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.
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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
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]
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
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
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
where \(q\in (0,1)\). The well-known Jackson integral of a real valued function Ψ is given by the following series expansion:
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
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
Definition 1.3
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
If \(k_{1}=0\), then
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
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
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
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
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
and
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
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:
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
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
and
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
and
Example 1.1
All the \(q_{1}q_{2}\)-integrals are different for general functions. For instance,
Furthermore,
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
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
or
Integrating the inequality over \([A,B]\), we have
Now, by the convexity of \(\Psi _{v}\), by the Theorem 1.3, we have
or
Evaluating the average integral over \([C,D]\), we have
Adding (2.5) and (2.6), we get
Again, by the convexity and applying the first inequality of Theorem 1.3, we have
and
Similarly, applying the second inequality of Theorem 1.3 and convexity of \(\Psi _{u}\) and \(\Psi _{v}\), we have
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
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
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
Proof
It suffices to see that we have, due to the coordinated convexity of Ψ,
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
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
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
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
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
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):
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.
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This work was supported by the Taif University Researchers Supporting Project (No. TURSP-2020/155), Taif University, Taif, Saudi Arabia, and the Deanship of Scientific Research at Princess Nourah Bint Abdulrahman University through the Fast-Track Research Funding Program.
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Alqudah, M.A., Kashuri, A., Mohammed, P.O. et al. Hermite–Hadamard integral inequalities on coordinated convex functions in quantum calculus. Adv Differ Equ 2021, 264 (2021). https://doi.org/10.1186/s13662-021-03420-x
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DOI: https://doi.org/10.1186/s13662-021-03420-x