Some integral inequalities for co-ordinated harmonically convex functions via fractional integrals

1. Introduction and Preliminaries

For a convex mapping \(\prod:I\rightarrow \mathbb{R}\) on a real interval, for all \(f_1,f_2\in I\) and \(t\in[0,1]\), the inequality

is known as the Hermite-Hadamard inequality [1]. The inequality (1) has been established for several generalized convex functions [2,3,4,5,6,7,8,9]. Dragomir [10] and Sarikaya [11] calculated Hermite-Hadamard inequality for co-ordinated convex functions. They define co-ordinated convex function as:

Definition 1. [10] A function \(\prod:\Delta=[f_1,f_2]\times [g_1,g_2]\subseteq \mathbb{R}^{2} \rightarrow \mathbb{R}\) is called co-ordinate convex on \(\Delta\) with \(f_1 < f_2\) and \(g_1 < g_2\), if the partial functions \begin{equation} \prod_{y}:[f_1,f_2] \rightarrow \mathbb{R}, \prod_{y}(u)=\prod(u,y), and \prod_{x}:[g_1,g_2] \rightarrow \mathbb{R}, \prod_{x}(v)=\prod(x,v), \end{equation} are convex for all \(x\in[f_1,f_2]\) and \(y\in [g_1,g_2]\).

Sarikaya [11] define the co-ordinated convex function as:

Definition 2. [11] A function \(\prod:\Delta=[f_1,f_2]\times [g_1,g_2]\subseteq \mathbb{R}^{2}\rightarrow \mathbb{R}\) is called coordinate convex on \(\Delta\) with \(f_1< f_2\) and \(g_1< g_2\), if \begin{align*} \begin{split} &\prod(t_1x+(1-t_1)z,t_2 y+(1-t_2)w) \\ &\leq t_1t_2\prod(x,y)+t_1(1-t_2)\prod(x,w)+(1-t_1)t_2 \prod(z,y)+(1-t_1)(1-t_2)\prod(z,w), \end{split} \end{align*} holds for all \(t_1,t_2\in [0,1]\) and \((x,y),(z,w)\in\Delta\).

Every convex function is co-ordinated convex but not conversely [10].

Theorem 3. [10] Let \(\prod:\Delta=[f_1,f_2]\times [g_1,g_2]\subseteq \mathbb{R}^{2} \rightarrow \mathbb{R}\) be convex on \(\Delta\) with \(f_1< f_2\) and \(g_1< g_2\). Then

Definition 4. [12] A function \(\prod:\Delta=[f_1,f_2]\times [g_1,g_2]\subseteq \mathbb{R}^{2} \rightarrow \mathbb{R}\) is called harmonically convex on \(\Delta\) with \(f_1< f_2\) and \(g_1< g_2\), if \begin{equation*} \prod\left(\frac{xz}{t_1x+(1-t_1)z},\frac{yw}{t_2 y+(1-t_2)w}\right) \leq t_1t_2 \prod(x,y)+(1-t_1)(1-t_2)\prod(z,w), \end{equation*} holds for all \(t_1,t_2\in [0,1]\) and \((x,y),(z,w)\in\Delta\).

Definition 5. [12] A function \(\prod:\Delta=[f_1,f_2]\times [g_1,g_2]\subseteq (0,\infty)\times (0,\infty) \rightarrow \mathbb{R}\) is called coordinated harmonically convex on \(\Delta\) with \(f_1< f_2\) and \(g_1< g_2\), if \begin{align*} \begin{split} &\prod\left(\frac{xz}{t_1x+(1-t_1)z},\frac{yw}{t_2 y+(1-t_2)w}\right) \\ &\leq t_1t_2 \prod(x,y)+t_1(1-t_2)\prod(x,w)+(1-t_1)t_2 \prod(z,y)+(1-t_1)(1-t_2)\prod(z,w), \end{split} \end{align*} holds for all \(t_1,t_2\in [0,1]\) and \((x,y),(z,w)\in\Delta\).

Note that, a function \(\prod:\Delta=[f_1,f_2]\times [g_1,g_2]\subseteq (0,\infty)\times (0,\infty) \rightarrow \mathbb{R}\) is called coordinated harmonically convex on \(\Delta\) with \(f_1< f_2\) and \(g_1< g_2\), if the partial functions

\begin{equation} \prod_{y}:[f_1,f_2] \rightarrow \mathbb{R}, \prod_{y}(u)=\prod(u,y), \prod_{x}:[g_1,g_2] \rightarrow \mathbb{R}, \prod_{x}(v)=\prod(x,v), \end{equation} are harmonically convex for all \(x\in[f_1,f_2]\) and \(y\in [g_1,g_2]\), (for more detail, see [9,12]).

Theorem 6. [12] Let \(\prod:\Delta=[f_1,f_2]\times [g_1,g_2]\subseteq (0,\infty)\times (0,\infty) \rightarrow \mathbb{R}\) be co-ordinated harmonically convex on \(\Delta\) with \(f_1< f_2\) and \(g_1< g_2\). Then

Definition 7. [13] Let \(\prod \in L[f_1,f_2]\). The right-hand side and left-hand side Riemann- Liouville fractional integrals \(J^{\alpha}_{f_1+}\prod\) and \(J^{\alpha}_{f_2-}\prod\) of order \(\alpha > 0\) with \(f_2 > f_1\geq 0\) are defined by \begin{equation*} J^{\alpha}_{f_1+}\prod(x)=\frac{1}{\Gamma(\alpha)}\int_{f_1}^{x}(x-t)^{\alpha-1}\prod(t)dt,\ x>f_1, \end{equation*} and \begin{equation*} J^{\alpha}_{f_2-}\prod(x)=\frac{1}{\Gamma(\alpha)}\int_{x}^{f_2}(t-x)^{\alpha-1}\prod(t)dt,\ x< f_2, \end{equation*} respectively, where \(\Gamma(\alpha)\) is the Gamma function defined by \(\Gamma(\alpha)=\int_{0}^{\infty}e^{-t}t^{\alpha-1}dt\).

Theorem 8. [14] Let \(\prod:I\subseteq (0,\infty)\rightarrow \mathbb{R}\) be a function such that \(\prod\in L_1(f_1,f_2)\) where \(f_1,f_2\in I\) with \(f_1< f_2\). If \(\prod\) is harmonocally convex function on \([f_1,f_2]\), then following inequality for fractional integral hold:

where \(\alpha>0\) and \(\Omega(x)=\frac{1}{x}\).

Definition 9. [11] Let \(\prod\in L_{1}([f_1,f_2]\times [g_1,g_2])\). The Riemann-Liouville integrals \(J^{\alpha,\beta}_{f_1+,g_1+}\), \(J^{\alpha,\beta}_{f_1+,g_2-}\), \(J^{\alpha,\beta}_{f_2-,g_1+}\) and \(J^{\alpha,\beta}_{f_2-,g_2-}\) of order \(\alpha,\beta>0\) with \(f_1,g_1\geq 0\) are defined by \begin{equation*} J^{\alpha,\beta}_{f_1+,g_1+}\prod(x,y)=\frac{1}{\Gamma(\alpha)\Gamma(\beta)}\int_{f_1}^{x}\int_{g_1}^{y}(x-t)^{\alpha-1}(y-s)^{\beta-1}\prod(t,s)dsdt,\ x>f_1 \ y>g_1, \end{equation*} \begin{equation*} J^{\alpha,\beta}_{f_1+,g_2-}\prod(x,y)=\frac{1}{\Gamma(\alpha)\Gamma(\beta)}\int_{f_1}^{x}\int_{y}^{g_2}(x-t)^{\alpha-1}(y-s)^{\beta-1}\prod(t,s)dsdt,\ x>f_1 \ y< g_2, \end{equation*} \begin{equation*} J^{\alpha,\beta}_{f_2-,g_1+}\prod(x,y)=\frac{1}{\Gamma(\alpha)\Gamma(\beta)}\int_{x}^{f_2}\int_{g_1}^{y}(x-t)^{\alpha-1}(y-s)^{\beta-1}\prod(t,s)dsdt,\ x< f_2 \ y>g_1, \end{equation*} and \begin{equation*} J^{\alpha,\beta}_{f_2-,g_2-}\prod(x,y)=\frac{1}{\Gamma(\alpha)\Gamma(\beta)}\int_{x}^{f_2}\int_{y}^{g_2}(x-t)^{\alpha-1}(y-s)^{\beta-1}\prod(t,s)dsdt,\ x< f_2 \ y< g_2, \end{equation*} respectively. Here \(\Gamma\) is the Gamma function.

Theorem 10. [11] Let \(\prod:\Delta=[f_1,f_2]\times [g_1,g_2]\subseteq \mathbb{R}^{2} \rightarrow \mathbb{R}\) be convex on \(\Delta\) with \(f_1< f_2\) and \(g_1< g_2\) and \(\prod\in L_{1}(\Delta)\). Then

In this paper, we gave integral results for co-ordinated harmonically convex functions via fractional integrals.

2. Main Results

In this section, our aim is to prove some Hermite-Hadamard type ineqalities for co-ordinated harmonically convex functions in fractional integrals.

Theorem 11. Let \(\prod:\Delta=[f_1,f_2]\times [g_1,g_2]\subseteq (0,\infty)\times (0,\infty) \rightarrow \mathbb{R}\) be harmonically convex on \(\Delta\) with \(f_1< f_2\) and \(g_1< g_2\) and \(\prod\in L_{1}(\Delta)\). Then

where \(\Omega(x,y)=\left(\frac{1}{x},\frac{1}{y} \right) \) for all \((x,y)\in ([\frac{1}{f_2},\frac{1}{f_1}],[\frac{1}{g_2},\frac{1}{g_1}])\).

Proof. Let \((x,y),(z,w)\in \Delta\) and \(t_1,t_2 \in [0,1]\). Since \(\prod\) is co-ordinated harmonically convex on \(\Delta\), we have

By taking \(x=\frac{f_1f_2}{t_1f_1+(1-t_1)f_2}\), \(z=\frac{f_1f_2}{t_1f_2+(1-t_1)f_1}\), \(y=\frac{g_1g_2}{t_2 g_1+(1-t_2)g_2}\), \(w=\frac{g_1g_2}{t_2 g_2+(1-t_2)g_1}\) and \(t_1=t_2=\frac{1}{2}\) in (7), we get Multiplying both sides of (8) by \(t_1^{\alpha-1}t_2^{\beta-1}\) and then integrating with respect to \((t_1,t_2)\) over \([0,1]\times [0,1]\), we get Applying change of variable, we find Then by multiplying and dividing by \(\Gamma(\alpha)\Gamma(\beta)\) on right hand side of inequality (10), we get the first inequality of (6). For the second inequality of (6) we use the co-ordinated harmonically convexity of \(\prod\) as: \begin{align*} \begin{split} &\prod\left( \frac{f_1f_2}{t_1f_1+(1-t_1)f_2},\frac{g_1g_2}{t_2 g_1+(1-t_2)g_2}\right) \\ &\leq t_1t_2 \prod(f_1,g_1)+t_1(1-t_2)\prod(f_1,g_2)+(1-t_1)t_2 \prod(f_2,g_1)+(1-t_1)(1-t_2)\prod(f_2,g_2), \end{split} \end{align*} \begin{align*} \begin{split} &\prod\left(\frac{f_1f_2}{t_1f_1+(1-t_1)f_2},\frac{g_1g_2}{t_2 g_2+(1-t_2)g_1}\right) \\ &\leq t_1t_2 \prod(f_1,g_2)+t_1(1-t_2)\prod(f_1,g_1)+(1-t_1)t_2 \prod(f_2,g_2)+(1-t_1)(1-t_2)\prod(f_2,g_1), \end{split} \end{align*} \begin{align*} \begin{split} &\prod\left(\frac{f_1f_2}{t_1f_2+(1-t_1)f_1},\frac{g_1g_2}{t_2 g_1+(1-t_2)g_2}\right) \\ &\leq t_1t_2 \prod(f_2,g_1)+t_1(1-t_2)\prod(f_2,g_2)+(1-t_1)t_2 \prod(f_1,g_1)+(1-t_1)(1-t_2)\prod(f_1,g_2), \end{split} \end{align*} and \begin{align*} \begin{split} &\prod\left(\frac{f_1f_2}{t_1f_2+(1-t_1)f_1},\frac{g_1g_2}{t_2 g_2+(1-t_2)g_1}\right) \\ &\leq t_1t_2 \prod(f_2,g_2)+t_1(1-t_2)\prod(f_2,g_1)+(1-t_1)t_2 \prod(f_1,g_2)+(1-t_1)(1-t_2)\prod(f_1,g_1). \end{split} \end{align*} Then by adding above inequalities, we get Thus by multiplying (11) by \(t_1^{\alpha-1}t_2^{\beta-1}\) and then integrating with respect to \((t_1,t_2)\) over \([0,1]\times [0,1]\), we get the second inequality of (6). Hence the proof is completed.

Remark 1. In Theorem 11, if one takes \(\alpha=\beta=1\) and using change of variable \(u=1/x\) and \(v=1/y\), then one has Theorem in [12].

Theorem 12. Let \(\prod:\Delta=[f_1,f_2]\times [g_1,g_2] \subseteq (0,\infty)\times (0,\infty)\rightarrow \mathbb{R}\) be harmonically convex on \(\Delta\) with \(f_1< f_2\) and \(g_1< g_2\) and \(\varPsi\in L_{1}(\Delta)\). Then

where \(\Omega(x,y)=\left(\frac{1}{x},\frac{1}{y} \right) \), \(\Omega_{1}(x,y)=\left(\frac{1}{x},y \right) \) and \(\Omega_{2}(x,y)=\left(x,\frac{1}{y} \right) \) for all \((x,y)\in \left( [\frac{1}{f_2},\frac{1}{f_1}],[\frac{1}{g_2},\frac{1}{g_1}]\right) \).

Proof. Since \(\prod\) is co-ordinated harmonically convex on \(\Delta\) then we have \(\prod_{\frac{1}{x}}:[f_1,f_2]\rightarrow \mathbb{R}\), \(\prod_{\frac{1}{x}}(y)=\prod(\frac{1}{x},y)\), is harmonically convex on \([g_1,g_2]\) for all \(x\in \left[ \frac{1}{f_2},\frac{1}{f_1}\right] \). Then from inequality (4), we have

In other words, for all \(x\in\left[ \frac{1}{f_2},\frac{1}{f_1}\right] \). Now by multiplying (14) by \(\frac{\alpha(x-1/f_2)^{\alpha-1}}{2}\left( \frac{f_1f_2}{f_2-f_1}\right) ^{\alpha}\) and \(\frac{\alpha(1/f_1-x)^{\alpha-1}}{2}\left( \frac{f_1f_2}{f_2-f_1}\right) ^{\alpha}\), and then integrating with respect to \(x\) over \([1/f_2,1/f_1]\), respectively, we find and Again by similar arguments for \(\prod_{\frac{1}{y}}:[f_1,f_2]\rightarrow \mathbb{R}\), \(\prod_{\frac{1}{y}}(x)=\prod(x,\frac{1}{y})\), we get and By adding inequalities (15)-(18), we have This completes the second and third inequality of (12). Now again using (4), we have Adding (20) and (21), we get This completes the first inequality of (12). For the last inequality by using (4), we have \begin{align*} \begin{split} &\frac{\alpha}{2}\left( \frac{f_1f_2}{f_2-f_1}\right) ^{\alpha}\left[\int_{1/f_2}^{1/f_1}\left( \frac{1}{f_1}-x\right) ^{\alpha-1}\prod\left( \frac{1}{x},g_1\right) {\text d}x+\int_{1/f_2}^{1/f_1}\left( x-\frac{1}{f_2}\right) ^{\beta-1}\prod\left( \frac{1}{x},g_1\right) {\text d}x\right] \\ &\leq \frac{\prod(f_1,g_1)+\prod(f_2,g_1)}{2},\\ &\frac{\alpha}{2}\left( \frac{f_1f_2}{f_2-f_1}\right) ^{\alpha}\left[\int_{1/f_2}^{1/f_1}\left( \frac{1}{f_1}-x\right) ^{\alpha-1}\prod\left( \frac{1}{x},g_2\right) {\text d}x+\int_{1/f_2}^{1/f_1}\left( x-\frac{1}{f_2}\right) ^{\beta-1}\prod\left( \frac{1}{x},g_2\right) {\text d}x\right] \\ &\leq \frac{\prod(f_1,g_2)+\prod(f_2,g_2)}{2},\\ &\frac{\beta}{2}\left( \frac{g_1g_2}{g_2-g_1}\right) ^{\beta}\left[\int_{1/g_2}^{1/g_1}\left( \frac{1}{g_1}-y\right) ^{\beta-1}\prod\left( f_1,\frac{1}{y}\right) {\text d}y+\int_{1/g_2}^{1/g_1}\left( y-\frac{1}{g_2}\right) ^{\beta-1}\prod\left( f_1,\frac{1}{y}\right) {\text d}y\right] \\ &\leq \frac{\prod(f_1,g_1)+\prod(f_1,g_2)}{2},\\ &\frac{\beta}{2}\left( \frac{g_1g_2}{g_2-g_1}\right) ^{\beta}\left[\int_{1/g_2}^{1/g_1}\left( \frac{1}{g_1}-y\right) ^{\beta-1}\prod\left( f_2,\frac{1}{y}\right) {\text d}y+\int_{1/g_2}^{1/g_1}\left( y-\frac{1}{g_2}\right) ^{\beta-1}\prod\left( f_2,\frac{1}{y}\right) {\text d}y\right] \\ &\leq \frac{\prod(f_2,g_1)+\prod(f_2,g_2)}{2}. \end{split} \end{align*} Thus by adding all above inequalities, we get the last inequality of (12). Hence the proof is completed.

Lemma 1. Let \(\prod:\Delta=[f_1,f_2]\times [g_1,g_2]\subseteq (0,\infty)\times (0,\infty) \rightarrow \mathbb{R}\) be a partial differentiable mapping on \(\Delta\) with \(0< f_1 < f_2\) and \(0< g_1< g_2\). If \(\partial^{2} \prod/\partial t_1\partial t_2\in L_1(\Delta)\), then following holds:

where and \(A_{t_1}=t_1f_1+(1-t_1)f_2\), \(B_{t_2}=t_2 c+(1-t_2)d\). Also, \(g(x,y)=(\frac{1}{x},\frac{1}{y})\), \(g_{1}(x,y)=(\frac{1}{x},y)\), and \(g_{2}(x,y)=(x,\frac{1}{y})\) for all \((x,y)\in \Delta\).

Proof. By integration by parts and using the change of variable \(x=\frac{A_{t_1}}{f_1f_2}\) and \(y=\frac{B_{t_2}}{g_1g_2}\), we find that

Similarly, we can have Thus from equalities (25)-(28), we have Multiplying both sides of equality (29) by \(\frac{f_1f_2g_1g_2(f_2-f_1)(g_2-g_1)}{4}\), we get the desired equality (23).

Theorem 13. Let \(\prod:\Delta=[f_1,f_2]\times [g_1,g_2]\subseteq (0,\infty)\times (0,\infty) \rightarrow \mathbb{R}\) be a partial differentiable mapping on \(\Delta\) with \(0< f_1< f_2\) and \(0< g_1< g_2\). If \(\left| \partial^{2} \prod/\partial t_1\partial t_2\right| \) is a harmonically convex on the co-ordinates on \(\Delta\), then following holds:

where

Proof. Using Lemma 1, we have

Now using co-ordinated harmonically convexity of \(\left| \frac{\partial^{2}\prod}{\partial t_1\partial t_2}\right| \), we get After calculating above integrations, we get the required result.

Theorem 14. Let \(\prod:\Delta=[f_1,f_2]\times [g_1,g_2]\subseteq (0,\infty)\times (0,\infty) \rightarrow \mathbb{R}\) be a partial differentiable mapping on \(\Delta\) with \(0< f_1< f_2\) and \(0< g_1< g_2\). If \(\left| \partial^{2} \prod/\partial t_1\partial t_2\right|^{q} \), \(q>1\), is a harmonically convex on the co-ordinates on \(\Delta\), then following holds:

where

Proof. Applying the Holder's inequality for double integrals in (35), we get

Using co-ordinated harmonically convexity of \(\left| \frac{\partial^{2}\prod}{\partial t_1\partial t_2}\right|^{q} \), we get By calculating all integrals, we get the required result (37).

3. Conclusion

In Theorem 11 and 12, we have proved some new Hermite-Hadamard type inequalities for co-ordinated harmonically convex on a rectangle via Riemann-Liouville fractional integrals. In Lemma 1, we have proved a fractional integral identity and then with the help of this Lemma 1 we proved some fractional Hermite-Hadamard type inequalities on the co-ordinates.

Acknowledgments

The present investigation is supported by National University of Science and Technology(NUST), Islamabad, Pakistan.

Authors Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of interest

The authors declare no conflict of interest.