac{1}{ [ 2 ] _{q_{1}}} \biggl( \frac{b-x }{b-a} \biggr) ^{2} \biggr) \frac{1}{ [ 2 ] _{q_{2}}} \biggl( \frac{d-y}{d-c} \biggr) ^{2} \\& \qquad {}+ \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \biggl( \frac{q_{1}}{ [ 2 ] _{q_{1}}}-\frac{b-x }{b-a}+ \frac{1}{ [ 2 ] _{q_{1}}} \biggl( \frac{b-x }{b-a} \biggr) ^{2} \biggr) \\& \qquad {} \times \biggl( \frac{d-y}{d-c}- \frac{1}{ [ 2 ] _{q_{2}}} \biggl( \frac{d-y}{d-c} \biggr) ^{2} \biggr) \biggr\} ^{\frac{1}{p_{1}}} \\& \qquad {}+ \biggl( \biggl( \frac{b-x }{b-a} \biggr) ^{1+\frac{1}{r_{1}}} \biggl( \frac{q_{1}}{ [ r_{1}+1 ] _{q_{1}}} \biggr) ^{\frac{1}{r_{1}}} \biggl( \int _{\frac{d-y}{d-c}}^{1} ( 1-q_{2}s ) ^{r_{1}}\,d_{q_{2}}s \biggr) ^{\frac{1}{r_{1}}} \biggr) \\& \qquad {}\times \biggl\{ \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \frac{1}{ [ 2 ] _{q_{1}} [ 2 ] _{q_{2}}} \biggl( \frac{b-x }{b-a} \biggr) ^{2} \biggl( 1- \biggl( \frac{d-y}{d-c} \biggr) ^{2} \biggr) \\& \qquad {}+ \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \frac{1}{ [ 2 ] _{q_{1}}} \biggl( \frac{b-x }{b-a} \biggr) ^{2} \biggl( \frac{q_{2}}{ [ 2 ] _{q_{2}}}-\frac{d-y}{d-c}+ \frac{1}{ [ 2 ] _{q_{2}}} \biggl( \frac{d-y}{d-c} \biggr) ^{2} \biggr) \\& \qquad {}+ \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \frac{1}{ [ 2 ] _{q_{2}}} \biggl( 1- \biggl( \frac{d-y}{d-c} \biggr) ^{2} \biggr) \biggl( \frac{b-x }{b-a}- \frac{1}{ [ 2 ] _{q_{1}}} \biggl( \frac{b-x }{b-a} \biggr) ^{2} \biggr) \\& \qquad {}+ \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \biggl( \frac{b-x }{b-a}-\frac{1}{ [ 2 ] _{q_{1}}} \biggl( \frac{b-x }{b-a} \biggr) ^{2} \biggr) \\& \qquad {} \times \biggl( \frac{q_{2}}{ [ 2 ] _{q_{2}}}- \frac{d-y}{d-c}+ \frac{1}{ [ 2 ] _{q_{2}}} \biggl( \frac{d-y}{d-c} \biggr) ^{2} \biggr) \biggr\} ^{\frac{1}{p_{1}}} \\& \qquad {}+ \biggl( \biggl( \int _{\frac{b-x }{b-a}}^{1} ( 1-q_{1}t ) ^{r_{1}}\,d_{q_{1}}t \biggr) ^{\frac{1}{r_{1}}} \biggl( \int _{\frac{d-y}{d-c}}^{1} ( 1-q_{2}s ) ^{r_{1}}\,d_{q_{2}}s \biggr) ^{\frac{1}{r_{1}}} \biggr) \\& \qquad {}\times \biggl[ \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \frac{1}{ [ 2 ] _{q_{1}} [ 2 ] _{q_{2}}} \biggl( 1- \biggl( \frac{b-x }{b-a} \biggr) ^{2} \biggr) \biggl( 1- \biggl( \frac{d-y}{d-c} \biggr) ^{2} \biggr) \\& \qquad {}+ \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \frac{1}{ [ 2 ] _{q_{1}}} \biggl( 1- \biggl( \frac{b-x }{b-a} \biggr) ^{2} \biggr) \biggl( \frac{q_{2}}{ [ 2 ] _{q_{2}}}- \frac{d-y}{d-c}+\frac{1}{ [ 2 ] _{q_{2}}} \biggl( \frac{d-y}{d-c} \biggr) ^{2} \biggr) \\& \qquad {}+ \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \frac{1}{ [ 2 ] _{q_{2}}} \biggl( 1- \biggl( \frac{d-y}{d-c} \biggr) ^{2} \biggr) \biggl( \frac{q_{1}}{ [ 2 ] _{q_{1}}}- \frac{b-x }{b-a}+\frac{1}{ [ 2 ] _{q_{1}}} \biggl( \frac{b-x }{b-a} \biggr) ^{2} \biggr) \\& \qquad {}+ \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \biggl( \frac{q_{1}}{ [ 2 ] _{q_{1}}}-\frac{b-x }{b-a}+ \frac{1}{ [ 2 ] _{q_{1}}} \biggl( \frac{b-x }{b-a} \biggr) ^{2} \biggr) \\& \qquad {} \times \biggl( \frac{q_{2}}{ [ 2 ] _{q_{2}}}-\frac{d-y}{d-c}+ \frac{1}{ [ 2 ] _{q_{2}}} \biggl( \frac{d-y}{d-c} \biggr) ^{2} \biggr) \biggr] ^{\frac{1}{p_{1}}} \biggr], \end{aligned}$$
(4.12)

where \(q_{1},q_{2}\in ( 0,1 ) \) and \(\frac{1}{r_{1}}+\frac{1}{p_{1}}=1\), \(p_{1}>1\).

Proof

Applying the well-known Hölder inequality for \(q_{1}q_{2}\)-integrals to the integrals in the right side of (4.2), we find

$$\begin{aligned}& \biggl\vert \frac{1}{ ( b-a ) ( d-c ) } \int _{a}^{b} \int _{c}^{d}F ( t,s ) \,{}^{b}d_{q_{1}}t \,{}^{d}d_{q_{2}}s-\frac{1}{b-a} \int _{a}^{b}F ( t,y ) \,{}^{b}d_{q_{1}}x \\& \qquad {} -\frac{1}{d-c} \int _{c}^{d}F ( x,s ) \,{}^{d}d_{q_{2}}s+F ( x,y ) \biggr\vert \\& \quad \leq ( b-a ) ( d-c ) \biggl[ \biggl( \int _{0}^{\frac{b-x}{b-a}} \int _{0}^{\frac{d-y}{d-c}} ( q_{1}q_{2}ts ) ^{r_{1}} \,d_{q_{1}}t \,d_{q_{2}}s \biggr) ^{\frac{1}{r_{1}}} \\& \qquad {}\times \biggl( \int _{0}^{\frac{b-x}{b-a}} \int _{0}^{\frac{d-y}{d-c}} \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( ta+ ( 1-t ) b,sc+ ( 1-s ) d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}}\,d_{q_{1}}t\,d_{q_{2}}s \biggr) ^{\frac{1}{p_{1}}} \\& \qquad {}+ \biggl( \int _{0}^{\frac{b-x}{b-a}} \int _{\frac{d-y}{d-c}}^{1} \bigl( q_{1}t ( 1-q_{2}s ) \bigr) ^{r_{1}}\,d_{q_{1}}t \,d_{q_{2}}s \biggr) ^{\frac{1}{r_{1}}} \\& \qquad {}\times \biggl( \int _{0}^{\frac{b-x}{b-a}} \int _{\frac{d-y}{d-c}}^{1} \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( ta+ ( 1-t ) b,sc+ ( 1-s ) d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}}\,d_{q_{1}}t\,d_{q_{2}}s \biggr) ^{\frac{1}{p_{1}}} \\& \qquad {}+ \biggl( \int _{\frac{b-x}{b-a}}^{1} \int _{0}^{\frac{d-y}{d-c}} \bigl( ( 1-q_{1}t ) q_{2}s \bigr) ^{r_{1}}\,d_{q_{1}}t \,d_{q_{2}}s \biggr) ^{\frac{1}{r_{1}}} \\& \qquad {}\times \biggl( \int _{\frac{b-x}{b-a}}^{1} \int _{0}^{\frac{d-y}{d-c}} \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( ta+ ( 1-t ) b,sc+ ( 1-s ) d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}}\,d_{q_{1}}t\,d_{q_{2}}s \biggr) ^{\frac{1}{p_{1}}} \\& \qquad {}+ \biggl( \int _{\frac{b-x}{b-a}}^{1} \int _{\frac{d-y}{d-c}}^{1} \bigl( ( 1-q_{1}t ) ( 1-q_{2}s ) \bigr) ^{r_{1}}\,d_{q_{1}}t \,d_{q_{2}}s \biggr) ^{\frac{1}{r_{1}}} \\& \qquad {} \times \biggl( \int _{\frac{b-x}{b-a}}^{1} \int _{\frac{d-y}{d-c}}^{1} \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( ta+ ( 1-t ) b,sc+ ( 1-s ) d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}}\,d_{q_{1}}t\,d_{q_{2}}s \biggr) ^{\frac{1}{p_{1}}} \biggr] . \end{aligned}$$
(4.13)

Now applying the convexity of \(\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( t,s ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \vert ^{p_{1}}\), we obtain that

$$\begin{aligned}& \biggl( \int _{0}^{\frac{b-x}{b-a}} \int _{0}^{\frac{d-y}{d-c}} ( q_{1}q_{2}ts ) ^{r_{1}} \,d_{q_{1}}t \,d_{q_{2}}s \biggr) ^{\frac{1}{r_{1}}} \end{aligned}$$
(4.14)
$$\begin{aligned}& \qquad {}\times \biggl( \int _{0}^{\frac{b-x}{b-a}} \int _{0}^{\frac{d-y}{d-c}} \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( ta+ ( 1-t ) b,sc+ ( 1-s ) d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}}\,d_{q_{1}}t\,d_{q_{2}}s \biggr) ^{\frac{1}{p_{1}}} \\& \quad \leq \biggl( \biggl( \frac{b-x}{b-a} \biggr) ^{1+\frac{1}{r_{1}}} \biggl( \frac{d-y}{d-c} \biggr) ^{1+\frac{1}{r_{1}}} \biggl( \frac{q_{1}}{ [ r_{1}+1 ] _{q_{1}}} \biggr) ^{\frac{1}{r_{1}}} \biggl( \frac{q_{2}}{ [ r_{1}+1 ] _{q_{2}}} \biggr) ^{\frac{1}{r_{1}}} \biggr) \\& \qquad {}\times \biggl[ \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \frac{1}{ [ 2 ] _{q_{1}} [ 2 ] _{q_{2}}} \biggl( \frac{b-x}{b-a} \biggr) ^{2} \biggl( \frac{d-y}{d-c} \biggr) ^{2} \\& \qquad {}+ \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \frac{1}{ [ 2 ] _{q_{1}}} \biggl( \frac{b-x}{b-a} \biggr) ^{2} \biggl( \frac{d-y}{d-c}-\frac{1}{ [ 2 ] _{q_{2}}} \biggl( \frac{d-y}{d-c} \biggr) ^{2} \biggr) \\& \qquad {}+ \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \frac{1}{ [ 2 ] _{q_{2}}} \biggl( \frac{d-y}{d-c} \biggr) ^{2} \biggl( \frac{b-x}{b-a}-\frac{1}{ [ 2 ] _{q_{1}}} \biggl( \frac{b-x}{b-a} \biggr) ^{2} \biggr) \\& \qquad {} + \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \\& \qquad {}\times\biggl( \frac{b-x}{b-a}- \frac{1}{ [ 2 ] _{q_{1}}} \biggl( \frac{b-x}{b-a} \biggr) ^{2} \biggr) \biggl( \frac{d-y}{d-c}- \frac{1}{ [ 2 ] _{q_{2}}} \biggl( \frac{d-y}{d-c} \biggr) ^{2} \biggr) \biggr] ^{\frac{1}{p_{1}}}, \\& \biggl( \int _{\frac{b-x}{b-a}}^{1} \int _{0}^{\frac{d-y}{d-c}} \bigl( q_{1}t ( 1-q_{2}s ) \bigr) ^{r_{1}}\,d_{q_{1}}t \,d_{q_{2}}s \biggr) ^{\frac{1}{r_{1}}} \end{aligned}$$
(4.15)
$$\begin{aligned}& \qquad {}\times \biggl( \int _{\frac{b-x}{b-a}}^{1} \int _{0}^{\frac{d-y}{d-c}} \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( ta+ ( 1-t ) b,sc+ ( 1-s ) d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}}\,d_{q_{1}}t\,d_{q_{2}}s \biggr) ^{\frac{1}{p_{1}}} \\& \quad \leq \biggl( \biggl( \frac{d-y}{d-c} \biggr) ^{1+\frac{1}{r_{1}}} \biggl( \frac{q_{2}}{ [ r_{1}+1 ] _{q_{2}}} \biggr) ^{\frac{1}{r_{1}}} \biggl( \int _{\frac{b-x}{b-a}}^{1} ( 1-q_{1}t ) ^{r_{1}}\,d_{q_{1}}t \biggr) ^{\frac{1}{r_{1}}} \biggr) \\& \qquad {}\times \biggl[ \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \frac{1}{ [ 2 ] _{q_{1}} [ 2 ] _{q_{2}}} \biggl( \frac{d-y}{d-c} \biggr) ^{2} \biggl( 1- \biggl( \frac{b-x}{b-a} \biggr) ^{2} \biggr) \\& \qquad {}+ \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \frac{1}{ [ 2 ] _{q_{1}}} \biggl( 1- \biggl( \frac{b-x}{b-a} \biggr) ^{2} \biggr) \biggl( \frac{d-y}{d-c}-\frac{1}{ [ 2 ] _{q_{2}}} \biggl( \frac{d-y}{d-c} \biggr) ^{2} \biggr) \\& \qquad {}+ \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \biggl( \frac{q_{1}}{ [ 2 ] _{q_{1}}}-\frac{b-x}{b-a}+ \frac{1}{ [ 2 ] _{q_{1}}} \biggl( \frac{b-x}{b-a} \biggr) ^{2} \biggr) \\& \qquad {}+ \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \biggl( \frac{q_{1}}{ [ 2 ] _{q_{1}}}-\frac{b-x}{b-a}+ \frac{1}{ [ 2 ] _{q_{1}}} \biggl( \frac{b-x}{b-a} \biggr) ^{2} \biggr) \frac{1}{ [ 2 ] _{q_{2}}} \biggl( \frac{d-y}{d-c} \biggr) ^{2} \\& \qquad {} \times \biggl( \frac{d-y}{d-c}- \frac{1}{ [ 2 ] _{q_{2}}} \biggl( \frac{d-y}{d-c} \biggr) ^{2} \biggr) \biggr] ^{\frac{1}{p_{1}}}, \\& \biggl( \int _{0}^{\frac{b-x}{b-a}} \int _{\frac{d-y}{d-c}}^{1} \bigl( q_{1}t ( 1-q_{2}s ) \bigr) ^{r_{1}}\,d_{q_{1}}t \,d_{q_{2}}s \biggr) ^{\frac{1}{r_{1}}} \end{aligned}$$
(4.16)
$$\begin{aligned}& \qquad {}\times \biggl( \int _{0}^{\frac{b-x}{b-a}} \int _{\frac{d-y}{d-c}}^{1} \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( ta+ ( 1-t ) b,sc+ ( 1-s ) d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}}\,d_{q_{1}}t\,d_{q_{2}}s \biggr) ^{\frac{1}{p_{1}}} \\& \quad \leq \biggl( \biggl( \frac{b-x}{b-a} \biggr) ^{1+\frac{1}{r_{1}}} \biggl( \frac{q_{1}}{ [ r_{1}+1 ] _{q_{1}}} \biggr) ^{\frac{1}{r_{1}}} \biggl( \int _{\frac{d-y}{d-c}}^{1} ( 1-q_{2}s ) ^{r_{1}}\,d_{q_{2}}s \biggr) ^{\frac{1}{r_{1}}} \biggr) \\& \qquad {}\times \biggl[ \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \frac{1}{ [ 2 ] _{q_{1}} [ 2 ] _{q_{2}}} \biggl( \frac{b-x}{b-a} \biggr) ^{2} \biggl( 1- \biggl( \frac{d-y}{d-c} \biggr) ^{2} \biggr) \\& \qquad {}+ \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \frac{1}{ [ 2 ] _{q_{1}}} \biggl( \frac{b-x}{b-a} \biggr) ^{2} \biggl( \frac{q_{2}}{ [ 2 ] _{q_{2}}}-\frac{d-y}{d-c}+ \frac{1}{ [ 2 ] _{q_{2}}} \biggl( \frac{d-y}{d-c} \biggr) ^{2} \biggr) \\& \qquad {}+ \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \frac{1}{ [ 2 ] _{q_{2}}} \biggl( 1- \biggl( \frac{d-y}{d-c} \biggr) ^{2} \biggr) \biggl( \frac{b-x}{b-a}-\frac{1}{ [ 2 ] _{q_{1}}} \biggl( \frac{b-x}{b-a} \biggr) ^{2} \biggr) \\& \qquad {}+ \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \biggl( \frac{b-x}{b-a}-\frac{1}{ [ 2 ] _{q_{1}}} \biggl( \frac{b-x}{b-a} \biggr) ^{2} \biggr) \\& \qquad {} \times \biggl( \frac{q_{2}}{ [ 2 ] _{q_{2}}}- \frac{d-y}{d-c}+ \frac{1}{ [ 2 ] _{q_{2}}} \biggl( \frac{d-y}{d-c} \biggr) ^{2} \biggr) \biggr] ^{\frac{1}{p_{1}}}, \\& \biggl( \int _{\frac{b-x}{b-a}}^{1} \int _{\frac{d-y}{d-c}}^{1} \bigl( ( 1-q_{1}t ) ( 1-q_{2}s ) \bigr) ^{r_{1}}\,d_{q_{1}}t \,d_{q_{2}}s \biggr) ^{\frac{1}{r_{1}}} \\& \qquad {}\times \biggl( \int _{\frac{b-x}{b-a}}^{1} \int _{\frac{d-y}{d-c}}^{1} \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( ta+ ( 1-t ) b,sc+ ( 1-s ) d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}}\,d_{q_{1}}t\,d_{q_{2}}s \biggr) ^{\frac{1}{p_{1}}} \\& \quad \leq \biggl( \biggl( \int _{\frac{b-x}{b-a}}^{1} ( 1-q_{1}t ) ^{r_{1}}\,d_{q_{1}}t \biggr) ^{\frac{1}{r_{1}}} \biggl( \int _{\frac{d-y}{d-c}}^{1} ( 1-q_{2}s ) ^{r_{1}}\,d_{q_{2}}s \biggr) ^{\frac{1}{r_{1}}} \biggr) \\& \qquad {}\times \biggl[ \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \frac{1}{ [ 2 ] _{q_{1}} [ 2 ] _{q_{2}}} \biggl( 1- \biggl( \frac{b-x}{b-a} \biggr) ^{2} \biggr) \biggl( 1- \biggl( \frac{d-y}{d-c} \biggr) ^{2} \biggr) \\& \qquad {}+ \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \frac{1}{ [ 2 ] _{q_{1}}} \biggl( 1- \biggl( \frac{b-x}{b-a} \biggr) ^{2} \biggr) \biggl( \frac{q_{2}}{ [ 2 ] _{q_{2}}}- \frac{d-y}{d-c}+\frac{1}{ [ 2 ] _{q_{2}}} \biggl( \frac{d-y}{d-c} \biggr) ^{2} \biggr) \\& \qquad {}+ \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \frac{1}{ [ 2 ] _{q_{2}}} \biggl( 1- \biggl( \frac{d-y}{d-c} \biggr) ^{2} \biggr) \biggl( \frac{q_{1}}{ [ 2 ] _{q_{1}}}- \frac{b-x}{b-a}+\frac{1}{ [ 2 ] _{q_{1}}} \biggl( \frac{b-x}{b-a} \biggr) ^{2} \biggr) \\& \qquad {}+ \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \biggl( \frac{q_{1}}{ [ 2 ] _{q_{1}}}-\frac{b-x}{b-a}+ \frac{1}{ [ 2 ] _{q_{1}}} \biggl( \frac{b-x}{b-a} \biggr) ^{2} \biggr) \\& \qquad {} \times \biggl( \frac{q_{2}}{ [ 2 ] _{q_{2}}}- \frac{d-y}{d-c}+ \frac{1}{ [ 2 ] _{q_{2}}} \biggl( \frac{d-y}{d-c} \biggr) ^{2} \biggr) \biggr] ^{\frac{1}{p_{1}}}. \end{aligned}$$
(4.17)

From (4.13)–(4.17) we get the desired inequality, and the proof is accomplished. □

Some particular cases

In this section, we present some particular cases of the results given in Sect. 4.

Remark 1

In Theorem 5,

(i) By taking \(p_{1}=1\) and \(\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( t,s ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \vert \leq M\), we have the following new Ostrowski-type inequality:

$$\begin{aligned}& \biggl\vert \frac{1}{ ( b-a ) ( d-c ) } \int _{a}^{b} \int _{c}^{d}F ( t,s ) \,{}^{b}d_{q_{1}}t \,{}^{d}d_{q_{2}}s-\frac{1}{b-a} \int _{a}^{b}F ( t,y ) \,{}^{b}d_{q_{1}}t \\& \qquad {} -\frac{1}{d-c} \int _{c}^{d}F ( x,s ) \,{}^{d}d_{q_{2}}s+F ( x,y ) \biggr\vert \\& \quad \leq \frac{M}{ [ 2 ] _{q_{1}} [ 2 ] _{q_{2}}} \bigl[ q_{1} ( x-a ) ^{2}+ ( 1-q_{1} ) ( x-a ) ( b-a ) +q_{1} ( b-x ) ^{2} \bigr] \\& \qquad {}\times \bigl[ q_{2} ( y-a ) ^{2}+ ( 1-q_{2} ) ( y-c ) ( d-c ) +q_{2} ( d-y ) ^{2} \bigr] . \end{aligned}$$
(5.1)

Particularly, taking the limit as \(q_{1},q_{2}\rightarrow 1^{-}\) in (5.1), we reduce inequality (5.1) to (1.4).

(ii) By taking \(x=\frac{a+q_{1}b}{ [ 2 ] _{q_{1}}}\) and \(y=\frac{c+q_{2}d}{ [ 2 ] _{q_{2}}}\) we obtain the following new midpoint inequality:

$$\begin{aligned}& \biggl\vert \frac{1}{ ( b-a ) ( d-c ) } \int _{a}^{b} \int _{c}^{d}F ( t,s ) \,{}^{b}d_{q_{1}}t \,{}^{d}d_{q_{2}}s-\frac{1}{b-a} \int _{a}^{b}F \biggl( t, \frac{c+q_{2}d}{ [ 2 ] _{q_{2}}} \biggr) \,{}^{b}d_{q_{1}}t \\& \qquad {} -\frac{1}{d-c} \int _{c}^{d}F \biggl( \frac{a+q_{1}b}{ [ 2 ] _{q_{1}}},s \biggr) \,{}^{d}d_{q_{2}}s+F \biggl( \frac{a+q_{1}b}{ [ 2 ] _{q_{1}}}, \frac{c+q_{2}d}{ [ 2 ] _{q_{2}}} \biggr) \biggr\vert \\& \quad \leq ( b-a ) ( d-c ) \biggl[ B_{1}^{1- \frac{1}{p_{1}}} ( q_{1} ) B_{1}^{1-\frac{1}{p_{1}}} ( q_{2} ) \\& \qquad {}\times \biggl\{ B_{2} ( q_{1} ) \biggl( B_{2} ( q_{2} ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}}+B_{3} ( q_{2} ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \biggr) \\& \qquad {} +B_{3} ( q_{1} ) \biggl( B_{2} ( q_{2} ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}}+B_{3} ( q_{2} ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \biggr) \biggr\} ^{\frac{1}{p_{1}}} \\& \qquad {}+B_{1}^{1-\frac{1}{p_{1}}} ( q_{1} ) B_{1}^{1- \frac{1}{p_{1}}} ( q_{2} ) \\& \qquad {}\times \biggl\{ B_{2} ( q_{1} ) \biggl( B_{4} ( q_{2} ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}}+B_{5} ( q_{2} ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \biggr) \\& \qquad {} +B_{3} ( q_{1} ) \biggl( B_{4} ( q_{2} ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}}+B_{5} ( q_{2} ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \biggr) \biggr\} ^{\frac{1}{p_{1}}} \\& \qquad {}+B_{1}^{1-\frac{1}{p_{1}}} ( q_{1} ) B_{1}^{1- \frac{1}{p_{1}}} ( q_{2} ) \\& \qquad {}\times \biggl\{ B_{4} ( q_{1} ) \biggl( B_{2} ( q_{2} ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}}+B_{3} ( q_{2} ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \biggr) \\& \qquad {} +B_{5} ( q_{1} ) \biggl( B_{2} ( q_{2} ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}}+B_{3} ( q_{2} ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \biggr) \biggr\} ^{\frac{1}{p_{1}}} \\& \qquad {}+B_{1}^{1-\frac{1}{p_{1}}} ( q_{1} ) B_{1}^{1- \frac{1}{p_{1}}} ( q_{2} ) \\& \qquad {}\times \biggl\{ B_{4} ( q_{1} ) \biggl( B_{4} ( q_{2} ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}}+B_{5} ( q_{2} ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \biggr) \\& \qquad {} +B_{5} ( q_{1} ) \biggl( B_{4} ( q_{2} ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}}+B_{5} ( q_{2} ) \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \biggr) \biggr\} ^{\frac{1}{p_{1}}} \biggr], \end{aligned}$$
(5.2)

where

$$\begin{aligned}& B_{1} ( q ) =\frac{q}{ [ 2 ] _{q}^{3}}, \qquad B_{2} ( q ) = \frac{q}{ [ 2 ] _{q}^{3} [ 3 ] _{q}}, \qquad B_{3} ( q ) = \frac{q^{2}}{ [ 2 ] _{q}^{2} [ 3 ] _{q}}, \\& B_{4} ( q ) =\frac{2q}{ [ 2 ] _{q}^{3} [ 3 ] _{q}}, \qquad B_{5} ( q ) = \frac{-q+q^{2}+q^{3}}{ [ 2 ] _{q}^{3} [ 3 ] _{q}}, \end{aligned}$$

and \(q_{1},q_{2}\in ( 0,1 ) \). Particularly, taking the limit as \(q_{1},q_{2}\rightarrow 1^{-}\) in (5.2), we reduce inequality (5.2) to [37, Theorem 2, inequality (2.4)]

$$\begin{aligned}& \biggl\vert \frac{1}{ ( b-a ) ( d-c ) } \int _{a}^{b} \int _{c}^{d}F ( t,s ) \,dt \,ds-\frac{1}{b-a} \int _{a}^{b}F \biggl( t,\frac{c+d}{2} \biggr) \,dt \\& \qquad {} -\frac{1}{d-c} \int _{c}^{d}F \biggl( \frac{a+b}{2},s \biggr) \,ds+F \biggl( \frac{a+b}{2},\frac{c+d}{2} \biggr) \biggr\vert \\& \quad \leq \frac{ ( b-a ) ( d-c ) }{64} \biggl( \frac{2}{3} \biggr) ^{\frac{2}{p_{1}}} \\& \qquad {}\times \biggl\{ \biggl[ \frac{\frac{\partial ^{2}F(a,c)}{\partial s\,\partial t}+2\frac{\partial ^{2}F(a,d)}{\partial s\,\partial t}+2\frac{\partial ^{2}F(b,c)}{\partial s\,\partial t}+4\frac{\partial ^{2}F(b,d)}{\partial s\,\partial t}}{4} \biggr] ^{\frac{1}{p_{1}}} \\& \qquad {}+ \biggl[ \frac{2\frac{\partial ^{2}F(a,c)}{\partial s\,\partial t}+\frac{\partial ^{2}F(a,d)}{\partial s\,\partial t}+4\frac{\partial ^{2}F(b,c)}{\partial s\,\partial t}+2\frac{\partial ^{2}F(b,d)}{\partial s\,\partial t}}{4} \biggr] ^{\frac{1}{p_{1}}} \\& \qquad {}+ \biggl[ \frac{2\frac{\partial ^{2}F(a,c)}{\partial s\,\partial t}+4\frac{\partial ^{2}F(a,d)}{\partial s\,\partial t}+\frac{\partial ^{2}F(b,c)}{\partial s\,\partial t}+2\frac{\partial ^{2}F(b,d)}{\partial s\,\partial t}}{4} \biggr] ^{\frac{1}{p_{1}}} \\& \qquad {} + \biggl[ \frac{4\frac{\partial ^{2}F(a,c)}{\partial s\,\partial t}+2\frac{\partial ^{2}F(a,d)}{\partial s\,\partial t}+2\frac{\partial ^{2}F(b,c)}{\partial s\,\partial t}+\frac{\partial ^{2}F(b,d)}{\partial s\,\partial t}}{4} \biggr] ^{\frac{1}{p_{1}}} \biggr\} . \end{aligned}$$
(5.3)

(iii) By taking \(p_{1}=1\) we have the following inequality:

$$\begin{aligned}& \biggl\vert \frac{1}{ ( b-a ) ( d-c ) } \int _{a}^{b} \int _{c}^{d}F ( t,s ) \,{}^{b}d_{q_{1}}t \,{}^{d}d_{q_{2}}s-\frac{1}{b-a} \int _{a}^{b}F ( t,y ) \,{}^{b}d_{q_{1}}x \\& \qquad {} -\frac{1}{d-c} \int _{c}^{d}F ( x ,s ) \,{}^{d}d_{q_{2}}s+F ( x ,y ) \biggr\vert \\& \quad \leq ( b-a ) ( d-c ) \biggl[ \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert \bigl\{ \bigl( A_{2} ( a,b,q_{1},x ) +A_{5} ( a,b,q_{1},x ) \bigr) \\& \qquad {} \times \bigl( A_{2} ( c,d,q_{2},y ) + \bigl( A_{5} ( c,d,q_{2},y ) \bigr) \bigr) \bigr\} \biggr] \\& \qquad {}+ \biggl[ \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert \bigl\{ \bigl( A_{2} ( a,b,q_{1},x ) +A_{5} ( a,b,q_{1},x ) \bigr) \\& \qquad {} \times \bigl( A_{3} ( c,d,q_{2},y ) + \bigl( A_{6} ( c,d,q_{2},y ) \bigr) \bigr) \bigr\} \biggr] \\& \qquad {}+ \biggl[ \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert \bigl\{ \bigl( A_{3} ( a,b,q_{1},x ) +A_{6} ( a,b,q_{1},x ) \bigr) \\& \qquad {} \times \bigl( A_{2} ( c,d,q_{2},y ) + \bigl( A_{5} ( c,d,q_{2},y ) \bigr) \bigr) \bigr\} \biggr] \\& \qquad {}+ \biggl[ \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert \bigl\{ \bigl( A_{3} ( a,b,q_{1},x ) +A_{6} ( a,b,q_{1},x ) \bigr) \\& \qquad {} \times \bigl( A_{3} ( c,d,q_{2},y ) + \bigl( A_{6} ( c,d,q_{2},y ) \bigr) \bigr) \bigr\} \biggr] . \end{aligned}$$
(5.4)

Particularly, taking the limit as \(q_{1},q_{2}\rightarrow 1^{-}\) in (5.4), we reduce inequality (5.4) reduces to [40, Theorem 2].

Remark 2

Consider Theorem 6.

(i) If \(\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( t,s ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \vert \leq M\), then as \(q_{1},q_{2}\rightarrow 1^{-}\), we obtain inequality (1.5).

(ii) Taking \(x =\frac{a+q_{1}b}{ [ 2 ] _{q_{1}}}\) and \(y=\frac{c+q_{2}d}{ [ 2 ] _{q_{2}}}\), we obtain the following new midpoint inequality:

$$\begin{aligned}& \biggl\vert \frac{1}{ ( b-a ) ( d-c ) } \int _{a}^{b} \int _{c}^{d}F ( t,s ) \,{}^{b}d_{q_{1}}t \,{}^{d}d_{q_{2}}s-\frac{1}{b-a} \int _{a}^{b}F \biggl( t, \frac{c+q_{2}d}{ [ 2 ] _{q_{2}}} \biggr) \,{}^{b}d_{q_{1}}t \\& \qquad {} -\frac{1}{d-c} \int _{c}^{d}F \biggl( \frac{a+q_{1}b}{ [ 2 ] _{q_{1}}},s \biggr) \,{}^{d}d_{q_{2}}s+F \biggl( \frac{a+q_{1}b}{ [ 2 ] _{q_{1}}}, \frac{c+q_{2}d}{ [ 2 ] _{q_{2}}} \biggr) \biggr\vert \\& \quad \leq ( b-a ) ( d-c ) \\& \qquad {}\times \biggl[ \biggl( \biggl( \frac{1}{ [ 2 ] _{q_{1}}} \biggr) ^{1+\frac{1}{r_{1}}} \biggl( \frac{1}{ [ 2 ] _{q_{2}}} \biggr) ^{1+ \frac{1}{r_{1}}} \biggl( \frac{q_{1}}{ [ r_{1}+1 ] _{q_{1}}} \biggr) ^{\frac{1}{r_{1}}} \biggl( \frac{q_{2}}{ [ r_{1}+1 ] _{q_{2}}} \biggr) ^{\frac{1}{r_{1}}} \biggr) \\& \qquad {}\times \biggl\{ \frac{1}{ [ 2 ] _{q_{1}}^{3} [ 2 ] _{q_{2}}^{3}} \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}}+ \frac{2q_{2}+q_{2}^{2}}{ [ 2 ] _{q_{1}}^{3} [ 2 ] _{q_{2}}^{3}} \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \\& \qquad {} + \frac{2q_{1}+q_{1}^{2}}{ [ 2 ] _{q_{1}}^{3} [ 2 ] _{q_{2}}^{3}} \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}}+ \frac{ ( 2q_{1}+q_{1}^{2} ) ( 2q_{2}+q_{2}^{2} )}{ [ 2 ] _{q_{1}}^{3} [ 2 ] _{q_{2}}^{3}} \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \biggr\} ^{\frac{1}{p_{1}}} \\& \qquad {}+ \biggl( \biggl( \frac{1}{ [ 2 ] _{q_{2}}} \biggr) ^{1+ \frac{1}{r_{1}}} \biggl( \frac{q_{2}}{ [ r_{1}+1 ] _{q_{2}}} \biggr) ^{\frac{1}{r_{1}}} \biggl( \int _{\frac{1}{ [ 2 ] _{q_{1}}}}^{1} ( 1-q_{1}t ) ^{r_{1}}\,d_{q_{1}}t \biggr) ^{\frac{1}{r_{1}}} \biggr) \\& \qquad {}\times \biggl\{ \frac{2q_{1}+q_{1}^{2}}{ [ 2 ] _{q_{1}}^{3} [ 2 ] _{q_{2}}^{3}} \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}}+ \frac{ ( 2q_{1}+q_{1}^{2} ) ( 2q_{2}+q_{2}^{2} ) }{ [ 2 ] _{q_{1}}^{3} [ 2 ] _{q_{2}}^{3}} \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \\& \qquad {}+ \frac{ ( q_{1}^{3}+q_{1}^{2}-q_{1} ) }{ [ 2 ] _{q_{1}}^{3} [ 2 ] _{q_{2}}^{3}} \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}}+ \frac{ ( q_{1}^{3}+q_{1}^{2}-q_{1} ) ( 2q_{2}+q_{2}^{2} ) }{ [ 2 ] _{q_{1}}^{3} [ 2 ] _{q_{2}}^{3}} \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \\& \qquad {} + \frac{ ( q_{1}^{3}+q_{1}^{2}-q_{1} ) }{ [ 2 ] _{q_{1}}^{3} [ 2 ] _{q_{2}}^{3}} \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}}\\& \qquad {}+ \frac{ ( q_{1}^{3}+q_{1}^{2}-q_{1} ) ( 2q_{2}+q_{2}^{2} ) }{ [ 2 ] _{q_{1}}^{3} [ 2 ] _{q_{2}}^{3}} \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \biggr\} ^{\frac{1}{p_{1}}} \\& \qquad {}+ \biggl( \biggl( \frac{1}{ [ 2 ] _{q_{1}}} \biggr) ^{1+ \frac{1}{r_{1}}} \biggl( \frac{q_{1}}{ [ r_{1}+1 ] _{q_{1}}} \biggr) ^{\frac{1}{r_{1}}} \biggl( \int _{\frac{1}{ [ 2 ] _{q_{2}}}}^{1} ( 1-q_{2}s ) ^{r_{1}}\,d_{q_{2}}s \biggr) ^{\frac{1}{r_{1}}} \biggr) \\& \qquad {}\times \biggl\{ \frac{ ( 2q_{1}+q_{1}^{2} ) }{ [ 2 ] _{q_{1}}^{3} [ 2 ] _{q_{2}}^{3}} \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}}+ \frac{ ( q_{2}^{3}+q_{2}^{2}-q_{2} ) ( 2q_{2}+q_{2}^{2} ) }{ [ 2 ] _{q_{1}}^{3} [ 2 ] _{q_{2}}^{3}} \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \\& \qquad {} + \frac{ ( 2q_{1}+q_{1}^{2} ) ( 2q_{2}+q_{2}^{2} ) }{ [ 2 ] _{q_{1}}^{3} [ 2 ] _{q_{2}}^{3}} \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}}\\& \qquad {}+ \frac{ ( q_{2}^{3}+q_{2}^{2}-q_{2} ) ( 2q_{2}+q_{2}^{2} ) }{ [ 2 ] _{q_{1}}^{3} [ 2 ] _{q_{2}}^{3}} \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \biggr\} ^{\frac{1}{p_{1}}} \\& \qquad {}+ \biggl( \biggl( \int _{\frac{1}{ [ 2 ] _{q_{1}}}}^{1} ( 1-q_{1}t ) ^{r_{1}}\,d_{q_{1}}t \biggr) ^{\frac{1}{r_{1}}} \biggl( \int _{\frac{1}{ [ 2 ] _{q_{2}}}}^{1} ( 1-q_{2}s ) ^{r_{1}}\,d_{q_{2}}s \biggr) ^{\frac{1}{r_{1}}} \biggr) \\& \qquad {}\times \biggl[ \frac{ ( 2q_{1}+q_{1}^{2} ) ( 2q_{2}+q_{2}^{2} ) }{ [ 2 ] _{q_{1}}^{3} [ 2 ] _{q_{2}}^{3}} \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}}\\& \qquad {}+ \frac{ ( 2q_{1}+q_{1}^{2} ) ( q_{2}^{3}+q_{2}^{2}-q_{2} ) }{ [ 2 ] _{q_{1}}^{3} [ 2 ] _{q_{2}}^{3}} \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( a,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \\& \qquad {}+ \frac{ ( q_{1}^{3}+q_{1}^{2}-q_{1} ) ( 2q_{2}+q_{2}^{2} ) }{ [ 2 ] _{q_{1}}^{3} [ 2 ] _{q_{2}}^{3}} \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,c ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \\& \qquad {} + \frac{ ( q_{1}^{3}+q_{1}^{2}-q_{1} ) ( q_{2}^{3}+q_{2}^{2}-q_{2} ) }{ [ 2 ] _{q_{1}}^{3} [ 2 ] _{q_{2}}^{3}} \biggl\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( b,d ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \biggr\vert ^{p_{1}} \biggr] ^{\frac{1}{p_{1}}} \biggr] . \end{aligned}$$

Conclusion

In this research, we proved some new quantum Ostrowski-type inequalities for \(q_{1}q_{2}\)-differentiable coordinated convex functions using the \(q_{1}q_{2}\)-integrals. We also showed that the results proved in this research transformed into some new and known inequalities by considering the limits as \(q_{1},q_{2}\rightarrow 1^{-}\) in the main results. It is interesting that the forthcoming researchers can offer similar inequalities for different kinds of convexities in their future work.

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Acknowledgements

The authors would like to express their sincere thanks to the editor and the anonymous reviewers for their helpful comments and suggestions.

Funding

The work was supported by the Natural Science Foundation of China (Grant Nos. 61673169, 11301127, 11701176, 11626101, 11601485, 11971241).

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Correspondence to Yu-Ming Chu.

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Ali, M.A., Chu, YM., Budak, H. et al. Quantum variant of Montgomery identity and Ostrowski-type inequalities for the mappings of two variables. Adv Differ Equ 2021, 25 (2021). https://doi.org/10.1186/s13662-020-03195-7

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Keywords

  • Ostrowski inequality
  • \(q_{1}q_{2}\)-integral
  • Quantum calculus
  • Coordinated convex function
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