Hermite–Hadamard integral inequalities on coordinated convex functions in quantum calculus

Abstract

At first, we recall the q-operators in the context of q-calculus and by examining these operators we will introduce new definitions of the partial q-operators. Then, we investigate some new refinements inequalities of Hermite–Hadamard ($H-H$) type on the coordinated convex functions involving the new defined partial q-operators. From our main results, we establish several specific inequalities and we point out the existing results which had already been obtained in the literature.

1 Introduction and Preliminaries

Let Ψ be defined on an interval $J\subseteq \mathbb{R}_{e}$ ($\mathbb{R}_{e}$ is the set of real numbers), then Ψ is convex if

$$ \Psi \bigl( \lambda u+ ( 1-\lambda ) w \bigr) \leq \lambda \Psi ( u )+ (1- \lambda ) \Psi ( w ), $$

(1.1)

for all u, w in J and for any λ in $[0,1]$.

Convex functions plays a vital role in the development of many fields of mathematics and significant applications are found in a variety of applied sciences such as optimization theory, number theory, combinatorics, special means theory, approximation theory and numerical analysis, see [1–5]. The well-known Hermite–Hadamard ($H-H$) inequality has central part in this development as it gives the criterion for convex functions. The $H-H$ inequality is given as follows: Let $\Psi : [ A,B ]\subset \mathbb{R}_{e}\to \mathbb{R}_{e}$ be a convex function with $A< B$, then the $H-H$ inequality is given as [6]

$$ \Psi \biggl(\frac{A+B}{2} \biggr) \leq \frac{1}{B-A} \int _{A}^{B} \Psi (\lambda ) \,d\lambda \leq \frac{\Psi ( A )+\Psi (B )}{2}. $$

(1.2)

The $H-H$ inequality (1.2) is converted into three equivalent integral inequalities by the help of Riemann–Liouville fractional operators; see [7–9]. Also, it has been generalized and converted into many other integral inequalities by the help of other types of fractional operators, see [10–13].

In 2001, Dragomir [14] introduced the notion of coordinated convex functions in a rectangle from of the plane $\mathbb{R}_{e}^{2}$.

Definition 1.1

([14])

Let $\Delta :=[A,B]\times [C,D]$ be a bi-dimensional interval such that $0\leq A< B<\infty$, $0\leq C< D<\infty$. Then, a function $\Psi :\Delta \rightarrow \mathbb{R}_{e}$ is called coordinated convex on Δ, if the partial mappings $\Psi _{v}:[A,B]\rightarrow \mathbb{R}_{e}$, $\Psi _{v}(x)=\Psi (x,v)$ and $\Psi _{u}:[C,D]\rightarrow \mathbb{R}_{e}$, $\Psi _{u}(y)=\Psi (u,y)$ are convex for each $y,v\in [C,D]$ and $x,u\in [A,B]$. Also, Ψ satisfies the inequality

$$ \Psi \bigl( \lambda u+(1-\lambda )w,\lambda v+(1-\lambda )z \bigr) \leq \lambda \Psi (u,v)+(1-\lambda )\Psi (w,z), $$

(1.3)

for all $(u,v),(w,z)\in \Delta$ and $\lambda \in [0,1]$.

By the help of above definition, Dragomir [14] established the following $H-H$ type inequalities similar to the one dimensional case.

Theorem 1.1

([14])

Suppose that $\Psi :\Delta \rightarrow \mathbb{R}_{e}$ is a coordinated convex function on Δ. Then we have

$$\begin{aligned}& \Psi \biggl( \frac{A+B}{2},\frac{C+D}{2} \biggr) \\& \quad \leq \frac{1}{2} \biggl[ \frac{1}{B-A} \int _{A}^{B}\Psi \biggl( u, \frac{C+D}{2} \biggr) \,du+\frac{1}{D-C} \int _{C}^{D}\Psi \biggl( \frac{A+B}{2},v \biggr) \,dv \biggr] \\& \quad \leq \frac{1}{ ( B-A ) ( D-C ) } \int _{A}^{B} \int _{C}^{D}\Psi ( u,v ) \,du\,dv \\& \quad \leq \frac{1}{2} \biggl[ \frac{1}{B-A} \int _{A}^{B} \bigl[ \Psi (u,C ) +\Psi ( u,D ) \bigr] \,du+ \frac{1}{D-C} \int _{C}^{D} \bigl[ \Psi ( A,v )+ \Psi ( B,v ) \bigr] \,dv \biggr] \\& \quad \leq \frac{\Psi ( A,C )+\Psi ( A,D ) +\Psi ( B,C )+\Psi ( B,D ) }{4}. \end{aligned}$$

(1.4)

Many relevant results have been reported in this direction with different class of convex functions; see [15–19] and the references therein.

In the early 20th century, Jackson [20, 21] worked on the classical notion of a derivative without limit allowing for easier study of number theory and ordinary calculus in his investigations. Jackson got the credit of the q-analogue of the various well-known results of calculus; see [20–24].

For a real valued function Ψ, the q-derivative is characterized by

$$ \mathbf{D}_{q}\Psi (\varsigma )= \frac{\Psi (q\varsigma )-\Psi (\varsigma )}{q\varsigma -\varsigma }, $$

(1.5)

where $q\in (0,1)$. The well-known Jackson integral of a real valued function Ψ is given by the following series expansion:

$$ \int _{0}^{\mu } \Psi (\varsigma )\,d_{q}\varsigma =(1-q)\mu \sum_{r=0}^{\infty }q^{r} \Psi \bigl(q^{r}\mu \bigr). $$

(1.6)

References [25, 26] discuss the notion of q-derivatives and q-integrals over the finite interval $[k_{1},k_{2}]$ of real numbers and defined the $q_{k_{1}}$-derivative and $q_{k_{1}}$-integral.

Definition 1.2

([25, 26])

For any continuous function $\Psi :[k_{1},k_{2}]\rightarrow \mathbb{R}_{e}$ and $q\in (0,1)$, the $q_{k_{1}}$-derivative of Ψ at $\varsigma \in [k_{1},k_{2}]$ is defined by

$$ _{k_{1}}\mathbf{D}_{q}\Psi (\varsigma )= \frac{\Psi (\varsigma )-\Psi (q\varsigma +(1-q)k_{1} )}{(1-q)(\varsigma -k_{1})}, \quad \varsigma \neq k_{1}. $$

(1.7)

Definition 1.3

([25, 26])

For a continuous function $\Psi :[k_{1},k_{2}]\rightarrow \mathbb{R}_{e}$ and $q\in (0,1)$, the $q_{k_{1}}$-integral of Ψ at $\varsigma \in [k_{1},k_{2}]$ is defined by

$$ \int _{k_{1}}^{k} \Psi (\varsigma )_{k_{1}}\,d_{q}\varsigma =(1-q) (k-k_{1}) \sum _{h=0}^{\infty }q^{h}\Psi \bigl(q^{h}k+\bigl(1-q^{h}\bigr)k_{1}\bigr), \quad k\in [k_{1},k_{2}]. $$

(1.8)

If $k_{1}=0$, then

$$ \int _{0}^{k} \Psi (\varsigma )_{0}\,d_{q}\varsigma =(1-q)k \sum _{h=0}^{\infty }q^{h}\Psi \bigl(q^{h}k\bigr)= \int _{0}^{k} \Psi ( \varsigma )\,d_{q}\varsigma , $$

(1.9)

which is the classical q-integral (1.6).

In view of the above definitions, the authors [27] established the following inequality.

Theorem 1.2

([27])

Let $\Psi : [A, B]\rightarrow \mathbb{R}_{e}$ be a convex differentiable function on $[A, B]$ and $q\in (0,1)$. Then we have

$$\begin{aligned} \Psi \biggl( \frac{qA+B}{1+q} \biggr)\leq \frac{1}{B-A} \int _{A}^{B} \Psi (\varsigma )\textit{ }_{A}\,d_{q}\varsigma \leq \frac{q\Psi (A)+\Psi (B)}{1+q}. \end{aligned}$$

(1.10)

In [28], the authors generalize the notion of q-derivatives and q-integrals by introducing the $q^{k_{2}}$-derivative and $q^{k_{2}}$-integral over the finite real interval $[k_{1},k_{2}]$.

Definition 1.4

([28])

For any continuous function $\Psi :[k_{1},k_{2}]\rightarrow \mathbb{R}_{e}$ and $q\in (0,1)$, the $q^{k_{2}}$-derivative of Ψ at $\varsigma \in [k_{1},k_{2}]$ is defined by

$$ ^{k_{2}}\mathbf{D}_{q}\Psi (\varsigma )= \frac{\Psi (\varsigma )-\Psi (q\varsigma +(1-q)k_{2} )}{(1-q)(\varsigma -k_{2})}, \quad \varsigma \neq k_{2}. $$

(1.11)

Definition 1.5

([28])

For any continuous function $\Psi :[k_{1},k_{2}]\rightarrow \mathbb{R}_{e}$ and $q\in (0,1)$, the $q^{k_{2}}$-integral of Ψ at $\varsigma \in [k_{1},k_{2}]$ is defined by

$$ \int _{k}^{k_{2}} \Psi (\varsigma )\text{ }^{k_{2}}\,d_{q}\varsigma =(1-q) (k_{2}-k)\sum_{h=0}^{\infty }q^{h} \Psi \bigl(q^{h}k+\bigl(1-q^{h}\bigr)k_{2} \bigr), \quad k\in [k_{1},k_{2}]. $$

(1.12)

In [28], the authors established the following counterpart for integrals (1.10).

Theorem 1.3

([28])

Let $\Psi :[A,B]\rightarrow \mathbb{R}_{e}$ be a convex differentiable function on $[A, B]$ and $q\in (0,1)$. Then we have

$$\begin{aligned} \Psi \biggl( \frac{A+qB}{1+q} \biggr)\leq \frac{1}{B-A} \int _{A}^{B} \Psi (\varsigma )\textit{ }^{B}\,d_{q}\varsigma \leq \frac{\Psi (A)+q\Psi (B)}{1+q}. \end{aligned}$$

(1.13)

In [29], the authors extended the definition of q-derivative and q-integral to the case of two variables.

Definition 1.6

([29])

Let $\Psi :\Delta\rightarrow \mathbb{R}_{e}$ be a continuous function of two variables and $0< q_{1}<1$, $0< q_{2}<1$. Then, the partial $q_{1A}$-derivative, partial $q_{2C}$-derivative and partial $q_{1A}q_{2C}$- derivatives are defined by

$$\begin{aligned}& \frac{_{A}\partial _{q_{1}}\Psi (u,v)}{_{A}\partial _{q_{1}}u}= \frac{\Psi (u,v)-\Psi (q_{1}u+(1-q_{1})A,v)}{(1-q_{1})(u-A)},\quad u \neq A, \end{aligned}$$

(1.14)

$$\begin{aligned}& \frac{_{C}\partial _{q_{2}}\Psi (u,v)}{_{C}\partial _{q_{2}}u}= \frac{\Psi (u,v)-\Psi (u,q_{2}v+(1-q_{2})C)}{(1-q_{2})(v-C)},\quad v \neq C, \end{aligned}$$

(1.15)

and

$$\begin{aligned} \frac{_{A,C}\partial _{q_{1}q_{2}}\Psi (u,v)}{_{A}\partial _{q_{1}}u_{C}\partial _{q_{2}}v} =& \frac{1}{(1-q_{1})(u-A)(1-q_{2})(v-C)} \bigl[\Psi \bigl(q_{1}u+(1-q_{1})A,q_{2}v+(1-q_{2})C \bigr) \\ & {}-\Psi \bigl(q_{1}u+(1-q_{1})A,v\bigr)-\Psi \bigl(u,q_{2}v+(1-q_{2})C\bigr) \\ & {}+\Psi (u,v) \bigr],\quad u \neq A, v\neq C. \end{aligned}$$

(1.16)

Definition 1.7

([29])

Let $\Psi :\Delta\rightarrow \mathbb{R}_{e}$ be a continuous function of two variables and $0< q_{1}<1$, $0< q_{2}<1$, Then the definite $q_{1A}q_{2C}$-integral on Δ is defined by

$$\begin{aligned}& \int _{A}^{\nu } \int _{C}^{\mu } \Psi (u,v)\text{ }_{C}\,d_{q_{2}}v_{A}\,d_{q_{1}}u \\& \quad =(1-q_{1}) (1-q_{2}) ( \nu -A) (\mu -C) \\& \qquad {}\times \sum_{h=0}^{\infty }\sum _{k=0}^{\infty }q_{1}^{k}q_{2}^{h} \Psi \bigl(q_{1}^{k}\nu +\bigl(1-q_{1}^{k} \bigr)A,q_{2}^{h}\mu +\bigl(1-q_{2}^{h} \bigr)C\bigr), \end{aligned}$$

(1.17)

for all $(\nu ,\mu )\in \Delta$.

Meanwhile, the authors [29] proved the following inequality.

Theorem 1.4

([29])

Let $\Psi :\Delta\rightarrow \mathbb{R}_{e}$ be a coordinated convex function on Δ, then the following inequalities hold:

$$\begin{aligned}& \Psi \biggl( \frac{A+B}{2},\frac{C+D}{2} \biggr) \\& \quad \leq \frac{1}{2} \biggl[ \frac{1}{B-A} \int _{A}^{B}\Psi \biggl( u, \frac{C+D}{2} \biggr) \textit{ }_{A}\,d_{q_{1}}u+ \frac{1}{D-C} \int _{C}^{D}\Psi \biggl( \frac{A+B}{2},v \biggr) \textit{ }_{C}\,d_{q_{2}}v \biggr] \\& \quad \leq \frac{1}{ ( B-A ) ( D-C ) } \int _{A}^{B} \int _{C}^{D}\Psi ( u,v ) \textit{ }_{C}\,d_{q_{2}}v \textit{ }_{A}\,d_{q_{1}}u \\& \quad \leq \frac{q_{2}}{2(1+q_{2})(B-A)} \int _{A}^{B} \Psi ( u,C ) \textit{ }_{A}\,d_{q_{1}}u \\& \qquad {}+\frac{1}{2(1+q_{2})(B-A)} \int _{A}^{B}\Psi ( u,D ) \textit{ }_{A}\,d_{q_{1}}u +\frac{q_{1}}{2(1+q_{1})(D-C)} \int _{C}^{D}\Psi ( A,v ) \textit{ }_{C}\,d_{q_{2}}v \\& \qquad {}+\frac{1}{2(1+q_{1})(D-C)} \int _{C}^{D}\Psi ( B,v ) \textit{ }_{C}\,d_{q_{2}}v \\& \quad \leq \frac{q_{1}q_{2}\Psi ( A,C ) +q_{1}\Psi ( A,D ) +q_{2}\Psi ( B,C ) +\Psi ( B,D ) }{(1+q_{1})(1+q_{2})}. \end{aligned}$$

(1.18)

Recently, in [30], the authors disproved the inequality (1.18) by giving a counter example and proved the following correct $H-H$ inequality.

Theorem 1.5

([30])

Let $\Psi :\Delta \rightarrow \mathbb{R}_{e}$ be a coordinated convex function on Δ, then for all $q_{1},q_{2}\in (0,1)$, we have

$$\begin{aligned}& \Psi \biggl( \frac{q_{1}A+B}{1+q_{1}},\frac{q_{2}C+D}{1+q_{2}} \biggr) \\& \quad \leq \frac{1}{2} \biggl[ \frac{1}{B-A} \int _{A}^{B}\Psi \biggl( u, \frac{q_{2}C+D}{1+q_{2}} \biggr) \textit{ }_{A}d_{q_{1}}u+ \frac{1}{D-C} \int _{C}^{D}\Psi \biggl( \frac{q_{1}A+B}{1+q_{1}},v \biggr) \textit{ }_{C}\,d_{q_{2}}v \biggr] \\& \quad \leq \frac{1}{ ( B-A ) ( D-C ) } \int _{A}^{B} \int _{C}^{D}\Psi ( u,v ) \textit{ }_{C}\,d_{q_{2}}v \textit{ }_{A}\,d_{q_{1}}u \\& \quad \leq \frac{q_{2}}{2(1+q_{2})(B-A)} \int _{A}^{B} \Psi (u,C ) \textit{ }_{A}\,d_{q_{1}}u \\& \qquad {}+\frac{1}{2(1+q_{2})(B-A)} \int _{A}^{B}\Psi ( u,D ) \textit{ }_{A}\,d_{q_{1}}u +\frac{q_{1}}{2(1+q_{1})(D-C)} \int _{C}^{D}\Psi ( A,v ) \textit{ }_{C}\,d_{q_{2}}v \\& \qquad {}+\frac{1}{2(1+q_{1})(D-C)} \int _{C}^{D}\Psi ( B,v ) \textit{ }_{C}\,d_{q_{2}}v \\& \quad \leq \frac{q_{1}q_{2}\Psi ( A,C ) +q_{1}\Psi ( A,D )+q_{2}\Psi ( B,C )+\Psi ( B,D ) }{(1+q_{1})(1+q_{2})}. \end{aligned}$$

(1.19)

Now, by combining the two concepts of q-derivatives and q-integrals given in Definitions 1.2–1.7, we present the following mixed types of partial derivatives and integrals in the two variables along with some examples.

Definition 1.8

Let $\Psi :\Delta\rightarrow \mathbb{R}_{e}$ be a continuous function of two variables and $0< q_{1}<1$, $0< q_{2}<1$, Then the partial $q_{1}^{B}$-derivative, partial $q_{2}^{D}$-derivative and partial $q_{1}^{B}q_{2}^{D}$, $q_{1}^{B}q_{2C}$ and $q_{1A}q_{2}^{D}$ derivatives are defined, respectively, by

$$\begin{aligned}& \frac{^{B}\partial _{q_{1}}\Psi (u,v)}{^{B}\partial _{q_{1}}u}= \frac{\Psi (u,v)-\Psi (q_{1}u+(1-q_{1})B,v)}{(1-q_{1})(u-B)},\quad u \neq B, \end{aligned}$$

(1.20)

$$\begin{aligned}& \frac{^{D}\partial _{q_{2}}\Psi (u,v)}{^{D}\partial _{q_{2}}u}= \frac{\Psi (u,v)-\Psi (u,q_{2}v+(1-q_{2})D)}{(1-q_{2})(v-D)},\quad v \neq D, \end{aligned}$$

(1.21)

$$\begin{aligned}& \frac{_{B,D}\partial _{q_{1}q_{2}}\Psi (u,v)}{^{B}\partial _{q_{1}}u^{D}\partial _{q_{2}}v} \\& \quad = \frac{1}{(1-q_{1})(u-B)(1-q_{2})(v-D)} \bigl[\Psi \bigl(q_{1}u+(1-q_{1})B,q_{2}v+(1-q_{2})D \bigr) \\& \qquad {}-\Psi \bigl(q_{1}u+(1-q_{1})B,v\bigr)-\Psi \bigl(u,q_{2}v+(1-q_{2})D\bigr) \\& \qquad {}+\Psi (u,v) \bigr],\quad u \neq B,v\neq D, \end{aligned}$$

(1.22)

$$\begin{aligned}& \frac{^{B}_{C}\partial _{q_{1}q_{2}}\Psi (u,v)}{^{B}\partial _{q_{1}}u\text{ }_{C}\partial _{q_{2}}v} \\& \quad = \frac{1}{(1-q_{1})(u-B)(1-q_{2})(v-C)} \bigl[\Psi \bigl(q_{1}u+(1-q_{1})B,q_{2}v+(1-q_{2})C \bigr) \\& \qquad {}-\Psi \bigl(q_{1}u+(1-q_{1})B,v\bigr)-\Psi \bigl(u,q_{2}v+(1-q_{2})C\bigr) \\& \qquad {}+\Psi (u,v) \bigr],\quad u \neq B,v\neq C, \end{aligned}$$

(1.23)

and

$$\begin{aligned}& \frac{_{A}^{D}\partial _{q_{1}q_{2}}\Psi (u,v)}{_{A}\partial _{q_{1}}u\text{ }^{D}\partial _{q_{2}}v} \\& \quad = \frac{1}{(1-q_{1})(u-A)(1-q_{2})(v-D)} \bigl[\Psi \bigl(q_{1}u+(1-q_{1})A,q_{2}v+(1-q_{2})D \bigr) \\& \qquad {}-\Psi \bigl(q_{1}u+(1-q_{1})A,v\bigr)-\Psi \bigl(u,q_{2}v+(1-q_{2})D\bigr) \\& \qquad {}+\Psi (u,v) \bigr],\quad u \neq A,v\neq D. \end{aligned}$$

(1.24)

Definition 1.9

Let $\Psi :\Delta\rightarrow \mathbb{R}_{e}$ be a continuous function of two variables and $0< q_{1}<1$, $0< q_{2}<1$, Then the definite $q_{1}^{B}q_{2}^{D}$-integral, $q_{1A}q_{2}^{D}$-integral and $q_{1}^{B}q_{2C}$-integral on Δ are defined by

$$\begin{aligned}& \int _{\nu }^{B} \int _{\mu }^{D} \Psi (u,v)\text{ }^{D}\,d_{q_{2}}v ^{B}\,d_{q_{1}}u \\& \quad =(1-q_{1}) (1-q_{2}) (B-\nu ) (D-\mu ) \\& \qquad {}\times \sum_{h=0}^{\infty }\sum _{k=0}^{\infty }q_{1}^{k}q_{2}^{h} \Psi \bigl(q_{1}^{k}\nu +\bigl(1-q_{1}^{k} \bigr)B,q_{2}^{h}\mu +\bigl(1-q_{2}^{h} \bigr)D\bigr), \quad \forall (\nu ,\mu )\in \Delta , \end{aligned}$$

(1.25)

$$\begin{aligned}& \int ^{\nu }_{A} \int _{\mu }^{D} \Psi (u,v)\text{ }^{D}\,d_{q_{2}}v \text{ }_{A}\,d_{q_{1}}u \\& \quad =(1-q_{1}) (1-q_{2}) (\nu -A) (D-\mu ) \\& \qquad {}\times \sum_{h=0}^{\infty }\sum _{k=0}^{\infty }q_{1}^{k}q_{2}^{h} \Psi \bigl(q_{1}^{k}\nu +\bigl(1-q_{1}^{k} \bigr)A,q_{2}^{h}\mu +\bigl(1-q_{2}^{h} \bigr)D\bigr), \quad \forall (\nu ,\mu )\in \Delta , \end{aligned}$$

(1.26)

and

$$\begin{aligned}& \int _{\nu }^{B} \int ^{\mu }_{C} \Psi (u,v)\text{ }_{C}\,d_{q_{2}}v \text{ }^{B}\,d_{q_{1}}u \\& \quad =(1-q_{1}) (1-q_{2}) (B-\nu ) (\mu -C) \\& \qquad {}\times \sum_{h=0}^{\infty }\sum _{k=0}^{\infty }q_{1}^{k}q_{2}^{h} \Psi \bigl(q_{1}^{k}\nu +\bigl(1-q_{1}^{k} \bigr)B,q_{2}^{h}\mu +\bigl(1-q_{2}^{h} \bigr)C\bigr), \quad \forall (\nu ,\mu )\in \Delta . \end{aligned}$$

(1.27)

Example 1.1

All the $q_{1}q_{2}$-integrals are different for general functions. For instance,

$$\begin{aligned} &\int _{A}^{B} \int _{C}^{D}uv\text{ }^{D}\,d_{q_{2}}v \text{ }^{B}\,d_{q_{1}}u= \frac{(B-A)(A+q_{1}B)(D-C)(C+q_{2}D)}{(1+q_{1})(1+q_{2})}, \\ &\int _{A}^{B} \int _{C}^{D}uv\text{ }^{D}\,d_{q_{2}}v \text{ }_{A}\,d_{q_{1}}u= \frac{(B-A)(q_{1}A+B)(D-C)(C+q_{2}D)}{(1+q_{1})(1+q_{2})}, \\ &\int _{A}^{B} \int _{C}^{D}uv\text{ }_{C}\,d_{q_{2}}v \text{ }^{B}\,d_{q_{1}}u= \frac{(B-A)(A+q_{1}B)(D-C)(q_{2}C+D)}{(1+q_{1})(1+q_{2})}, \\ &\int _{A}^{B} \int _{C}^{D}uv\text{ }_{C}\,d_{q_{2}}v \text{ }_{A}\,d_{q_{1}}u= \frac{(B-A)(q_{1}A+B)(D-C)(q_{2}C+D)}{(1+q_{1})(1+q_{2})}. \end{aligned}$$

Furthermore,

$$\begin{aligned} \int _{C}^{B} \int _{C}^{D}uv\,dv\,du= \frac{(B^{2}-A^{2})(D^{2}-C^{2})}{4}, \end{aligned}$$

subject to the condition that both $q_{1},q_{2}\to 1^{-}$.

Now, we obtain midpoint type inequalities from the inequality (1.19).

Remark 1.1

We have four special cases for this inequality at midpoint.

• If $A=a$, $B=\frac{q_{1}a+b}{1+q_{1}}$, $C=c$, $D=\frac{q_{2}d+c}{1+q_{2}}$, then

  $$\begin{aligned}& \Psi \biggl( \frac{(2q_{1}+q_{1}^{2})a+b}{(1+q_{1})^{2}}, \frac{(1+q_{2}+q_{2}^{2})c+q_{2}d}{(1+q_{2})^{2}} \biggr) \\& \quad \leq \frac{1+q_{1}}{2(b-a)} \int _{a}^{\frac{q_{1}a+b}{1+q_{1}}} \Psi \biggl( u, \frac{(1+q_{2}+q_{2}^{2})c+q_{2}d}{(1+q_{2})^{2}} \biggr) \text{ }_{a}\,d_{q_{1}}u \\& \qquad {}+\frac{1+q_{2}}{2q_{2}(d-c)} \int _{c}^{ \frac{q_{2}d+c}{1+q_{2}}}\Psi \biggl( \frac{(2q_{1}+q_{1}^{2})a+b}{(1+q_{1})^{2}},v \biggr) \text{ }_{c}\,d_{q_{2}}v \\& \quad \leq \frac{(1+q_{1})(1+q_{2})}{q_{2} ( b-a ) ( d-c ) } \int _{a}^{\frac{q_{1}a+b}{1+q_{1}}}\int _{c}^{\frac{q_{2}d+c}{1+q_{2}}}\Psi ( u,v ) \text{ }_{c}\,d_{q_{2}}v\text{ }_{a}\,d_{q_{1}}u \\& \quad \leq \frac{q_{2}(1+q_{1})}{2(1+q_{2})(b-a)} \int _{a}^{ \frac{q_{1}a+b}{1+q_{1}}} \Psi (u,c ) \text{ }_{a}\,d_{q_{1}}u \\& \qquad {}+ \frac{1+q_{1}}{2(1+q_{2})(b-a)} \int _{a}^{ \frac{q_{1}a+b}{1+q_{1}}}\Psi \biggl( u, \frac{q_{2}d+c}{1+q_{2}} \biggr) \text{ }_{a}\,d_{q_{1}}u \\& \qquad {}+\frac{q_{1}(1+q_{2})}{2q_{2}(1+q_{1})(d-c)} \int _{c}^{ \frac{q_{2}d+c}{1+q_{2}}} \Psi ( a,v ) \text{ }_{c}\,d_{q_{2}}v \\& \qquad {}+ \frac{1+q_{2}}{2q_{2}(1+q_{1})(d-c)} \int _{c}^{ \frac{q_{2}d+c}{1+q_{2}}}\Psi \biggl( \frac{q_{1}a+b}{1+q_{1}},v \biggr) \text{ }_{c}\,d_{q_{2}}v \\& \quad \leq \frac{q_{1}q_{2}\Psi ( a,c ) +q_{1}\Psi ( a,\frac{q_{2}d+c}{1+q_{2}} ) +q_{2}\Psi ( \frac{q_{1}a+b}{1+q_{1}},c )+\Psi ( \frac{q_{1}a+b}{1+q_{1}},\frac{q_{2}d+c}{1+q_{2}} ) }{(1+q_{1})(1+q_{2})}. \end{aligned}$$

  (1.28)

• If $A=a$, $B=\frac{q_{1}b+a}{1+q_{1}}$, $C=c$, $D=\frac{q_{2}d+c}{1+q_{2}}$, then

  $$\begin{aligned}& \Psi \biggl( \frac{(1+q_{1}+q_{1}^{2})a+q_{1}b}{(1+q_{1})^{2}}, \frac{(1+q_{2}+q_{2}^{2})c+q_{2}d}{(1+q_{2})^{2}} \biggr) \\& \quad \leq \frac{1+q_{1}}{2q_{1}(b-a)} \int _{a}^{ \frac{q_{1}b+a}{1+q_{1}}}\Psi \biggl( u, \frac{(1+q_{2}+q_{2}^{2})c+q_{2}d}{(1+q_{2})^{2}} \biggr) \text{ }_{a}\,d_{q_{1}}u \\& \qquad {}+\frac{1+q_{2}}{2q_{2}(d-c)} \int _{c}^{ \frac{q_{2}d+c}{1+q_{2}}}\Psi \biggl( \frac{(1+q_{1}+q_{1}^{2})a+q_{1}b}{(1+q_{1})^{2}},v \biggr) \text{ }_{c}\,d_{q_{2}}v \\& \quad \leq \frac{(1+q_{1})(1+q_{2})}{q_{1}q_{2} ( b-a ) ( d-c ) } \int _{a}^{\frac{q_{1}b+a}{1+q_{1}}} \int _{c}^{ \frac{q_{2}d+c}{1+q_{2}}}f ( u,v ) \text{ }_{c}\,d_{q_{2}}v \text{ }_{a}\,d_{q_{1}}u \\& \quad \leq \frac{q_{2}(1+q_{1})}{2q_{1}(1+q_{2})(b-a)} \int _{a}^{ \frac{q_{1}b+a}{1+q_{1}}} \Psi ( u,c ) \text{ }_{a}\,d_{q_{1}}u \\& \qquad {}+ \frac{1+q_{1}}{2q_{1}(1+q_{2})(b-a)} \int _{a}^{ \frac{q_{1}b+a}{1+q_{1}}}\Psi \biggl( u, \frac{q_{2}d+c}{1+q_{2}} \biggr) \text{ }_{a}\,d_{q_{1}}u \\& \qquad {}+\frac{q_{1}(1+q_{2})}{2q_{2}(1+q_{1})(d-c)} \int _{c}^{ \frac{q_{2}d+c}{1+q_{2}}}\Psi ( a,v ) \text{ }_{c}\,d_{q_{2}}v \\& \qquad {}+ \frac{1+q_{2}}{2q_{2}(1+q_{1})(d-c)} \int _{c}^{ \frac{q_{2}d+c}{1+q_{2}}} \Psi \biggl( \frac{q_{1}a+b}{1+q_{1}},v \biggr) \text{ }_{c}\,d_{q_{2}}v \\& \quad \leq \frac{q_{1}q_{2}\Psi ( a,c ) +q_{1}\Psi ( a,\frac{q_{2}d+c}{1+q_{2}} ) +q_{2}\Psi ( \frac{q_{1}b+a}{1+q_{1}},c )+\Psi ( \frac{q_{1}b+a}{1+q_{1}},\frac{q_{2}d+c}{1+q_{2}} ) }{(1+q_{1})(1+q_{2})}. \end{aligned}$$

  (1.29)

• If $A=a$, $B=\frac{q_{1}a+b}{1+q_{1}}$, $C=c$, $D=\frac{q_{2}c+d}{1+q_{2}}$, then

  $$\begin{aligned}& \Psi \biggl( \frac{(2q_{1}+q_{1}^{2})a+b}{(1+q_{1})^{2}}, \frac{(2q_{2}+q_{2}^{2})c+d}{(1+q_{2})^{2}} \biggr) \\& \quad \leq \frac{1+q_{1}}{2(b-a)} \int _{a}^{\frac{q_{1}a+b}{1+q_{1}}} \Psi \biggl( u, \frac{(2q_{2}+q_{2}^{2})c+d}{(1+q_{2})^{2}} \biggr) \text{ }_{a}\,d_{q_{1}}u \\& \qquad {}+\frac{1+q_{2}}{2(d-c)} \int _{c}^{\frac{q_{2}c+d}{1+q_{2}}} \Psi \biggl( \frac{(2q_{1}+q_{1}^{2})a+b}{(1+q_{1})^{2}},v \biggr) \text{ }_{c}\,d_{q_{2}}v \\& \quad \leq \frac{(1+q_{1})(1+q_{2})}{ ( b-a ) ( d-c ) } \int _{a}^{\frac{q_{1}a+b}{1+q_{1}}} \int _{c}^{ \frac{q_{2}c+d}{1+q_{2}}}f ( u,v ) \text{ }_{c}\,d_{q_{2}}v \text{ }_{a}\,d_{q_{1}}u \\& \quad \leq \frac{q_{2}(1+q_{1})}{2(1+q_{2})(b-a)} \int _{a}^{ \frac{q_{1}a+b}{1+q_{1}}} \Psi ( u,c ) \text{ }_{a}\,d_{q_{1}}u \\& \qquad {}+ \frac{1+q_{1}}{2(1+q_{2})(b-a)} \int _{a}^{ \frac{q_{1}a+b}{1+q_{1}}}\Psi \biggl( u, \frac{q_{2}d+c}{1+q_{2}} \biggr) \text{ }_{a}\,d_{q_{1}}u \\& \qquad {}+\frac{q_{1}(1+q_{2})}{2(1+q_{1})(d-c)} \int _{c}^{ \frac{q_{2}c+d}{1+q_{2}}} \Psi ( a,v ) \text{ }_{c}\,d_{q_{2}}v \\& \qquad {}+ \frac{1+q_{2}}{2(1+q_{1})(d-c)} \int _{c}^{ \frac{q_{2}c+d}{1+q_{2}}} \Psi \biggl( \frac{q_{1}a+b}{1+q_{1}},v \biggr) \text{ }_{c}\,d_{q_{2}}v \\& \quad \leq \frac{q_{1}q_{2}\Psi ( a,c ) +q_{1}\Psi ( a,\frac{q_{2}c+d}{1+q_{2}} )+q_{2}\Psi ( \frac{q_{1}a+b}{1+q_{1}},c ) +\Psi ( \frac{q_{1}a+b}{1+q_{1}},\frac{q_{2}c+d}{1+q_{2}} ) }{(1+q_{1})(1+q_{2})}. \end{aligned}$$

  (1.30)

• If $A=a$, $B=\frac{q_{1}b+a}{1+q_{1}}$, $C=c$, $D=\frac{q_{2}c+d}{1+q_{2}}$, then

  $$\begin{aligned}& \Psi \biggl( \frac{(1+q_{1}+q_{1}^{2})a+q_{1}b}{(1+q_{1})^{2}}, \frac{(2q_{2}+q_{2}^{2})c+d}{(1+q_{2})^{2}} \biggr) \\& \quad \leq \frac{1+q_{1}}{2q_{1}(b-a)} \int _{a}^{ \frac{q_{1}b+a}{1+q_{1}}}\Psi \biggl( u, \frac{(2q_{2}+q_{2}^{2})c+d}{(1+q_{2})^{2}} \biggr) \text{ }_{a}\,d_{q_{1}}u \\& \qquad {}+\frac{1+q_{2}}{2(d-c)} \int _{c}^{\frac{q_{2}c+d}{1+q_{2}}} \Psi \biggl(\frac{(1+q_{1}+q_{1}^{2})a+q_{1}b}{(1+q_{1})^{2}},v \biggr) \text{ }_{c}\,d_{q_{2}}v \\& \quad \leq \frac{(1+q_{1})(1+q_{2})}{q_{1} ( b-a ) ( d-c ) } \int _{a}^{\frac{q_{1}b+a}{1+q_{1}}} \int _{c}^{ \frac{q_{2}c+d}{1+q_{2}}}\Psi ( u,v ) \text{ }_{c}\,d_{q_{2}}v \text{ }_{a}\,d_{q_{1}}u \\& \quad \leq \frac{q_{2}(1+q_{1})}{2q_{1}(1+q_{2})(b-a)} \int _{a}^{ \frac{q_{1}b+a}{1+q_{1}}} \Psi ( u,c ) \text{ }_{a}\,d_{q_{1}}u \\& \qquad {}+ \frac{1+q_{1}}{2q_{1}(1+q_{2})(b-a)} \int _{a}^{ \frac{q_{1}b+a}{1+q_{1}}}\Psi \biggl( u, \frac{q_{2}d+c}{1+q_{2}} \biggr) \text{ }_{a}\,d_{q_{1}}u \\& \qquad {}+\frac{q_{1}(1+q_{2})}{2(1+q_{1})(d-c)} \int _{c}^{ \frac{q_{2}c+d}{1+q_{2}}} \Psi ( a,v ) \text{ }_{c}\,d_{q_{2}}v \\& \qquad {}+ \frac{1+q_{2}}{2(1+q_{1})(d-c)} \int _{c}^{ \frac{q_{2}c+d}{1+q_{2}}} \Psi \biggl( \frac{q_{1}b+a}{1+q_{1}},v \biggr) \text{ }_{c}\,d_{q_{2}}v \\& \quad \leq \frac{q_{1}q_{2}\Psi ( a,c ) +q_{1}\Psi ( a,\frac{q_{2}c+d}{1+q_{2}} )+q_{2}\Psi ( \frac{q_{1}b+a}{1+q_{1}},c ) +\Psi ( \frac{q_{1}b+a}{1+q_{1}},\frac{q_{2}c+d}{1+q_{2}} ) }{(1+q_{1})(1+q_{2})}. \end{aligned}$$

  (1.31)

In view of the above results and literatures, and following this tendency of the newly introduced q-derivatives and q-integrals, the aim of this paper is to establish some new refinements of the $H-H$ inequality in the quantum domain using coordinated convex functions. Several special cases from our main results will be given in detail and many well-known results will be recaptured. At the end, we provide a briefly conclusion as well.

2 Main results

By utilizing Theorem 1.3, we have the new result.

Theorem 2.1

Suppose that $\Psi :\Delta \rightarrow \mathbb{R}_{e}$ is a coordinated convex function on Δ and $\Psi \in L_{1}(\Delta )$. Then we have

$$\begin{aligned} &\Psi \biggl( \frac{A+q_{1}B}{1+q_{1}},\frac{C+q_{2}D}{1+q_{2}} \biggr) \\ &\quad \leq \frac{1}{2} \biggl[ \frac{1}{B-A} \int _{A}^{B}\Psi \biggl( u, \frac{C+q_{2}D}{1+q_{2}} \biggr) \textit{ }^{B}\,d_{q_{1}}u+ \frac{1}{D-C} \int _{C}^{D}\Psi \biggl( \frac{A+q_{1}B}{1+q_{1}},v \biggr) \textit{ }^{D}\,d_{q_{2}}v \biggr] \\ &\quad \leq \frac{1}{ ( B-A ) ( D-C ) } \int _{A}^{B} \int _{C}^{D}\Psi ( u,v ) \textit{ }^{D}\,d_{q_{2}}v \textit{ }^{B}\,d_{q_{1}}u \\ &\quad \leq \frac{1}{2(1+q_{2})(B-A)} \int _{A}^{B} \Psi (u,C )\textit{ }^{B}\,d_{q_{1}}u +\frac{q_{2}}{2(1+q_{2})(B-A)} \int _{A}^{B}\Psi ( u,D ) \textit{ }^{B}\,d_{q_{1}}u \\ &\qquad {}+\frac{1}{2(1+q_{1})(D-C)} \int _{C}^{D} \Psi ( A,v )\textit{ }^{D}\,d_{q_{2}}v +\frac{q_{1}}{2(1+q_{1})(D-C)} \int _{C}^{D}\Psi ( B,v ) \textit{ }^{D}\,d_{q_{2}}v \\ &\quad \leq \frac{\Psi (A,C)+q_{2}\Psi (A,D)+q_{1}\Psi (B,C)+q_{1}q_{2}\Psi (B,D) }{(1+q_{1})(1+q_{2})}. \end{aligned}$$

(2.1)

Proof

Due to the coordinated convexity of $\Psi :\Delta \rightarrow \mathbb{R}_{e}$, the partial mapping $\Psi _{u}: [C,D ] \rightarrow \mathbb{R}_{e}$ defined by $\Psi _{u} ( y )=\Psi (u,y )$ for all $u\in [A,B ]$ will be convex on $[ C,D ]$. Analogously, $\Psi _{v}: [ A,B ] \rightarrow \mathbb{R}_{e}$ defined by $\Psi _{v} ( x ) =\Psi ( x,v )$ for all $v\in [C,D ]$ is convex on $[ A,B ]$. Then, by the Theorem 1.3, we get

$$\begin{aligned} \Psi _{u} \biggl( \frac{C+q_{2}D}{1+q_{2}} \biggr)\leq \frac{1}{D-C} \int _{C}^{D}\Psi _{u}(v)\text{ }^{D}\,d_{q_{2}}v \leq \frac{\Psi _{u}(C)+q_{2}\Psi _{u}(D)}{1+q_{2}}, \end{aligned}$$

or

$$\begin{aligned} \Psi \biggl(u, \frac{C+q_{2}D}{1+q_{2}} \biggr)\leq \frac{1}{D-C} \int _{C}^{D}\Psi (u,v)\text{ }^{D}\,d_{q_{2}}v \leq \frac{\Psi (u,C)+q_{2}\Psi (u,D)}{1+q_{2}}. \end{aligned}$$

(2.2)

Integrating the inequality over $[A,B]$, we have

$$\begin{aligned} \frac{1}{B-A} \int _{A}^{B}\Psi \biggl(u, \frac{C+q_{2}D}{1+q_{2}} \biggr)\text{ }^{B}\,d_{q_{1}}u \leq& \frac{1}{(B-A)(D-C)} \int _{A}^{B} \int _{C}^{D}\Psi (u,v) ^{D}\,d_{q_{2}}v\text{ }^{B}\,d_{q_{1}}u \\ \leq& \frac{1}{(1+q_{2})(B-A)} \int _{A}^{B}\Psi (u,C)^{B}\,d_{q_{1}}u \\ &{}+ \frac{q_{2}}{(1+q_{2})(B-A)} \int _{A}^{B}\Psi (u,D)\text{ }^{B}\,d_{q_{1}}u. \end{aligned}$$

(2.3)

Now, by the convexity of $\Psi _{v}$, by the Theorem 1.3, we have

$$\begin{aligned} \Psi _{v} \biggl( \frac{A+q_{1}B}{1+q_{1}} \biggr) \leq \frac{1}{B-A} \int _{A}^{B}\Psi _{v}(u)\text{ }^{B}\,d_{q_{1}}u \leq \frac{\Psi _{v}(A)+q_{1}\Psi _{v}(B)}{1+q_{1}}, \end{aligned}$$

(2.4)

or

$$\begin{aligned} \Psi \biggl( \frac{A+q_{1}B}{1+q_{1}},v \biggr)\leq \frac{1}{B-A} \int _{A}^{B}\Psi (u,v)\text{ }^{B}\,d_{q_{1}}u \leq \frac{\Psi (A,v)+q_{1}\Psi (B,v)}{1+q_{1}}. \end{aligned}$$

(2.5)

Evaluating the average integral over $[C,D]$, we have

$$\begin{aligned}& \frac{1}{D-C} \int _{C}^{D}\Psi \biggl( \frac{A+q_{1}B}{1+q_{2}},v \biggr)\text{ }^{D}\,d_{q_{2}}v \\& \quad \leq \frac{1}{(B-A)(D-C)} \int _{A}^{B} \int _{C}^{D}\Psi (u,v) \text{ }^{D}\,d_{q_{2}}v\text{ }^{B}\,d_{q_{1}}u \\& \quad \leq \frac{1}{(1+q_{1})(D-C)} \int _{C}^{D}\Psi (A,v)\text{ }^{D}\,d_{q_{2}}v+ \frac{q_{1}}{(1+q_{1})(D-C)} \int _{C}^{D}\Psi (B,v)\text{ }^{D}\,d_{q_{2}}v. \end{aligned}$$

(2.6)

Adding (2.5) and (2.6), we get

$$\begin{aligned} &\frac{1}{B-A} \int _{A}^{B}\Psi \biggl(u, \frac{C+q_{2}D}{1+q_{2}} \biggr)\text{ }^{B}\,d_{q_{1}}u+ \frac{1}{D-C} \int _{C}^{D}\Psi \biggl( \frac{A+q_{1}B}{1+q_{2}},v \biggr) \text{ }^{D}\,d_{q_{2}}v \\ &\quad \leq \frac{2}{(B-A)(D-C)} \int _{A}^{B} \int _{C}^{D} \Psi (u,v)\text{ }^{D}\,d_{q_{2}}v\text{ }^{B}\,d_{q_{1}}u \\ &\quad \leq \frac{1}{(1+q_{2})(B-A)} \int _{A}^{B}\Psi (u,C)\text{ }^{B}\,d_{q_{1}}u+ \frac{q_{2}}{(1+q_{2})(B-A)} \int _{A}^{B}\Psi (u,D)\text{ }^{B}\,d_{q_{1}}u \\ &\qquad {}+\frac{1}{(1+q_{1})(D-C)} \int _{C}^{D}\Psi (A,v)\text{ }^{D}\,d_{q_{2}}v+ \frac{q_{1}}{(1+q_{1})(D-C)} \int _{C}^{D}\Psi (B,v)\text{ }^{D}\,d_{q_{2}}v. \end{aligned}$$

(2.7)

Again, by the convexity and applying the first inequality of Theorem 1.3, we have

$$\begin{aligned} \Psi \biggl(\frac{A+q_{1}B}{1+q_{2}}, \frac{C+q_{2}D}{1+q_{2}} \biggr) \leq \frac{1}{D-C} \int _{C}^{D}\Psi \biggl( \frac{A+q_{1}B}{1+q_{2}},v \biggr) \text{ }^{D}\,d_{q_{2}}v \end{aligned}$$

(2.8)

and

$$\begin{aligned} \Psi \biggl(\frac{A+q_{1}B}{1+q_{2}}, \frac{C+q_{2}D}{1+q_{2}} \biggr) \leq \frac{1}{B-A} \int _{A}^{B}\Psi \biggl(u, \frac{C+q_{2}D}{1+q_{2}} \biggr) \text{ }^{B}\,d_{q_{1}}u. \end{aligned}$$

(2.9)

Similarly, applying the second inequality of Theorem 1.3 and convexity of $\Psi _{u}$ and $\Psi _{v}$, we have

$$\begin{aligned} &\frac{1}{(1+q_{2})(B-A)} \int _{A}^{B}\Psi (u,C)\text{ }^{B}\,d_{q_{1}}u+ \frac{q_{2}}{(1+q_{2})(B-A)} \int _{A}^{B}\Psi (u,D)\text{ }^{B}\,d_{q_{1}}u \\ &\qquad {}+\frac{1}{(1+q_{1})(D-C)} \int _{C}^{D}\Psi (A,v)\text{ }^{D}\,d_{q_{2}}v+ \frac{q_{1}}{(1+q_{1})(D-C)} \int _{C}^{D}\Psi (B,v)\text{ }^{D}\,d_{q_{2}}v \\ &\quad \leq \frac{2}{(1+q_{1})(1+q_{2})} \bigl[\Psi (A,C)+q_{2}\Psi (A,D)+q_{1} \Psi (B,C)+q_{1}q_{2}\Psi (B,D) \bigr]. \end{aligned}$$

(2.10)

Combining inequalities (2.7), (2.8), (2.9) and (2.10), we directly obtain our desired inequality. □

Remark 2.1

From Theorem 2.1, we can deduce the following midpoint special cases.

• If $A=\frac{q_{1}a+b}{1+q_{1}}$, $B=b$, $C=\frac{q_{2}d+c}{1+q_{2}}$, $D=d$, then

  $$\begin{aligned} &\Psi \biggl( \frac{q_{1}a+(1+q_{1}+q_{1}^{2})b}{(1+q_{1})^{2}}, \frac{c+(2q_{2}+q_{2}^{2})d}{(1+q_{2})^{2}} \biggr) \\ &\quad \leq \frac{1+q_{1}}{2q_{1}(b-a)} \int _{ \frac{q_{1}a+b}{1+q_{1}}}^{b}\Psi \biggl( u, \frac{c+(2q_{2}+q_{2}^{2})d}{(1+q_{2})^{2}} \biggr) \text{ }^{b}\,d_{q_{1}}u \\ &\qquad {}+ \frac{1+q_{2}}{2(d-c)}\int _{\frac{q_{2}d+c}{1+q_{2}}}^{d}\Psi \biggl( \frac{q_{1}a+(1+q_{1}+q_{1}^{2})b}{(1+q_{1})^{2}},v \biggr) \text{ }^{d}\,d_{q_{2}}v \\ &\quad \leq \frac{(1+q_{1})(1+q_{2})}{q_{1} ( b-a ) ( d-c ) } \int _{\frac{q_{1}a+b}{1+q_{1}}}^{b}\int _{\frac{q_{2}d+c}{1+q_{2}}}^{d}\Psi ( u,v ) \text{ }^{d}\,d_{q_{2}}v\text{ }^{b}\,d_{q_{1}}u \\ &\quad \leq \frac{1+q_{1}}{2q_{1}(1+q_{2})(b-a)} \int _{ \frac{q_{1}a+b}{1+q_{1}}}^{b} \Psi \biggl( u, \frac{q_{2}d+c}{1+q_{2}} \biggr)\text{ }^{b}\,d_{q_{1}}u \\ &\qquad {} + \frac{q_{2}(1+q_{1})}{2q_{1}(1+q_{2})(b-a)} \int _{ \frac{q_{1}a+b}{1+q_{1}}}^{b}\Psi ( u,d ) \text{ }^{b}\,d_{q_{1}}u \\ &\qquad {}+\frac{1+q_{2}}{2(1+q_{1})(d-c)} \int _{ \frac{q_{2}d+c}{1+q_{2}}}^{d}\Psi \biggl( \frac{q_{1}a+b}{1+q_{1}},v \biggr)\text{ }^{d}\,d_{q_{2}}v \\ &\qquad {} + \frac{q_{1}(1+q_{2})}{2(1+q_{1})(d-c)} \int _{ \frac{q_{2}d+c}{1+q_{2}}}^{d}\Psi ( b,v ) \text{ }^{d}\,d_{q_{2}}v \\ &\quad \leq \frac{\Psi (\frac{q_{1}a+b}{1+q_{1}},\frac{q_{2}d+c}{1+q_{2}} )+q_{2}\Psi (\frac{q_{1}a+b}{1+q_{1}},d )+q_{1}\Psi (b,\frac{q_{2}d+c}{1+q_{2}} )+q_{1}q_{2}\Psi (b,d ) }{(1+q_{1})(1+q_{2})}. \end{aligned}$$

  (2.11)

• If $A=\frac{q_{1}b+a}{1+q_{1}}$, $B=b$, $C=\frac{q_{2}d+c}{1+q_{2}}$, $D=d$, then

  $$\begin{aligned} &\Psi \biggl( \frac{a+(2q_{1}+q_{1}^{2})b}{(1+q_{1})^{2}}, \frac{c+(2q_{2}+q_{2}^{2})d}{(1+q_{2})^{2}} \biggr) \\ &\quad \leq \frac{1+q_{1}}{2(b-a)} \int _{\frac{q_{1}b+a}{1+q_{1}}}^{b} \Psi \biggl( u, \frac{c+(2q_{2}+q_{2}^{2})d}{(1+q_{2})^{2}} \biggr) \text{ }^{b}\,d_{q_{1}}u \\ &\qquad {}+ \frac{1+q_{2}}{2(d-c)}\int _{\frac{q_{2}d+c}{1+q_{2}}}^{d}\Psi \biggl( \frac{a+(2q_{1}+q_{1}^{2})b}{(1+q_{1})^{2}},v \biggr) \text{ }^{d}\,d_{q_{2}}v \\ &\quad \leq \frac{(1+q_{1})(1+q_{2})}{ ( b-a ) ( d-c ) } \int _{\frac{q_{1}b+a}{1+q_{1}}}^{b}\int _{\frac{q_{2}d+c}{1+q_{2}}}^{d}\Psi ( u,v ) \text{ }^{d}\,d_{q_{2}}v\text{ }^{b}\,d_{q_{1}}u \\ &\quad \leq \frac{1+q_{1}}{2(1+q_{2})(b-a)} \int _{ \frac{q_{1}b+a}{1+q_{1}}}^{b} \Psi \biggl(u, \frac{q_{2}d+c}{1+q_{2}} \biggr)\text{ }^{b}\,d_{q_{1}}u \\ &\qquad {} +\frac{q_{2}(1+q_{1})}{2(1+q_{2})(b-a)} \int _{\frac{q_{1}b+a}{1+q_{1}}}^{b}\Psi ( u,d ) \text{ }^{b}\,d_{q_{1}}u \\ &\qquad {}+\frac{1+q_{2}}{2(1+q_{1})(d-c)} \int _{ \frac{q_{2}d+c}{1+q_{2}}}^{d} \Psi \biggl( \frac{q_{1}b+a}{1+q_{1}},v \biggr)\text{ }^{d}\,d_{q_{2}}v \\ &\qquad {} +\frac{q_{1}(1+q_{2})}{2(1+q_{1})(d-c)} \int _{\frac{q_{2}d+c}{1+q_{2}}}^{d}\Psi ( b,v ) \text{ }^{d}\,d_{q_{2}}v \\ &\quad \leq \frac{\Psi (\frac{q_{1}b+a}{1+q_{1}},\frac{q_{2}d+c}{1+q_{2}} )+q_{2}\Psi (\frac{q_{1}b+a}{1+q_{1}},d )+q_{1}\Psi (b,\frac{q_{2}d+c}{1+q_{2}} )+q_{1}q_{2}\Psi (b,d ) }{(1+q_{1})(1+q_{2})}. \end{aligned}$$

  (2.12)

• If $A=\frac{q_{1}a+b}{1+q_{1}}$, $B=b$, $C=\frac{q_{2}c+d}{1+q_{2}}$, $D=d$, then

  $$\begin{aligned} &\Psi \biggl( \frac{q_{1}a+(1+q_{1}+q_{1}^{2})b}{(1+q_{1})^{2}}, \frac{q_{2}c+(1+q_{2}+q_{2}^{2})d}{(1+q_{2})^{2}} \biggr) \\ &\quad \leq \frac{1+q_{1}}{2q_{1}(b-a)} \int _{ \frac{q_{1}a+b}{1+q_{1}}}^{b}\Psi \biggl( u, \frac{c+(2q_{2}+q_{2}^{2})d}{(1+q_{2})^{2}} \biggr) \text{ }^{b}\,d_{q_{1}}u \\ &\qquad {}+ \frac{1+q_{2}}{2q_{2}(d-c)} \int _{\frac{q_{2}c+d}{1+q_{2}}}^{d} \Psi \biggl( \frac{q_{1}a+(1+q_{1}+q_{1}^{2})b}{(1+q_{1})^{2}},v \biggr) \text{ }^{d}\,d_{q_{2}}v \\ &\quad \leq \frac{(1+q_{1})(1+q_{2})}{q_{1}q_{2} ( b-a ) ( d-c ) } \int _{\frac{q_{1}a+b}{1+q_{1}}}^{b} \int _{ \frac{q_{2}c+d}{1+q_{2}}}^{d}\Psi ( u,v ) \text{ }^{d}\,d_{q_{2}}v \text{ }^{b}\,d_{q_{1}}u \\ &\quad \leq \frac{1+q_{1}}{2q_{1}(1+q_{2})(b-a)} \int _{ \frac{q_{1}a+b}{1+q_{1}}}^{b} \Psi \biggl(u, \frac{q_{2}d+c}{1+q_{2}} \biggr)\text{ }^{b}\,d_{q_{1}}u \\ &\qquad {} + \frac{q_{2}(1+q_{1})}{2q_{1}(1+q_{2})(b-a)} \int _{ \frac{q_{1}a+b}{1+q_{1}}}^{b}\Psi ( u,d ) \text{ }^{b}\,d_{q_{1}}u \\ &\qquad {}+\frac{1+q_{2}}{2q_{2}(1+q_{1})(d-c)} \int _{ \frac{q_{2}c+d}{1+q_{2}}}^{d} \Psi \biggl( \frac{q_{1}a+b}{1+q_{1}},v \biggr)\text{ }^{d}\,d_{q_{2}}v \\ &\qquad {} + \frac{q_{1}(1+q_{2})}{2q_{2}(1+q_{1})(d-c)} \int _{ \frac{q_{2}c+d}{1+q_{2}}}^{d}\Psi ( b,v ) \text{ }^{d}\,d_{q_{2}}v \\ &\quad \leq \frac{\Psi (\frac{q_{1}a+b}{1+q_{1}},\frac{q_{2}c+d}{1+q_{2}} )+q_{2}\Psi (\frac{q_{1}a+b}{1+q_{1}},d )+q_{1}\Psi (b,\frac{q_{2}c+d}{1+q_{2}} )+q_{1}q_{2}\Psi (b,d ) }{(1+q_{1})(1+q_{2})}. \end{aligned}$$

  (2.13)

• If $A=\frac{q_{1}b+a}{1+q_{1}}$, $B=b$, $C=\frac{q_{2}c+d}{1+q_{2}}$, $D=d$, then

  $$\begin{aligned} &\Psi \biggl( \frac{a+(2q_{1}+q_{1}^{2})b}{(1+q_{1})^{2}}, \frac{q_{2}c+(1+q_{2}+q_{2}^{2})d}{(1+q_{2})^{2}} \biggr) \\ &\quad \leq \frac{1+q_{1}}{2(b-a)} \int _{\frac{q_{1}b+a}{1+q_{1}}}^{b} \Psi \biggl( u, \frac{q_{2}c+(1+q_{2}+q_{2}^{2})d}{(1+q_{2})^{2}} \biggr) \text{ }^{b}\,d_{q_{1}}u \\ &\qquad {}+ \frac{1+q_{2}}{2q_{2}(d-c)} \int _{\frac{q_{2}c+d}{1+q_{2}}}^{d}\Psi \biggl( \frac{a+(2q_{1}+q_{1}^{2})b}{(1+q_{1})^{2}},v \biggr) \text{ }^{d}\,d_{q_{2}}v \\ &\quad \leq \frac{(1+q_{1})(1+q_{2})}{q_{2} ( b-a ) ( d-c ) } \int _{\frac{q_{1}b+a}{1+q_{1}}}^{b}\int _{\frac{q_{2}c+d}{1+q_{2}}}^{d}\Psi ( u,v ) \text{ }^{d}\,d_{q_{2}}v\text{ }^{b}\,d_{q_{1}}u \\ &\quad \leq \frac{1+q_{1}}{2(1+q_{2})(b-a)} \int _{ \frac{q_{1}b+a}{1+q_{1}}}^{b} \Psi \biggl(u, \frac{q_{2}d+c}{1+q_{2}} \biggr)\text{ }^{b}\,d_{q_{1}}u \\ &\qquad {} +\frac{q_{2}(1+q_{1})}{2(1+q_{2})(b-a)} \int _{\frac{q_{1}b+a}{1+q_{1}}}^{b}\Psi ( u,d ) \text{ }^{b}\,d_{q_{1}}u \\ &\qquad {}+\frac{1+q_{2}}{2q_{2}(1+q_{1})(d-c)} \int _{ \frac{q_{2}c+d}{1+q_{2}}}^{d} \Psi \biggl( \frac{q_{1}b+a}{1+q_{1}},v \biggr)\text{ }^{d}\,d_{q_{2}}v \\ &\qquad {} + \frac{q_{1}(1+q_{2})}{2q_{2}(1+q_{1})(d-c)} \int _{ \frac{q_{2}c+d}{1+q_{2}}}^{d}\Psi ( b,v ) \text{ }^{d}\,d_{q_{2}}v \\ &\quad \leq \frac{\Psi (\frac{q_{1}b+a}{1+q_{1}},\frac{q_{2}c+d}{1+q_{2}} )+q_{2}\Psi (\frac{q_{1}b+a}{1+q_{1}},d )+q_{1}\Psi (b,\frac{q_{2}c+d}{1+q_{2}} )+q_{1}q_{2}\Psi (b,d ) }{(1+q_{1})(1+q_{2})}. \end{aligned}$$

  (2.14)

The application of the Theorems 1.2 and 1.3 leads to the following result.

Theorem 2.2

Suppose that $\Psi :\Delta \rightarrow \mathbb{R}_{e}$ is a coordinated convex function on Δ and $\Psi \in L_{1}(\Delta )$. Then we have

$$\begin{aligned}& \Psi \biggl( \frac{A+q_{1}B}{1+q_{1}},\frac{q_{2}C+D}{1+q_{2}} \biggr) \\& \quad \leq \frac{1}{2} \biggl[ \frac{1}{B-A} \int _{A}^{B}\Psi \biggl( u, \frac{q_{2}C+D}{1+q_{2}} \biggr) \textit{ }^{B}\,d_{q_{1}}u+ \frac{1}{D-C} \int _{C}^{D}\Psi \biggl( \frac{A+q_{1}B}{1+q_{1}},v \biggr) \textit{ }_{C}\,d_{q_{2}}v \biggr] \\& \quad \leq \frac{1}{ ( B-A ) ( D-C ) } \int _{A}^{B} \int _{C}^{D}\Psi ( u,v ) \textit{ }_{C}\,d_{q_{2}}v \textit{ }^{B}\,d_{q_{1}}u \\& \quad \leq \biggl[\frac{q_{2}}{2(1+q_{2})(B-A)} \int _{A}^{B} \Psi (u,C )\textit{ }^{B}\,d_{q_{1}}u +\frac{1}{2(1+q_{2})(B-A)} \int _{A}^{B}\Psi ( u,D ) \textit{ }^{B}\,d_{q_{1}}u \\& \qquad {}+\frac{1}{2(1+q_{1})(D-C)} \int _{C}^{D} \Psi ( A,v )\textit{ }_{C}\,d_{q_{2}}v +\frac{q_{1}}{2(1+q_{1})(D-C)} \int _{C}^{D}\Psi ( B,v ) \textit{ }_{C}\,d_{q_{2}}v \biggr] \\& \quad \leq \frac{q_{2}\Psi ( A,C ) +\Psi ( A,D ) +q_{1}q_{2}\Psi ( B,C ) +q_{1}\Psi ( B,D ) }{(1+q_{1})(1+q_{2})}. \end{aligned}$$

(2.15)

Proof

The proof is omitted. □

Remark 2.2

From Theorem 2.2, we can deduce the following midpoint special cases.

• If $A=\frac{q_{1}a+b}{1+q_{1}}$, $B=b$, $C=c$, $D=\frac{q_{2}d+c}{1+q_{2}}$, then

  $$\begin{aligned} &\Psi \biggl( \frac{q_{1}a+(1+q_{1}+q_{1}^{2})b}{(1+q_{1})^{2}}, \frac{(1+q_{2}+q_{2}^{2})c+q_{2}d}{(1+q_{2})^{2}} \biggr) \\ &\quad \leq \frac{1+q_{1}}{2q_{1}(b-a)} \int _{ \frac{q_{1}a+b}{1+q_{1}}}^{b}\Psi \biggl( u, \frac{(1+q_{2}+q_{2}^{2})c+q_{2}d}{(1+q_{2})^{2}} \biggr) \text{ }^{b}\,d_{q_{1}}u \\ &\qquad {}+ \frac{1+q_{2}}{2q_{2}(d-c)} \int _{c}^{ \frac{q_{2}d+c}{1+q_{2}}}\Psi \biggl( \frac{q_{1}a+(1+q_{1}+q_{1}^{2})b}{(1+q_{1})^{2}},v \biggr) \text{ }_{c}\,d_{q_{2}}v \\ &\quad \leq \frac{(1+q_{1})(1+q_{2})}{q_{1}q_{2} ( b-a ) ( d-c ) } \int _{\frac{q_{1}a+b}{1+q_{1}}}^{b} \int _{c}^{ \frac{q_{2}d+c}{1+q_{2}}}\Psi ( u,v ) \text{ }_{c}\,d_{q_{2}}v \text{ }^{b}\,d_{q_{1}}u \\ &\quad \leq \frac{q_{2}(1+q_{1})}{2q_{1}(1+q_{2})(b-a)} \int _{ \frac{q_{1}a+b}{1+q_{1}}}^{b} \Psi ( u,c )\text{ }^{b}\,d_{q_{1}}u \\ &\qquad {} +\frac{1+q_{1}}{2q_{1}(1+q_{2})(b-a)} \int _{ \frac{q_{1}a+b}{1+q_{1}}}^{b}\Psi \biggl( u, \frac{q_{2}d+c}{1+q_{2}} \biggr) \text{ }^{b}\,d_{q_{1}}u \\ &\qquad {}+\frac{1+q_{2}}{2q_{2}(1+q_{1})(d-c)} \int _{c}^{ \frac{q_{2}d+c}{1+q_{2}}} \Psi \biggl( \frac{q_{1}a+b}{1+q_{1}},v \biggr)\text{ }_{c}\,d_{q_{2}}v \\ &\qquad {} + \frac{q_{1}(1+q_{2})}{2q_{2}(1+q_{1})(d-c)} \int _{c}^{ \frac{q_{2}d+c}{1+q_{2}}}\Psi ( b,v ) \text{ }_{c}\,d_{q_{2}}v \\ &\quad \leq \frac{q_{2}\Psi ( \frac{q_{1}a+b}{1+q_{1}},c ) +\Psi ( \frac{q_{1}a+b}{1+q_{1}},\frac{q_{2}d+c}{1+q_{2}} ) +q_{1}q_{2}\Psi ( b,c )+q_{1}\Psi ( b,\frac{q_{2}d+c}{1+q_{2}} ) }{(1+q_{1})(1+q_{2})}. \end{aligned}$$

  (2.16)

• If $A=\frac{q_{1}b+a}{1+q_{1}}$, $B=b$, $C=c$, $D=\frac{q_{2}d+c}{1+q_{2}}$, then

  $$\begin{aligned} &\Psi \biggl( \frac{a+(2q_{1}+q_{1}^{2})b}{(1+q_{1})^{2}}, \frac{(1+q_{2}+q_{2}^{2})c+q_{2}d}{(1+q_{2})^{2}} \biggr) \\ &\quad \leq \frac{1+q_{1}}{2(b-a)} \int _{\frac{q_{1}b+a}{1+q_{1}}}^{b} \Psi \biggl( u, \frac{(1+q_{2}+q_{2}^{2})c+q_{2}d}{(1+q_{2})^{2}} \biggr) \text{ }^{b}\,d_{q_{1}}u \\ &\qquad {}+ \frac{1+q_{2}}{2q_{2}(d-c)} \int _{c}^{\frac{q_{2}d+c}{1+q_{2}}}\Psi \biggl( \frac{a+(2q_{1}+q_{1}^{2})b}{(1+q_{1})^{2}},v \biggr) \text{ }_{c}\,d_{q_{2}}v \\ &\quad \leq \frac{(1+q_{1})(1+q_{2})}{q_{2} ( b-a ) ( d-c ) } \int _{\frac{q_{1}b+a}{1+q_{1}}}^{b} \int _{c}^{ \frac{q_{2}d+c}{1+q_{2}}}\Psi ( u,v ) \text{ }_{c}\,d_{q_{2}}v \text{ }^{b}\,d_{q_{1}}u \\ &\quad \leq \frac{q_{2}(1+q_{1})}{2(1+q_{2})(b-a)} \int _{ \frac{q_{1}b+a}{1+q_{1}}}^{b} \Psi (u,c )\text{ }^{b}\,d_{q_{1}}u \\ &\qquad {} +\frac{1+q_{1}}{2(1+q_{2})(b-a)} \int _{ \frac{q_{1}b+a}{1+q_{1}}}^{b}\Psi \biggl( u, \frac{q_{2}d+c}{1+q_{2}} \biggr) \text{ }^{b}\,d_{q_{1}}u \\ &\qquad {}+\frac{1+q_{2}}{2q_{2}(1+q_{1})(d-c)} \int _{c}^{ \frac{q_{2}d+c}{1+q_{2}}} \Psi \biggl( \frac{q_{1}b+a}{1+q_{1}},v \biggr)\text{ }_{c}\,d_{q_{2}}v \\ &\qquad {} + \frac{q_{1}(1+q_{2})}{2q_{2}(1+q_{1})(d-c)} \int _{c}^{ \frac{q_{2}d+c}{1+q_{2}}}\Psi ( b,v ) \text{ }_{c}\,d_{q_{2}}v \\ &\quad \leq \frac{q_{2}\Psi ( \frac{q_{1}b+a}{1+q_{1}},c ) +\Psi ( \frac{q_{1}b+a}{1+q_{1}},\frac{q_{2}d+c}{1+q_{2}} ) +q_{1}q_{2}\Psi ( b,c )+q_{1}\Psi ( b,\frac{q_{2}d+c}{1+q_{2}} ) }{(1+q_{1})(1+q_{2})}. \end{aligned}$$

  (2.17)

• If $A=\frac{q_{1}a+b}{1+q_{1}}$, $B=b$, $C=c$, $D=\frac{q_{2}c+d}{1+q_{2}}$, then

  $$\begin{aligned} &\Psi \biggl( \frac{q_{1}a+(1+q_{1}+q_{1}^{2})b}{(1+q_{1})^{2}}, \frac{c(2q_{2}+q_{2}^{2})+d}{(1+q_{2})^{2}} \biggr) \\ &\quad \leq \frac{1+q_{1}}{2q_{1}(b-a)} \int _{ \frac{q_{1}a+b}{1+q_{1}}}^{b}\Psi \biggl( u, \frac{c(2q_{2}+q_{2}^{2})+d}{(1+q_{2})^{2}} \biggr) \text{ }^{b}\,d_{q_{1}}u \\ &\qquad {}+ \frac{1+q_{2}}{2(d-c)} \int _{c}^{\frac{q_{2}c+d}{1+q_{2}}} \Psi \biggl( \frac{q_{1}a+(1+q_{1}+q_{1}^{2})b}{(1+q_{1})^{2}},v \biggr) \text{ }_{c}\,d_{q_{2}}v \\ &\quad \leq \frac{(1+q_{1})(1+q_{2})}{q_{1} ( b-a ) ( d-c ) } \int _{\frac{q_{1}a+b}{1+q_{1}}}^{b} \int _{c}^{ \frac{q_{2}c+d}{1+q_{2}}}\Psi ( u,v ) \text{ }_{c}\,d_{q_{2}}v \text{ }^{b}\,d_{q_{1}}u \\ &\quad \leq \frac{q_{2}(1+q_{1})}{2q_{1}(1+q_{2})(b-a)} \int _{ \frac{q_{1}a+b}{1+q_{1}}}^{b} \Psi ( u,c )\text{ }^{b}\,d_{q_{1}}u \\ &\qquad {} +\frac{1+q_{1}}{2q_{1}(1+q_{2})(b-a)} \int _{ \frac{q_{1}a+b}{1+q_{1}}}^{b}\Psi \biggl( u, \frac{q_{2}d+c}{1+q_{2}} \biggr) \text{ }^{b}\,d_{q_{1}}u \\ &\qquad {}+\frac{1+q_{2}}{2(1+q_{1})(d-c)} \int _{c}^{ \frac{q_{2}c+d}{1+q_{2}}} \Psi \biggl( \frac{q_{1}a+b}{1+q_{1}},v \biggr)\text{ }_{c}\,d_{q_{2}}v \\ &\qquad {} +\frac{q_{1}(1+q_{2})}{2(1+q_{1})(d-c)} \int _{c}^{\frac{q_{2}c+d}{1+q_{2}}}\Psi ( b,v ) \text{ }_{c}\,d_{q_{2}}v \\ &\quad \leq \frac{q_{2}\Psi ( \frac{q_{1}a+b}{1+q_{1}},c )+\Psi ( \frac{q_{1}a+b}{1+q_{1}},\frac{q_{2}c+d}{1+q_{2}} ) +q_{1}q_{2}\Psi ( b,c )+q_{1}\Psi ( b,\frac{q_{2}c+d}{1+q_{2}} ) }{(1+q_{1})(1+q_{2})}. \end{aligned}$$

  (2.18)

• If $A=\frac{q_{1}b+a}{1+q_{1}}$, $B=b$, $C=c$, $D=\frac{q_{2}c+d}{1+q_{2}}$, then

  $$\begin{aligned} &\Psi \biggl( \frac{a+(2q_{1}+q_{1}^{2})b}{(1+q_{1})^{2}}, \frac{(2q_{2}+q_{2}^{2})c+d}{(1+q_{2})^{2}} \biggr) \\ &\quad \leq \frac{1+q_{1}}{2(b-a)} \int _{\frac{q_{1}b+a}{1+q_{1}}}^{b} \Psi \biggl( u, \frac{(2q_{2}+q_{2}^{2})c+d}{(1+q_{2})^{2}} \biggr) \text{ }^{b}\,d_{q_{1}}u \\ &\qquad {}+ \frac{1+q_{2}}{2(d-c)} \int _{c}^{ \frac{q_{2}c+d}{1+q_{2}}}\Psi \biggl( \frac{a+(2q_{1}+q_{1}^{2})b}{(1+q_{1})^{2}},v \biggr) \text{ }_{c}\,d_{q_{2}}v \\ &\quad \leq \frac{(1+q_{1})(1+q_{2})}{ ( b-a ) ( d-c ) } \int _{\frac{q_{1}b+a}{1+q_{1}}}^{b} \int _{c}^{ \frac{q_{2}c+d}{1+q_{2}}}\Psi ( u,v ) \text{ }_{c}\,d_{q_{2}}v \text{ }^{b}\,d_{q_{1}}u \\ &\quad \leq \frac{q_{2}(1+q_{1})}{2(1+q_{2})(b-a)} \int _{ \frac{q_{1}b+a}{1+q_{1}}}^{b} \Psi ( u,c )\text{ }^{b}\,d_{q_{1}}u \\ &\qquad {} +\frac{1+q_{1}}{2(1+q_{2})(b-a)} \int _{ \frac{q_{1}b+a}{1+q_{1}}}^{b}\Psi \biggl( u, \frac{q_{2}d+c}{1+q_{2}} \biggr) \text{ }^{b}\,d_{q_{1}}u \\ &\qquad {}+\frac{1+q_{2}}{2(1+q_{1})(d-c)} \int _{c}^{ \frac{q_{2}c+d}{1+q_{2}}} \Psi \biggl( \frac{q_{1}b+a}{1+q_{1}},v \biggr)\text{ }_{c}\,d_{q_{2}}v \\ &\qquad {} +\frac{q_{1}(1+q_{2})}{2(1+q_{1})(d-c)} \int _{c}^{\frac{q_{2}c+d}{1+q_{2}}}\Psi ( b,v ) \text{ }_{c}\,d_{q_{2}}v \\ &\quad \leq \frac{q_{2}\Psi ( \frac{q_{1}b+a}{1+q_{1}},c ) +\Psi ( \frac{q_{1}b+a}{1+q_{1}},\frac{q_{2}c+d}{1+q_{2}} ) +q_{1}q_{2}\Psi ( b,c )+q_{1}\Psi ( b,\frac{q_{2}c+d}{1+q_{2}} ) }{(1+q_{1})(1+q_{2})}. \end{aligned}$$

  (2.19)

Finally, we have the following inequalities by the utilizing Theorems 1.2 and 1.3.

Theorem 2.3

Let $\Psi :\Delta \rightarrow \mathbb{R}_{e}$ be a coordinated convex function on Δ and $q_{1},q_{2}\in (0,1)$. Then one has

$$\begin{aligned} &\Psi \biggl( \frac{q_{1}A+B}{1+q_{1}},\frac{C+q_{2}D}{1+q_{2}} \biggr) \\ &\quad \leq \frac{1}{2} \biggl[ \frac{1}{B-A} \int _{A}^{B}\Psi \biggl( u, \frac{C+q_{2}D}{1+q_{2}} \biggr) \textit{ }_{A}\,d_{q_{1}}u \\ &\qquad{}+ \frac{1}{D-C} \int _{C}^{D}\Psi \biggl( \frac{q_{1}A+B}{1+q_{1}},v \biggr) \textit{ }^{D}\,d_{q_{2}}v \biggr] \\ &\quad \leq \frac{1}{ ( B-A ) ( D-C ) } \int _{A}^{B} \int _{C}^{D}\Psi ( u,v ) \textit{ }^{D}\,d_{q_{2}}v \textit{ }_{A}\,d_{q_{1}}u \\ &\quad \leq \biggl[\frac{1}{2(1+q_{2})(B-A)} \int _{A}^{B} \Psi (u,C )\textit{ }_{A}\,d_{q_{1}}u \\ &\qquad{} + \frac{q_{2}}{2(1+q_{2})(B-A)} \int _{A}^{B}\Psi ( u,D ) \textit{ }_{A}\,d_{q_{1}}u \\ &\qquad {}+\frac{q_{1}}{2(1+q_{1})(D-C)} \int _{C}^{D} \Psi ( A,v )\textit{ }^{D}\,d_{q_{2}}v \\ &\qquad{} +\frac{1}{2(1+q_{1})(D-C)} \int _{C}^{D}\Psi ( B,v ) \textit{ }^{D}\,d_{q_{2}}v \biggr] \\ &\quad \leq \frac{q_{1}\Psi ( A,C ) +q_{1}q_{2}\Psi ( A,D ) +\Psi ( B,C ) +q_{2}\Psi ( B,D ) }{(1+q_{1})(1+q_{2})}. \end{aligned}$$

(2.20)

Proof

This proof is similar to our proof of Theorem 2.1, so we omit it. □

Remark 2.3

From Theorem 2.3, we can deduce the following midpoint special cases.

• If $A=a$, $B=\frac{q_{1}a+b}{1+q_{1}}$, $C=\frac{q_{2}d+c}{1+q_{2}}$, $D=d$, then

  $$\begin{aligned} &\Psi \biggl( \frac{(2q_{1}+q_{1}^{2})a+b}{(1+q_{1})^{2}}, \frac{c+(2q_{2}+q_{2}^{2})d}{(1+q_{2})^{2}} \biggr) \\ &\quad \leq \frac{1+q_{1}}{2(b-a)} \int _{a}^{ \frac{q_{1}a+b}{1+q_{1}}}\Psi \biggl( u, \frac{c+(2q_{2}+q_{2}^{2})d}{(1+q_{2})^{2}} \biggr) \text{ }_{a}\,d_{q_{1}}u \\ &\qquad {}+ \frac{1+q_{2}}{2(d-c)} \int _{\frac{q_{2}d+c}{1+q_{2}}}^{d} \Psi \biggl( \frac{(2q_{1}+q_{1}^{2})a+b}{(1+q_{1})^{2}},v \biggr) \text{ }^{d}\,d_{q_{2}}v \\ &\quad \leq \frac{(1+q_{1})(1+q_{2})}{ ( b-a ) ( d-c ) } \int _{a}^{\frac{q_{1}a+b}{1+q_{1}}} \int _{ \frac{q_{2}d+c}{1+q_{2}}}^{d}\Psi ( u,v ) \text{ }^{d}\,d_{q_{2}}v \text{ }_{a}\,d_{q_{1}}u \\ &\quad \leq \frac{1+q_{1}}{2(1+q_{2})(b-a)} \int _{a}^{ \frac{q_{1}a+b}{1+q_{1}}} \Psi \biggl( u, \frac{q_{2}d+c}{1+q_{2}} \biggr)\text{ }_{a}\,d_{q_{1}}u \\ &\qquad {} +\frac{q_{2}(1+q_{1})}{2(1+q_{2})(b-a)} \int _{a}^{\frac{q_{1}a+b}{1+q_{1}}}\Psi ( u,d ) \text{ }_{a}\,d_{q_{1}}u \\ &\qquad {}+\frac{q_{1}(1+q_{2})}{2(1+q_{1})(d-c)} \int _{ \frac{q_{2}d+c}{1+q_{2}}}^{d} \Psi ( a,v )\text{ }^{d}\,d_{q_{2}}v \\ &\qquad {} +\frac{1+q_{2}}{2(1+q_{1})(d-c)} \int _{ \frac{q_{2}d+c}{1+q_{2}}}^{d}\Psi \biggl( \frac{q_{1}a+b}{1+q_{1}},v \biggr) \text{ }^{d}\,d_{q_{2}}v \\ &\quad \leq \frac{q_{1}\Psi ( a,\frac{q_{2}d+c}{1+q_{2}} ) +q_{1}q_{2}\Psi ( a,d ) +\Psi ( \frac{q_{1}a+b}{1+q_{1}},\frac{q_{2}d+c}{1+q_{2}} ) +q_{2}\Psi ( \frac{q_{1}a+b}{1+q_{1}},d ) }{(1+q_{1})(1+q_{2})}. \end{aligned}$$

  (2.21)

• If $A=a$, $B=\frac{q_{1}b+a}{1+q_{1}}$, $C=\frac{q_{2}d+c}{1+q_{2}}$, $D=d$, then

  $$\begin{aligned} &\Psi \biggl( \frac{(1+q_{1}+q_{1}^{2})a+q_{1}b}{(1+q_{1})^{2}}, \frac{c+(2q_{2}+q_{2}^{2})d}{(1+q_{2})^{2}} \biggr) \\ &\quad \leq \frac{1+q_{1}}{2q_{1}(b-a)} \int _{a}^{ \frac{q_{1}b+a}{1+q_{1}}}\Psi \biggl( u, \frac{c+(2q_{2}+q_{2}^{2})d}{(1+q_{2})^{2}} \biggr) \text{ }_{a}\,d_{q_{1}}u \\ &\qquad {}+ \frac{1+q_{2}}{2(d-c)} \int _{\frac{q_{2}d+c}{1+q_{2}}}^{d} \Psi \biggl( \frac{(1+q_{1}+q_{1}^{2})a+q_{1}b}{(1+q_{1})^{2}},v \biggr) \text{ }^{d}\,d_{q_{2}}v \\ &\quad \leq \frac{(1+q_{1})(1+q_{2})}{q_{1} ( b-a ) ( d-c ) } \int _{a}^{\frac{q_{1}b+a}{1+q_{1}}} \int _{ \frac{q_{2}d+c}{1+q_{2}}}^{d}\Psi ( u,v ) \text{ }^{d}\,d_{q_{2}}v \text{ }_{a}\,d_{q_{1}}u \\ &\quad \leq \frac{1+q_{1}}{2q_{1}(1+q_{2})(b-a)} \int _{a}^{ \frac{q_{1}b+a}{1+q_{1}}} \Psi \biggl(u, \frac{q_{2}d+c}{1+q_{2}} \biggr)\text{ }_{a}\,d_{q_{1}}u \\ &\qquad {} + \frac{q_{2}(1+q_{1})}{2q_{1}(1+q_{2})(b-a)} \int _{a}^{ \frac{q_{1}b+a}{1+q_{1}}}\Psi ( u,d ) \text{ }_{a}\,d_{q_{1}}u \\ &\qquad {}+\frac{q_{1}(1+q_{2})}{2(1+q_{1})(d-c)} \int _{ \frac{q_{2}d+c}{1+q_{2}}}^{d} \Psi ( a,v )\text{ }^{d}\,d_{q_{2}}v \\ &\qquad {} +\frac{1+q_{2}}{2(1+q_{1})(d-c)} \int _{ \frac{q_{2}d+c}{1+q_{2}}}^{d}\Psi \biggl( \frac{q_{1}b+a}{1+q_{1}},v \biggr) \text{ }^{d}\,d_{q_{2}}v \\ &\quad \leq \frac{q_{1}\Psi ( a,\frac{q_{2}d+c}{1+q_{2}} ) +q_{1}q_{2}\Psi ( a,d ) +\Psi ( \frac{q_{1}b+a}{1+q_{1}},\frac{q_{2}d+c}{1+q_{2}} ) +q_{2}\Psi ( \frac{q_{1}b+a}{1+q_{1}},d ) }{(1+q_{1})(1+q_{2})}. \end{aligned}$$

  (2.22)

• If $A=a$, $B=\frac{q_{1}a+b}{1+q_{1}}$, $C=\frac{q_{2}c+d}{1+q_{2}}$, $D=d$, then

  $$\begin{aligned} &\Psi \biggl( \frac{(2q_{1}+q_{1}^{2})a+b}{(1+q_{1})^{2}}, \frac{q_{2}c+(1+q_{2}+q_{2}^{2})d}{(1+q_{2})^{2}} \biggr) \\ &\quad \leq \frac{1+q_{1}}{2(b-a)} \int _{a}^{ \frac{q_{1}a+b}{1+q_{1}}}\Psi \biggl( u, \frac{q_{2}c+(1+q_{2}+q_{2}^{2})d}{(1+q_{2})^{2}} \biggr) \text{ }_{a}\,d_{q_{1}}u \\ &\qquad {}+ \frac{1+q_{2}}{2q_{2}(d-c)} \int _{\frac{q_{2}c+d}{1+q_{2}}}^{d} \Psi \biggl( \frac{(2q_{1}+q_{1}^{2})a+b}{(1+q_{1})^{2}},v \biggr) \text{ }^{d}\,d_{q_{2}}v \\ &\quad \leq \frac{(1+q_{1})(1+q_{2})}{q_{2} ( b-a ) ( d-c ) } \int _{a}^{\frac{q_{1}a+b}{1+q_{1}}} \int _{ \frac{q_{2}c+d}{1+q_{2}}}^{d}\Psi ( u,v ) \text{ }^{d}\,d_{q_{2}}v \text{ }_{a}\,d_{q_{1}}u \\ &\quad \leq \frac{1+q_{1}}{2(1+q_{2})(b-a)} \int _{a}^{ \frac{q_{1}a+b}{1+q_{1}}} \Psi \biggl(u, \frac{q_{2}c+d}{1+q_{2}} \biggr)\text{ }_{a}\,d_{q_{1}}u \\ &\qquad {} +\frac{q_{2}(1+q_{1})}{2(1+q_{2})(b-a)} \int _{a}^{\frac{q_{1}a+b}{1+q_{1}}}\Psi ( u,d ) \text{ }_{a}\,d_{q_{1}}u \\ &\qquad {}+\frac{q_{1}(1+q_{2})}{2q_{2}(1+q_{1})(d-c)} \int _{ \frac{q_{2}c+d}{1+q_{2}}}^{d} \Psi ( a,v )\text{ }^{d}\,d_{q_{2}}v \\ &\qquad {} +\frac{1+q_{2}}{2q_{2}(1+q_{1})(d-c)} \int _{ \frac{q_{2}c+d}{1+q_{2}}}^{d}\Psi \biggl( \frac{q_{1}a+b}{1+q_{1}},v \biggr) \text{ }^{d}\,d_{q_{2}}v \\ &\quad \leq \frac{q_{1}\Psi ( a,\frac{q_{2}c+d}{1+q_{2}} ) +q_{1}q_{2}\Psi ( a,d ) +\Psi ( \frac{q_{1}a+b}{1+q_{1}},\frac{q_{2}c+d}{1+q_{2}} ) +q_{2}\Psi ( \frac{q_{1}a+b}{1+q_{1}},d ) }{(1+q_{1})(1+q_{2})}. \end{aligned}$$

  (2.23)

• If $A=a$, $B=\frac{q_{1}b+a}{1+q_{1}}$, $C=\frac{q_{2}c+d}{1+q_{2}}$, $D=d$, then

  $$\begin{aligned} &\Psi \biggl( \frac{(1+q_{1}+q_{1}^{2})a+q_{1}b}{(1+q_{1})^{2}}, \frac{q_{2}c+(1+q_{2}+q_{2}^{2})d}{(1+q_{2})^{2}} \biggr) \\ &\quad \leq \frac{1+q_{1}}{2q_{1}(b-a)} \int _{a}^{ \frac{q_{1}b+a}{1+q_{1}}}\Psi \biggl( u, \frac{q_{2}c+(1+q_{2}+q_{2}^{2})d}{(1+q_{2})^{2}} \biggr) \text{ }_{a}\,d_{q_{1}}u \\ &\qquad {}+ \frac{1+q_{2}}{2q_{2}(d-c)} \int _{\frac{q_{2}c+d}{1+q_{2}}}^{d} \Psi \biggl( \frac{(1+q_{1}+q_{1}^{2})a+q_{1}b}{(1+q_{1})^{2}},v \biggr) \text{ }^{d}\,d_{q_{2}}v \\ &\quad \leq \frac{(1+q_{1})(1+q_{2})}{q_{1}q_{2} ( b-a ) ( d-c ) } \int _{a}^{\frac{q_{1}b+a}{1+q_{1}}} \int _{ \frac{q_{2}c+d}{1+q_{2}}}^{d}\Psi ( u,v ) \text{ }^{d}\,d_{q_{2}}v \text{ }_{a}\,d_{q_{1}}u \\ &\quad \leq \frac{1+q_{1}}{2q_{1}(1+q_{2})(b-a)} \int _{a}^{ \frac{q_{1}b+a}{1+q_{1}}} \Psi \biggl(u, \frac{q_{2}c+d}{1+q_{2}} \biggr)\text{ }_{a}\,d_{q_{1}}u \\ &\qquad {} + \frac{q_{2}(1+q_{1})}{2q_{1}(1+q_{2})(b-a)} \int _{a}^{ \frac{q_{1}b+a}{1+q_{1}}}\Psi ( u,d ) \text{ }_{a}\,d_{q_{1}}u \\ &\qquad {}+\frac{q_{1}(1+q_{2})}{2q_{2}(1+q_{1})(d-c)} \int _{ \frac{q_{2}c+d}{1+q_{2}}}^{d} \Psi ( a,v )\text{ }^{d}\,d_{q_{2}}v \\ &\qquad {} +\frac{1+q_{2}}{2q_{2}(1+q_{1})(d-c)} \int _{ \frac{q_{2}c+d}{1+q_{2}}}^{d}\Psi \biggl( \frac{q_{1}b+a}{1+q_{1}},v \biggr) \text{ }^{d}\,d_{q_{2}}v \\ &\quad \leq \frac{q_{1}\Psi ( a,\frac{q_{2}c+d}{1+q_{2}} ) +q_{1}q_{2}\Psi ( a,d ) +\Psi ( \frac{q_{1}b+a}{1+q_{1}},\frac{q_{2}c+d}{1+q_{2}} ) +q_{2}\Psi ( \frac{q_{1}b+a}{1+q_{1}},d ) }{(1+q_{1})(1+q_{2})}. \end{aligned}$$

  (2.24)

By combining Theorems 1.5, 2.1, 2.2 and Theorem 2.3, we can deduce similar bounds to the inequalities in Theorem 1.1.

Theorem 2.4

Suppose that $\Psi :\Delta \rightarrow \mathbb{R}_{e}$ is a coordinated convex function on Δ and $\Psi\in L_{1}(\Delta )$. Then we have

$$\begin{aligned} &2\Psi \biggl( \frac{A+B}{2},\frac{C+D}{2} \biggr) \\ &\quad \leq \frac{1}{2(B-A)} \biggl[ \int _{A}^{B} \biggl(\Psi \biggl( u, \frac{C+q_{2}D}{1+q_{2}} \biggr)+\Psi \biggl( u, \frac{q_{2}C+D}{1+q_{2}} \biggr) \biggr) \textit{ }_{A}\,d_{q_{1}}u \\ &\qquad {}+ \int _{A}^{B} \biggl(\Psi \biggl( u, \frac{C+q_{2}D}{1+q_{2}} \biggr)+\Psi \biggl( u, \frac{q_{2}C+D}{1+q_{2}} \biggr) \biggr) \textit{ }^{B}\,d_{q_{1}}u \biggr] \\ &\qquad {}+ \frac{1}{2(D-C)} \biggl[ \int _{C}^{D} \biggl(\Psi \biggl( \frac{A+q_{1}B}{1+q_{1}},v \biggr)+\Psi \biggl( \frac{q_{1}A+B}{1+q_{1}},v \biggr) \biggr) \textit{ }_{C}\,d_{q_{2}}v \\ &\qquad {}+ \int _{C}^{D} \biggl(\Psi \biggl( \frac{A+q_{1}B}{1+q_{1}},v \biggr)+\Psi \biggl( \frac{q_{1}A+B}{1+q_{1}},v \biggr) \biggr) \textit{ }^{D}\,d_{q_{2}}v \biggr] \\ &\quad \leq \frac{1}{ ( B-A ) ( D-C ) } \biggl[ \int _{A}^{B} \int _{C}^{D}\Psi ( u,v ) \textit{ }^{D}\,d_{q_{2}}v\textit{ }_{A}\,d_{q_{1}}u+ \int _{A}^{B} \int _{C}^{D}\Psi ( u,v ) \textit{ }^{D}\,d_{q_{2}}v \textit{ }^{B}\,d_{q_{1}}u \\ &\qquad {}+ \int _{A}^{B} \int _{C}^{D}\Psi ( u,v ) \textit{ }_{C}\,d_{q_{2}}v\textit{ }^{B}\,d_{q_{1}}u + \int _{A}^{B} \int _{C}^{D}\Psi ( u,v ) \textit{ }_{C}\,d_{q_{2}}v \textit{ }_{A}\,d_{q_{1}}u\biggr] \\ &\quad \leq \frac{1}{2(B-A)} \int _{A}^{B} \bigl( \Psi (u,C )+\Psi (u,D ) \bigr)\textit{ }_{A}\,d_{q_{1}}u \\ &\qquad {}+ \frac{1}{2(B-A)} \int _{A}^{B} \bigl( \Psi (u,C )+ \Psi (u,D ) \bigr)\textit{ }^{B}\,d_{q_{1}}u \\ &\qquad {}+\frac{1}{2(D-C)} \int _{A}^{B} \bigl( \Psi (A,v )+\Psi (B,v ) \bigr)\textit{ }_{C}\,d_{q_{2}}v \\ &\qquad {}+ \frac{1}{2(D-C)} \int _{A}^{B} \bigl( \Psi (A,v )+ \Psi (B,v ) \bigr)\textit{ }^{D}\,d_{q_{2}}v \biggr] \\ &\quad \leq \Psi ( A,C ) +\Psi ( A,D ) +\Psi ( B,C )+\Psi ( B,D ) . \end{aligned}$$

(2.25)

Proof

It suffices to see that we have, due to the coordinated convexity of Ψ,

$$\begin{aligned} 4\Psi \biggl( \frac{A+B}{2},\frac{C+D}{2} \biggr) ={}&4\Psi \biggl( \frac{q_{1}A+B+A+q_{1}B}{2(1+q_{1})}, \frac{q_{2}C+D+C+q_{2}D}{2(1+q_{2})} \biggr) \\ \leq {}&\Psi \biggl( \frac{q_{1}A+B}{1+q_{1}},\frac{q_{2}C+D}{1+q_{2}} \biggr) +\Psi \biggl( \frac{q_{1}A+B}{1+q_{1}}, \frac{C+q_{2}D}{1+q_{2}} \biggr) \\ &{}+\Psi \biggl( \frac{A+q_{1}B}{1+q_{1}},\frac{q_{2}C+D}{1+q_{2}} \biggr) +\Psi \biggl( \frac{A+q_{1}B}{1+q_{1}}, \frac{C+q_{2}D}{1+q_{2}} \biggr). \end{aligned}$$

Then, we can obtain the desired inequality by utilizing Theorems 1.5, 2.1, 2.2 and Theorem 2.3. □

Remark 2.4

If $q_{1}, q_{2}\to 1^{-}$ in Theorem 2.4, then Theorem 2.4 reduces to Theorem 1.1.

Finally, by utilizing Remarks 1.1, 2.1, 2.2 and Remark 2.3, we obtain the following refinements for previous results.

Theorem 2.5

Let $\Psi :\Delta\rightarrow \mathbb{R}_{e}$ be a coordinated convex function on Δ and $q_{1},q_{2}\in (0,1)$, then we have

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

(2.26)

Proof

By summing up the inequalities (1.30), (2.12), (2.19) and (2.21), we can obtain the second, third, fourth and fifth inequalities of the desired inequality (2.26). The last inequality is the consequence of the coordinated convexity of the function Ψ.

For the first inequality, we note that

$$\begin{aligned} &4\Psi \biggl( \frac{a+b}{2},\frac{c+d}{2} \biggr) \\ &\quad =4\Psi \biggl( \frac{(2q_{1}+q_{1}^{2})a+b+a+(2q_{1}+q_{1}^{2})b}{2(1+q_{1}^{2})}, \frac{(2q_{2}+q_{2}^{2})c+d+c+(2q_{2}+q_{2}^{2})d}{2(1+q_{1}^{2})} \biggr) \\ &\quad \leq \Psi \biggl( \frac{(2q_{1}+q_{1}^{2})a+b}{(1+q_{1})^{2}}, \frac{(2q_{2}+q_{2}^{2})c+d}{(1+q_{2})^{2}} \biggr)+\Psi \biggl( \frac{(2q_{1}+q_{1}^{2})a+b}{(1+q_{1})^{2}}, \frac{(2q_{2}+q_{2}^{2})d+c}{(1+q_{2})^{2}} \biggr) \\ &\qquad {}+\Psi \biggl( \frac{(2q_{1}+q_{1}^{2})b+a}{(1+q_{1})^{2}}, \frac{(2q_{2}+q_{2}^{2})c+d}{(1+q_{2})^{2}} \biggr) \\ &\qquad {}+\Psi \biggl( \frac{(2q_{1}+q_{1}^{2})b+a}{(1+q_{1})^{2}}, \frac{(2q_{2}+q_{2}^{2})d+c}{(1+q_{2})^{2}} \biggr), \end{aligned}$$

(2.27)

which completes our proof. □

In a similar way, by using inequalities (1.29), (2.13), (2.16) and (2.24), we can deduce the following theorem.

Theorem 2.6

Let $\Psi :\Delta\rightarrow \mathbb{R}_{e}$ be a coordinated convex function on Δ and $q_{1},q_{2}\in (0,1)$, then we have

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

(2.28)

In a similar way, by using inequalities (1.31), (2.11), (2.18) and (2.22), we can deduce the following theorem.

Theorem 2.7

Let $\Psi :\Delta\rightarrow \mathbb{R}_{e}$ be Δ a coordinated convex function on Δ and $q_{1},q_{2}\in (0,1)$, then one has

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

Also, by using inequalities (1.28), (2.14), (2.17) and (2.23), we can deduce the following theorem.

Theorem 2.8

Let $\Psi :\Delta\rightarrow \mathbb{R}_{e}$ be a coordinated convex function on Δ and $q_{1},q_{2}\in (0,1)$, then one has

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

Remark 2.5

If $q_{1},q_{2}\to 1^{-}$, then Theorems 2.5–2.8 give to the following inequalities, which is the refinement for the classical inequalities (1.1):

$$\begin{aligned}& 4\Psi \biggl( \frac{a+b}{2},\frac{c+d}{2} \biggr) \\& \quad \leq \Psi \biggl( \frac{3a+b}{4},\frac{3c+d}{4} \biggr)+\Psi \biggl( \frac{3a+b}{4},\frac{3d+c}{4} \biggr) \\& \qquad {}+\Psi \biggl( \frac{a+3b}{4}, \frac{3c+d}{4} \biggr)+\Psi \biggl( \frac{a+3b}{4},\frac{c+3d}{4} \biggr) \\& \quad \leq \frac{1}{b-a} \biggl[ \int _{a}^{b}\Psi \biggl( u, \frac{3c+d}{4} \biggr) \,du+ \int _{a}^{b}\Psi \biggl( u, \frac{c+3d}{4} \biggr) \,du \biggr] \\& \qquad {}+\frac{1}{d-c} \biggl[ \int _{c}^{d}\Psi \biggl( \frac{3a+b}{4},v \biggr) \,dv+ \int _{c}^{d}\Psi \biggl( \frac{a+3b}{4},v \biggr) \,dv \biggr] \\& \leq \frac{4}{ ( b-a ) ( d-c ) } \int _{a}^{b} \int _{c}^{d}\Psi ( u,v ) \,dv\,du \\& \leq \frac{1}{2(b-a)} \biggl[ \int _{a}^{b} \Psi (u,c ) \,du+ 2 \int _{a}^{b}\Psi \biggl( u, \frac{c+d}{2} \biggr) \,du + \int _{a}^{b}\Psi ( u,d ) \,du \biggr] \\& \qquad {} + \frac{1}{2(d-c)} \biggl[ \int _{c}^{d}\Psi ( a,v ) \,dv+2 \int _{c}^{d} \Psi \biggl( \frac{a+b}{2},v \biggr) \,dv+ \int _{c}^{d} \Psi ( b,v ) \,dv \biggr] \\& \quad \leq \frac{ [\Psi ( a,c )+\Psi (a,d )+\Psi (b,c)+\Psi ( b,d ) ]}{4}+ \frac{2\Psi ( a,\frac{c+d}{2} )+2\Psi ( b,\frac{c+d}{2} ) }{4} \\& \qquad {}+ \frac{2\Psi ( \frac{a+b}{2},c )+2\Psi (\frac{a+b}{2},d ) }{4} +\frac{4\Psi ( \frac{a+b}{2},\frac{c+d}{2} )}{4} \\& \quad \leq \Psi ( a,c )+\Psi (a,d )+\Psi (b,c)+\Psi ( b,d ). \end{aligned}$$

(2.29)

3 Conclusion

In this study, we have extended the definition of q-derivatives and q-integrals over the interval $[A,B]$ of the real lines. We have considered new $H-H$ inequalities in the context of q-calculus. For the desired results, we have developed an inequality with the same lower and upper estimates as in classical Theorem 1.1. Also, we have established new midpoint $H-H$ type inequalities, which confirm the refinements to the previously known inequalities.

Our results suggest that two different partitions exist for midpoint type inequalities in the q-analogues. Indeed, for the interval $[A,B]$, we have $A\leq \frac{qA+B}{1+q}\leq B$ and $A\leq \frac{A+qB}{1+q}\leq B$ with $q\in (0,1)$. Similarly for $[C,D]$, we have $C\leq \frac{qC+D}{1+q}\leq D$ and $C\leq \frac{C+qD}{1+q}\leq D$ with $q\in (0,1)$. The last four inequalities use all partitions for the desired results. We believe that the results of this paper can be extended to establish new inequalities via different kinds of convex functions in the premises of q-calculus.