Hermite-Hadamard-Fejér type inequalities for co-ordinated harmonically convex functions via Katugampola fractional integrals

1. Introduction

A function \(f:\mathcal{K}\rightarrow \mathbb{R}\), where \(\mathcal{K}\) is an interval of real numbers, is called convex if the following inequality holds:

for all \(u_1,u_2\in \mathcal{K}\) and \(r\in[0,1]\). Function \(f\) is called concave if \(-f\) is convex.

The Hermite-Hadamard inequality [1] for convex functions \(f:\mathcal{K}\rightarrow \mathbb{R}\) on an interval of real line is:

where \(u_1,u_2\in \mathcal{K}\) with \(u_1< u_2\). Then Fejér [2] introduced the weighted generalization of (2) as follows where \(g:[u_1,u_2]\rightarrow \mathbb{R}\) is nonnegative, integrable and symmetric to \((u_1+u_2)/2\). For more results and details see [3,4,5,6,7,8,9,10,11,12,13,14].

Definition 1 ([15]). Let \(\mathcal{K}\subset\mathbb{R}\setminus\{0\}\) be a real interval. A function \(f:\mathcal{K}\rightarrow \mathbb{R}\) is said to be harmonically convex, if

for all \(u_1,u_2\in \mathcal{K}\) and \(r\in[0,1]\). If the inequality in (4) is reversed, then \(f\) is said to be harmonically concave.

Dragomir [16] gave the Hadamard's inequality for convex functions on the co-ordinate which is defined as:

Definition 2 ([16]). A function \(f:\Delta=[u_1,u_2]\times [v_1,v_2]\subseteq \mathbb{R}^{2} \rightarrow \mathbb{R}\) is called convex on the co-ordinate with \(u_1< u_2\) and \(v_1< v_2\) if the partial functions

\(f_{y}:[u_1,u_2] \rightarrow \mathbb{R}\), \(f_{y}(a)=f(a,y)\) and \(f_{x}:[v_1,v_2] \rightarrow \mathbb{R}\), \(f_{x}(c)=f(x,c)\) are convex for all \(x\in[u_1,u_2]\) and \(y\in [v_1,v_2]\).

Definition 3 ([17]). A function \(f:\Delta=[u_1,u_2]\times [v_1,v_2]\subseteq \mathbb{R}^{2} \rightarrow \mathbb{R}\) is called co-ordinate convex on \(\Delta\) with \(u_1< u_2\) and \(v_1< v_2\), if \begin{align*} &f(rx+(1-r)z,\tau y+(1-\tau)w)\leq r\tau f(x,y)+r(1-\tau)f(x,w)+(1-r)\tau f(z,y)+(1-r)(1-\tau)f(z,w), \end{align*} for all \(r,\tau\in [0,1]\) and \((x,y),(z,w)\in\Delta\). For more results and details see [16,17,18,19].

Definition 4 ([20]). A function \(f:\Delta=[u_1,u_2]\times [v_1,v_2]\subseteq (0,\infty)\times (0,\infty) \rightarrow \mathbb{R}\) is called co-ordinated harmonically convex on \(\Delta\) with \(u_1< u_2\) and \(v_1< v_2\), if \begin{align*} &f\left(\frac{xz}{rx+(1-r)z},\frac{yw}{\tau y+(1-\tau)w}\right)\leq r\tau f(x,y)+r(1-\tau)f(x,w)+(1-r)\tau f(z,y)+(1-r)(1-\tau)f(z,w), \end{align*} for all \(r,\tau\in [0,1]\) and \((x,y),(z,w)\in\Delta\).

Clearly, a function \(f:\Delta=[u_1,u_2]\times [v_1,v_2]\subseteq (0,\infty)\times (0,\infty) \rightarrow \mathbb{R}\) is called harmonically convex on the co-ordinate with \(u_1< u_2\) and \(v_1< v_2\) if the partial functions \(f_{y}:[u_1,u_2] \rightarrow \mathbb{R}\), \(f_{y}(a)=f(a,y)\) and \(f_{x}:[v_1,v_2] \rightarrow \mathbb{R}\), \(f_{x}(b)=f(x,b)\) are harmonically convex for all \(x\in[u_1,u_2]\) and \(y\in [v_1,v_2]\), see [21] for more details.

Definition 5 ([22]). Let \([u_1,u_2]\subset\mathbb{R}\) be a finite interval. The left- and right-side Katugampola fractional integrals of order \(\alpha(>0)\) of \(f\in X^{p}_{c}(u_1,u_2)\) are defined by,

\begin{equation*} ^{\rho}I^{\alpha}_{u_1+}f(x)=\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_{u_1}^{x}(x^{\rho}-t^{\rho})^{\alpha-1}t^{\rho-1}f(t)dt, \end{equation*} and \begin{equation*} ^{\rho}I^{\alpha}_{u_2-}f(x)=\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_{x}^{u_2}(t^{\rho}-x^{\rho})^{\alpha-1}t^{\rho-1}f(t)dt, \end{equation*} with \(u_1< x< u_2\) and \(\rho>0\), where \(X^{p}_{c}(u_1,u_2)\) \((c \in \mathbb{R}, 1\leq p\leq \infty)\) is the space of those complex valued Lebesgue measurable functions \(f\) on \([u_1,u_2]\) for which \(\|f\|_{X^{p}_{c}}< \infty\), where the norm is defined by \begin{equation*} \|f\|_{X^{p}_{c}}=\left( \int_{u_1}^{u_2}|t^{c}f(t)|^{p}\frac{dt}{t}\right)^{1/p}< \infty, \end{equation*} for \(1\leq p< \infty\), \(c\in \mathbb{R}\) and for the case \(p=\infty\), \begin{equation*} \|f\|_{X^{\infty}_{c}}= ess\ sup_{u_1\leq t\leq u_2}[t^{c}|f(t)|]. \end{equation*}

Definition 6 ([23]). Let \(f\in L_{1}([u_1,u_2]\times [v_1,v_2])\). The Katugampola fractional integrals \(^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{u_1+,v_1+}\), \(^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{u_1+,v_2-}\), \(^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{u_2-,v_1+}\) and \(^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{u_2-,v_2-}\) of order \(\alpha,\beta>0\) with \(a,c\geq 0\) are defined by

\begin{equation*} ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{u_1+,v_1+}f(x,y)=\frac{\rho^{1-\alpha}_{1}\rho^{1-\beta}_{2}}{\Gamma(\alpha)\Gamma(\beta)}\int_{u_1}^{x}\int_{v_1}^{y}(x^{\rho_{1}}-t^{\rho_{1}})^{\alpha-1}(y^{\rho_{2}}-s^{\rho_{2}})^{\beta-1}t^{\rho_{1}-1}s^{\rho_{2}-1}f(t,s)dsdt, \end{equation*} with \(x>u_1, \ y>v_1\), \begin{equation*} ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{u_1+,v_2-}f(x,y)=\frac{\rho^{1-\alpha}_{1}\rho^{1-\beta}_{2}}{\Gamma(\alpha)\Gamma(\beta)}\int_{u_1}^{x}\int_{y}^{v_2}(x^{\rho_{1}}-t^{\rho_{1}})^{\alpha-1}(s^{\rho_{2}}-y^{\rho_{2}})^{\beta-1}t^{\rho_{1}-1}s^{\rho_{2}-1}f(t,s)dsdt, \end{equation*} with \(x>u_1, \ y< v_2\), \begin{equation*} ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{u_2-,v_1+}f(x,y)=\frac{\rho^{1-\alpha}_{1}\rho^{1-\beta}_{2}}{\Gamma(\alpha)\Gamma(\beta)}\int_{x}^{u_2}\int_{v_1}^{y}(t^{\rho_{1}}-x^{\rho_{1}})^{\alpha-1}(y^{\rho_{2}}-s^{\rho_{2}})^{\beta-1}t^{\rho_{1}-1}s^{\rho_{2}-1}f(t,s)dsdt, \end{equation*} with \(x< u_2, \ y>v_1\), and \begin{equation*} ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{u_2-,v_2-}f(x,y)=\frac{\rho^{1-\alpha}_{1}\rho^{1-\beta}_{2}}{\Gamma(\alpha)\Gamma(\beta)}\int_{x}^{u_2}\int_{y}^{v_2}(t^{\rho_{1}}-x^{\rho_{1}})^{\alpha-1}(s^{\rho_{2}}-y^{\rho_{2}})^{\beta-1}t^{\rho_{1}-1}s^{\rho_{2}-1}f(t,s)dsdt, \end{equation*} with \(x< u_2, \ y< v_2\), respectively, where the Gamma function \(\Gamma\) is defined as \(\Gamma(\alpha)=\int_{0}^{\infty}e^{-t}t^{\alpha-1}dt\).

In the next section, we give result for harmonically convex functions in one dimension.

2. Hermite-Hadamard-Fejér type inequalities

In this section, we give Hermite-Hadamard-Fejér type inequalities for harmonically convex functions via Katugampola fractional integral in one dimension which will play a key role for the results in the next section. Latif et al., [24] defined following useful definition:

Definition 7 ([24]). A function \(h:[u_1,u_2]\subseteq\mathbb{R}\backslash \{0\}\rightarrow \mathbb{R} \) is said to be harmonically symmetric with respect to \(2u_1u_2/(u_1+u_2)\) if

\begin{equation*} h(x)=h\left(\frac{1}{\frac{1}{u_1}+\frac{1}{u_2}-\frac{1}{x}}\right) \end{equation*} holds for all \(x\in[u_1,u_2]\).

Lemma 1. Let \(\rho>0\). If \(h:[u_1^{\rho},u_2^{\rho}]\subseteq(0,\infty)\rightarrow \mathbb{R}\) is integrable and harmonically symmetric with respect to \(2u_1^{\rho}u_2^{\rho}/(u_1^{\rho}+u_2^{\rho})\), then

with \(\alpha>0\) and \(g(x^{\rho})=1/x^{\rho}\).

Proof. Since \(h\) is harmonically symmetric with respect to \(2u_1^{\rho}u_2^{\rho}/(u_1^{\rho}+u_2^{\rho})\), then by definition we have \(h(\frac{1}{x^{\rho}})=h\left(\frac{1}{\frac{1}{u_1^{\rho}}+\frac{1}{u_2^{\rho}}-x^{\rho}}\right)\), for all \(x^{\rho}\in\left[\frac{1}{u_2^{\rho}},\frac{1}{u_1^{\rho}}\right] \). In the following integral, by setting \(t^{\rho}=\frac{1}{u_1^{\rho}}+\frac{1}{u_2^{\rho}}-x^{\rho}\), we get \begin{align*} ^{\rho}I^{\alpha}_{1/u_2+}(h\circ g)(1/u_1^{\rho})&=\frac{\rho^{1-\alpha}}{\Gamma(\alpha)} \int^{\frac{1}{u_1}}_{\frac{1}{u_2}}\left(\frac{1}{u_1^{\rho}}-t^{\rho}\right)^{\alpha-1}t^{\rho-1}h\left( \frac{1}{t^{\rho}}\right) dt \\ &=\frac{\rho^{1-\alpha}}{\Gamma(\alpha)} \int^{\frac{1}{u_1}}_{\frac{1}{u_2}}\left(x^{\rho}-\frac{1}{u_2^{\rho}}\right)^{\alpha-1}x^{\rho-1}h\left( \frac{1}{\frac{1}{u_1^{\rho}}+\frac{1}{u_2^{\rho}}-x^{\rho}}\right) dx \\ &=\frac{\rho^{1-\alpha}}{\Gamma(\alpha)} \int^{\frac{1}{u_1}}_{\frac{1}{u_2}}\left(x^{\rho}-\frac{1}{u_2^{\rho}}\right)^{\alpha-1}x^{\rho-1}h\left( \frac{1}{x^{\rho}}\right) dx\\ &=\ ^{\rho}I^{\alpha}_{1/u_1-}(h\circ g)(1/u_2^{\rho}). \end{align*} This completes the proof.

Remark 1. In Lemma 1, if we take \(\rho\longmapsto 0\), we get Lemma 2 in [25].

Theorem 8. Let \(\rho>0\). Let \(f:[u_1^{\rho},u_2^{\rho}]\subseteq (0,\infty)\rightarrow\mathbb{R}\) be a harmonically convex with \(u_1< u_2\) and \(f\in L_1[u_1,u_2]\). If \(h:[u_1^{\rho},u_2^{\rho}]\subseteq(0,\infty)\rightarrow \mathbb{R}\) is nonnegative and harmonically symmetric with respect to \(2u_1^{\rho}u_2^{\rho}/(u_1^{\rho}+u_2^{\rho})\), then the following inequalities hold:

with \(\alpha>0\) and \(g(x^{\rho})=1/x^{\rho}\).

Proof. Since \(f\) is harmonically convex on \([u_1^{\rho},u_2^{\rho}]\), we have for all \(r\in[0,1]\)

Multiplying (7) by \(r^{\alpha\rho-1}h\left(\frac{u_1^{\rho}u_2^{\rho}}{r^{\rho}u_2^{\rho}+(1-r^{\rho})u_1^{\rho}}\right)\) on both sides and integrate with respect to \([0,1]\), we get \begin{align*} \begin{split} 2f\left(\frac{2u_1^{\rho}u_2^{\rho}}{u_1^{\rho}+u_2^{\rho}}\right)&\int_{0}^{1} r^{\alpha\rho-1}h\left(\frac{u_1^{\rho}u_2^{\rho}}{r^{\rho}u_2^{\rho}+(1-r^{\rho})u_1^{\rho}}\right)dr \\ &\leq \int_{0}^{1}r^{\alpha\rho-1}f\left(\frac{u_1^{\rho}u_2^{\rho}}{r^{\rho}u_1^{\rho}+(1-r^{\rho})u_2^{\rho}}\right) h\left(\frac{u_1^{\rho}u_2^{\rho}}{r^{\rho}u_2^{\rho}+(1-r^{\rho})u_1^{\rho}}\right)dr\\ &\;\;\;\;+\int_{0}^{1}r^{\alpha\rho-1}f\left(\frac{u_1^{\rho}u_2^{\rho}}{r^{\rho}u_2^{\rho}+(1-r^{\rho})u_1^{\rho}}\right) h\left(\frac{u_1^{\rho}u_2^{\rho}}{r^{\rho}u_2^{\rho}+(1-r^{\rho})u_1^{\rho}}\right)dr. \end{split} \end{align*} Since \(h\) is harmonically symmetric with respect to \(2u_1^{\rho}u_2^{\rho}/(u_1^{\rho}+u_2^{\rho})\). By setting \(x^{\rho}=\frac{r^{\rho}u_2^{\rho}+(1-r^{\rho})u_1^{\rho}}{u_1^{\rho}u_2^{\rho}}\), we get \begin{align*} \begin{split} 2\left(\frac{u_1^{\rho}u_2^{\rho}}{u_2^{\rho}-u_1^{\rho}}\right)^{\alpha}&f\left(\frac{2u_1^{\rho}u_2^{\rho}}{u_1^{\rho}+u_2^{\rho}}\right) \int_{1/u_2}^{1/u_1}\left(x^{p}-\frac{1}{u_2^{\rho}}\right)^{\alpha-1}h\left( \frac{1}{x^{\rho}}\right) dx \\ \leq&\left(\frac{u_1^{\rho}u_2^{\rho}}{u_2^{\rho}-u_1^{\rho}}\right)^{\alpha}\left[\int_{1/u_2}^{1/u_1}\left(x^{p}-\frac{1}{u_2^{\rho}}\right)^{\alpha-1} f\left(\frac{1}{\frac{1}{u_1^{\rho}}+\frac{1}{u_2^{\rho}}-x^{\rho}}\right)h\left( \frac{1}{x^{\rho}}\right) dx\right.\\&\left.+\int_{1/u_2}^{1/u_1}\left(x^{p}-\frac{1}{u_2^{\rho}}\right)^{\alpha-1} f\left( \frac{1}{x^{\rho}}\right) h\left( \frac{1}{x^{\rho}}\right) dx \right] \\ =&\left(\frac{u_1^{\rho}u_2^{\rho}}{u_2^{\rho}-u_1^{\rho}}\right)^{\alpha}\left[\int_{1/u_2}^{1/u_1}\left(\frac{1}{u_1^{\rho}}-x^{p}\right)^{\alpha-1} f\left( \frac{1}{x^{\rho}}\right) h\left(\frac{1}{\frac{1}{u_1^{\rho}}+\frac{1}{u_2^{\rho}}-x^{\rho}}\right)dx\right. \\ &\left.+\int_{1/u_2}^{1/u_1}\left(x^{p}-\frac{1}{u_2^{\rho}}\right)^{\alpha-1} f\left( \frac{1}{x^{\rho}}\right) h\left( \frac{1}{x^{\rho}}\right) dx \right]. \end{split} \end{align*} Then by Lemma 1, we have This completes the first inequality. For second inequality, we first note that if \(f\) is harmonically convex function, then we have Multiplying (8) by \(r^{\alpha\rho-1}h\left(\frac{u_1^{\rho}u_2^{\rho}}{r^{\rho}u_2^{\rho}+(1-r^{\rho})u_1^{\rho}}\right)\) on both sides and integrate with respect to \(r\in[0,1]\), we get \begin{align*} \begin{split} &\int_{0}^{1}r^{\alpha\rho-1}f\left(\frac{u_1^{\rho}u_2^{\rho}}{r^{\rho}u_1^{\rho}+(1-r^{\rho})u_2^{\rho}}\right) h\left(\frac{u_1^{\rho}u_2^{\rho}}{r^{\rho}u_2^{\rho}+(1-r^{\rho})u_1^{\rho}}\right)dr \\ &\;\;\;+\int_{0}^{1}r^{\alpha\rho-1}f\left(\frac{u_1^{\rho}u_2^{\rho}}{r^{\rho}u_2^{\rho}+(1-r^{\rho})u_1^{\rho}}\right) h\left(\frac{u_1^{\rho}u_2^{\rho}}{r^{\rho}u_2^{\rho}+(1-r^{\rho})u_1^{\rho}}\right)dr \\ &\leq (f(u_1^{\rho})+f(u_2^{\rho}))\int_{0}^{1}r^{\alpha\rho-1}h\left(\frac{u_1^{\rho}u_2^{\rho}}{r^{\rho}u_2^{\rho}+(1-r^{\rho})u_1^{\rho}}\right)dr, \end{split} \end{align*} i.e., \begin{align*} \begin{split} \left(\frac{u_1^{\rho}u_2^{\rho}}{u_2^{\rho}-u_1^{\rho}}\right)^{\alpha}&\rho^{\alpha-1}\Gamma(\alpha) \left[^{\rho}I^{\alpha}_{1/u_1-}(fh\circ g)(1/u_2^{\rho})+^{\rho}I^{\alpha}_{1/u_2+}(fh\circ g)(1/u_1^{\rho})\right] \\ &\leq\left(\frac{u_1^{\rho}u_2^{\rho}}{u_2^{\rho}-u_1^{\rho}}\right)^{\alpha}\rho^{\alpha-1}\Gamma(\alpha)\frac{f(u_1^{\rho})+f(u_2^{\rho})}{2} \left[^{\rho}I^{\alpha}_{1/u_1-}(h\circ g)(1/u_2^{\rho})+^{\rho}I^{\alpha}_{1/u_2+}(h\circ g)(1/u_1^{\rho})\right]. \end{split} \end{align*} This completes the proof.

Remark 2.

• 1)   In Theorem 8, if we take \(\rho\rightarrow 1\), we get Theorem 5 in [25].
• 2)   In Theorem 8, if we take \(\rho\rightarrow 1\) and \(\alpha=1\), we get Theorem 8 in [26].

3. Hermite-Hadamard-Fejér type inequalities on co-ordinates

In this section, we established some new results by using Katugampola fractional integrals on co-ordinates. First we give the following result:

Theorem 9. Let \(\alpha,\beta>0\) and \(\rho_{1},\rho_{2}>0\). Let \(f:\Delta=[u_1^{\rho_{1}},u_2^{\rho_{1}}]\times [v_1^{\rho_{2}},v_2^{\rho_{2}}]\subseteq (0,\infty)\times(0,\infty) \rightarrow \mathbb{R}\) be a co-ordinated harmonically convex on \(\Delta\), with \(0< u_1< u_2\), \(0< v_1< v_2\). If \(h:\Delta\rightarrow \mathbb{R}\) is nonnegative and harmonically symmetric with respect to \(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}}\), \(\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\) on \(\Delta\). Then

holds, where \(g(x^{\rho_{1}},y^{\rho_{2}})=\left(\frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}} \right) \).

Proof. Since \(f\) is co-ordinated harmonically convex on \(\Delta\), we have

Multiplying (11) by \(r^{\rho_{1}\alpha-1}\tau^{\rho_{2}\beta-1}h\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{r^{\rho_{1}}u_2^{\rho_{1}}+(1-r^{\rho_{1}})u_1^{\rho_{1}}},\frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{\tau^{\rho_{2}} v_2^{\rho_{2}}+(1-\tau^{\rho_{2}})v_1^{\rho_{2}}}\right)\) on both sides and then integrating with respect to \((r,\tau)\) over \([0,1]\times [0,1]\), we get \begin{align*} f\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right)& \int_{0}^{1}\int_{0}^{1}r^{\rho_{1}\alpha-1}\tau^{\rho_{2}\beta-1} h\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{r^{\rho_{1}}u_2^{\rho_{1}}+(1-r^{\rho_{1}})u_1^{\rho_{1}}},\frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{\tau^{\rho_{2}} v_2^{\rho_{2}}+(1-\tau^{\rho_{2}})v_1^{\rho_{2}}}\right)drd\tau \\ \leq& \frac{1}{4}\bigg[\int_{0}^{1}\int_{0}^{1} f\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{r^{\rho_{1}}u_1^{\rho_{1}}+(1-r^{\rho_{1}})u_2^{\rho_{1}}},\frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{\tau^{\rho_{2}} v_1^{\rho_{2}}+(1-\tau^{\rho_{2}})v_2^{\rho_{2}}}\right) \\ &\times h\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{r^{\rho_{1}}u_2^{\rho_{1}}+(1-r^{\rho_{1}})u_1^{\rho_{1}}},\frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{\tau^{\rho_{2}} v_2^{\rho_{2}}+(1-\tau^{\rho_{2}})v_1^{\rho_{2}}}\right)r^{\rho_{1}\alpha-1}\tau^{\rho_{2}\beta-1}drd\tau \\ & +\int_{0}^{1}\int_{0}^{1}f\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{r^{\rho_{1}}u_1^{\rho_{1}}+(1-r^{\rho_{1}})u_2^{\rho_{1}}},\frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{\tau^{\rho_{2}} v_2^{\rho_{2}}+(1-\tau^{\rho_{2}})v_1^{\rho_{2}}}\right) \\ &\times h\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{r^{\rho_{1}}u_2^{\rho_{1}}+(1-r^{\rho_{1}})u_1^{\rho_{1}}},\frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{\tau^{\rho_{2}} v_2^{\rho_{2}}+(1-\tau^{\rho_{2}})v_1^{\rho_{2}}}\right)r^{\rho_{1}\alpha-1}\tau^{\rho_{2}\beta-1}drd\tau \\ &+ \int_{0}^{1}\int_{0}^{1} f\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{r^{\rho_{1}}u_2^{\rho_{1}}+(1-r^{\rho_{1}})u_1^{\rho_{1}}},\frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{\tau^{\rho_{2}} v_1^{\rho_{2}}+(1-\tau^{\rho_{2}})v_2^{\rho_{2}}}\right) \\ &\times h\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{r^{\rho_{1}}u_2^{\rho_{1}}+(1-r^{\rho_{1}})u_1^{\rho_{1}}},\frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{\tau^{\rho_{2}} v_2^{\rho_{2}}+(1-\tau^{\rho_{2}})v_1^{\rho_{2}}}\right)r^{\rho_{1}\alpha-1}\tau^{\rho_{2}\beta-1}drd\tau \\ & +\int_{0}^{1}\int_{0}^{1}f\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{r^{\rho_{1}}u_2^{\rho_{1}}+(1-r^{\rho_{1}})u_1^{\rho_{1}}},\frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{\tau^{\rho_{2}} v_2^{\rho_{2}}+(1-\tau^{\rho_{2}})v_1^{\rho_{2}}}\right) \\ &\times h\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{r^{\rho_{1}}u_2^{\rho_{1}}+(1-r^{\rho_{1}})u_1^{\rho_{1}}},\frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{\tau^{\rho_{2}} v_2^{\rho_{2}}+(1-\tau^{\rho_{2}})v_1^{\rho_{2}}}\right)r^{\rho_{1}\alpha-1}\tau^{\rho_{2}\beta-1}drd\tau \bigg]. \end{align*} By change of variables \(x^{\rho_{1}}=\frac{r^{\rho_{1}}u_2^{\rho_{1}}+(1-r^{\rho_{1}})u_1^{\rho_{1}}}{u_1^{\rho_{1}}u_2^{\rho_{1}}}\) and \(y^{\rho_{2}}=\frac{\tau^{\rho_{2}} v_2^{\rho_{2}}+(1-\tau^{\rho_{2}})v_1^{\rho_{2}}}{v_1^{\rho_{2}}v_2^{\rho_{2}}}\) and using the symmetric property of \(h\), we find \begin{align*} &\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_2^{\rho_{1}}-u_1^{\rho_{1}}}\right)^{\alpha}\left( \frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_2^{\rho_{2}}-v_1^{\rho_{2}}}\right)^{\beta}f\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right) \\ &\;\;\;\;\times \int_{1/v_2}^{1/v_1}\int_{1/u_2}^{1/u_1} \left(x^{\rho_{1}}-\frac{1}{u_2^{\rho_{1}}} \right)^{\alpha-1}\left(y^{\rho_{2}}-\frac{1}{v_2^{\rho_{2}}} \right)^{\beta-1}x^{\rho_{1}-1}y^{\rho_{2}-1} h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dxdy \\ & \leq \frac{1}{4}\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_2^{\rho_{1}}-u_1^{\rho_{1}}}\right)^{\alpha}\left( \frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_2^{\rho_{2}}-v_1^{\rho_{2}}}\right)^{\beta}\left[\int_{1/v_2}^{1/v_1}\int_{1/u_2}^{1/u_1} \left(x^{\rho_{1}}-\frac{1}{u_2^{\rho_{1}}} \right)^{\alpha-1}\left(y^{\rho_{2}}-\frac{1}{v_2^{\rho_{2}}} \right)^{\beta-1}\right. \end{align*} \begin{align*} &\;\;\;\times x^{\rho_{1}-1}y^{\rho_{2}-1}f\left( \frac{1}{\frac{1}{u_1^{\rho_{1}}}+\frac{1}{u_2^{\rho_{1}}}-x^{\rho_{1}}},\frac{1}{\frac{1}{v_1^{\rho_{2}}}+\frac{2}{v_2^{\rho_{2}}}-y^{\rho_{2}}}\right) h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dxdy\\ &\;\;\;+\int_{1/v_2}^{1/v_1}\int_{1/u_2}^{1/u_1} \left(x^{\rho_{1}}-\frac{1}{u_2^{\rho_{1}}} \right)^{\alpha-1}\left(y^{\rho_{2}}-\frac{1}{v_2^{\rho_{2}}} \right)^{\beta-1} x^{\rho_{1}-1}y^{\rho_{2}-1}f\left( \frac{1}{\frac{1}{u_1^{\rho_{1}}}+\frac{1}{u_2^{\rho_{1}}}-x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right) h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dxdy \\ &\;\;\;+\int_{1/v_2}^{1/v_1}\int_{1/u_2}^{1/u_1} \left(x^{\rho_{1}}-\frac{1}{u_2^{\rho_{1}}} \right)^{\alpha-1}\left(y^{\rho_{2}}-\frac{1}{v_2^{\rho_{2}}} \right)^{\beta-1} x^{\rho_{1}-1}y^{\rho_{2}-1}f\left( \frac{1}{x^{\rho_{1}}},\frac{1}{\frac{1}{v_1^{\rho_{2}}}+\frac{2}{v_1^{\rho_{2}}}-y^{\rho_{2}}}\right) h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dxdy \\ &\;\;\;\left.+\int_{1/v_2}^{1/v_1}\int_{1/u_2}^{1/u_1} \left(x^{\rho_{1}}-\frac{1}{u_2^{\rho_{1}}} \right)^{\alpha-1}\left(y^{\rho_{2}}-\frac{1}{v_2^{\rho_{2}}} \right)^{\beta-1} x^{\rho_{1}-1}y^{\rho_{2}-1}f\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right) h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dxdy \right] \\ &= \frac{1}{4}\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_2^{\rho_{1}}-u_1^{\rho_{1}}}\right)^{\alpha}\left( \frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_2^{\rho_{2}}-v_1^{\rho_{2}}}\right)^{\beta}\left[\int_{1/v_2}^{1/v_1}\int_{1/u_2}^{1/u_1} \left(\frac{1}{u_1^{\rho_{1}}}-x^{\rho_{1}} \right)^{\alpha-1}\left(\frac{1}{v_1^{\rho_{2}}}-y^{\rho_{2}} \right)^{\beta-1}\right. \\ &\;\;\;\times x^{\rho_{1}-1}y^{\rho_{2}-1}f\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)h\left( \frac{1}{\frac{1}{u_1^{\rho_{1}}}+\frac{1}{u_2^{\rho_{1}}}-x^{\rho_{1}}},\frac{1}{\frac{1}{v_1^{\rho_{2}}}+\frac{2}{v_2^{\rho_{2}}}-y^{\rho_{2}}}\right)dxdy \\ &\;\;\;+\int_{1/v_2}^{1/v_1}\int_{1/u_2}^{1/u_1} \left(\frac{1}{u_1^{\rho_{1}}}-x^{\rho_{1}} \right)^{\alpha-1}\left(y^{\rho_{2}}-\frac{1}{v_2^{\rho_{2}}} \right)^{\beta-1} x^{\rho_{1}-1}y^{\rho_{2}-1}f\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)h\left( \frac{1}{\frac{1}{u_1^{\rho_{1}}}+\frac{1}{u_2^{\rho_{1}}}-x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dxdy \\ &\;\;\;+\int_{1/v_2}^{1/v_1}\int_{1/u_2}^{1/u_1} \left(x^{\rho_{1}}-\frac{1}{u_2^{\rho_{1}}} \right)^{\alpha-1}\left(\frac{1}{v_1^{\rho_{2}}}-y^{\rho_{2}} \right)^{\beta-1} x^{\rho_{1}-1}y^{\rho_{2}-1}f\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{\frac{1}{v_1^{\rho_{2}}}+\frac{2}{v_2^{\rho_{2}}}-y^{\rho_{2}}}\right)dxdy \\ &\;\;\;\left.+\int_{1/v_2}^{1/v_1}\int_{1/u_2}^{1/u_1} \left(x^{\rho_{1}}-\frac{1}{u_2^{\rho_{1}}} \right)^{\alpha-1}\left(y^{\rho_{2}}-\frac{1}{v_2^{\rho_{2}}} \right)^{\beta-1} x^{\rho_{1}-1}y^{\rho_{2}-1}f\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right) h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dxdy\right]. \end{align*} Thus, we get \begin{align*} \begin{split} \frac{\Gamma(\alpha)\Gamma(\beta)}{\rho_{1}^{1-\alpha}\rho_2^{1-\beta}}&\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_2^{\rho_{1}}-u_1^{\rho_{1}}}\right)^{\alpha}\left( \frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_2^{\rho_{2}}-v_1^{\rho_{2}}}\right)^{\beta}f\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right) \\ &\times\left[\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_1-,1/v_1-}h\circ g\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) +\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_1-,1/v_2+}h\circ g\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right)\right. \\ &\left.+\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_2+,1/v_1-}h\circ g\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) +\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_2+,1/v_2+}h\circ g\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right] \\ \leq& \frac{\Gamma(\alpha)\Gamma(\beta)}{4\rho_{1}^{1-\alpha}\rho_2^{1-\beta}}\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_2^{\rho_{1}}-u_1^{\rho_{1}}}\right)^{\alpha}\left( \frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_2^{\rho_{2}}-v_1^{\rho_{2}}}\right)^{\beta} \\ &\times \left[\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_1-,1/v_1-}(fh\circ g) \left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) +\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_1-,1/v_2+}(fh\circ g)\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right)\right. \\ &\left.+\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_2+,1/v_1-}(fh\circ g)\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) +\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_2+,1/v_2+}(fh\circ g)\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right]. \end{split} \end{align*} This completes the first inequality of (10). For the second inequality of (10) we use the co-ordinated harmonically convexity of \(f\) as: Thus multiplying (12) by \(r^{\rho_{1}\alpha-1}\tau^{\rho_{2}\beta-1}h\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{r^{\rho_{1}}u_2^{\rho_{1}}+(1-r^{\rho_{1}})u_1^{\rho_{1}}},\frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{\tau^{\rho_{2}} v_2^{\rho_{2}}+(1-\tau^{\rho_{2}})v_1^{\rho_{2}}}\right)\) and integrating with respect to \((r,\tau)\) over \([0,1]\times [0,1]\), we get the second inequality of (10). Hence the proof is completed.

Theorem 10. Let \(\alpha,\beta>0\) and \(\rho_{1},\rho_{2}>0\). Let \(f:\Delta=[u_1^{\rho_{1}},u_2^{\rho_{1}}]\times [v_1^{\rho_{2}},v_2^{\rho_{2}}]\subseteq (0,\infty)\times(0,\infty) \rightarrow \mathbb{R}\) be a co-ordinated harmonically convex on \(\Delta\), with \(0< u_1< u_2\), \(0< v_1< v_2\) and \(f\in L_1[\Delta]\). If \(h:\Delta\rightarrow \mathbb{R}\) is nonnegative and harmonically symmetric with respect to \(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}}\),\(\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\) on \(\Delta\). Then the following inequalities hold:

\begin{align*} f\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right)&\left[\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_1-,1/v_1-}h\circ g\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) +\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_1-,1/v_2+}h\circ g\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right)\right. \\ &\left.+\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_2+,1/v_1-}h\circ g\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) +\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_2+,1/v_2+}h\circ g\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right] \\ \leq& \ ^{\rho_{1}}I^{\alpha}_{1/u_1-}\left[ f\circ g_{1}\left( \frac{1}{u_2^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right)\ ^{\rho_{2}}I^{\beta}_{1/v_1-}h\circ g_2\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right] \\ &+\ ^{\rho_{1}}I^{\alpha}_{1/u_1-}\bigg[ f\circ g_{1}\left( \frac{1}{u_2^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right) \ ^{\rho_{2}}I^{\beta}_{1/v_2+}h\circ g_2\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \bigg] \\ &+\ ^{\rho_{1}}I^{\alpha}_{1/u_2+}\left[ f\circ g_{1}\left( \frac{1}{u_1^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right)\ ^{\rho_{2}}I^{\beta}_{1/v_1-}h\circ g_2\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right] \\ &+\ ^{\rho_{1}}I^{\alpha}_{1/u_2+}\left[ f\circ g_{1}\left( \frac{1}{u_1^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right)\ ^{\rho_{2}}I^{\beta}_{1/v_2+}h\circ g_2\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right] \\ &+\ ^{\rho_{2}}I^{\beta}_{1/v_1-}\bigg[ f\circ g_{2}\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}}, \frac{1}{v_2^{\rho_{2}}}\right) \ ^{\rho_{1}}I^{\alpha}_{1/u_1-}h\circ g_1\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \bigg] \\ &+ \ ^{\rho_{2}}I^{\beta}_{1/v_1-}\left[ f\circ g_{2}\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}}, \frac{1}{v_2^{\rho_{2}}}\right) \ ^{\rho_{1}}I^{\alpha}_{1/u_2+}h\circ g_1\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right] \\ &+\ ^{\rho_{2}}I^{\beta}_{1/v_2+}\left[ f\circ g_{2}\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}}, \frac{1}{v_1^{\rho_{2}}}\right) \ ^{\rho_{1}}I^{\alpha}_{1/u_1-}h\circ g_1\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right] \\ &+\ ^{\rho_{2}}I^{\beta}_{1/v_2+}\bigg[ f\circ g_{2}\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}}, \frac{1}{v_1^{\rho_{2}}}\right) \ ^{\rho_{1}}I^{\alpha}_{1/u_2+}h\circ g_1\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \bigg] \\ \leq& 2\left[\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_1-,1/v_1-}fh\circ g\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) +\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_1-,1/v_2+}fh\circ g\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right)\right. \\ &+\left.\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_2+,1/v_1-}fh\circ g\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) +\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_2+,1/v_2+}fh\circ g\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right] \\ \leq& \ ^{\rho_{1}}I^{\alpha}_{1/u_1-}\left[ f\circ g_{1}\left( \frac{1}{u_2^{\rho_{1}}},v_1^{\rho_{2}}\right)\ ^{\rho_{2}}I^{\beta}_{1/v_2+}h\circ g_2\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right] \\ &+\ ^{\rho_{1}}I^{\alpha}_{1/u_1-}\bigg[ f\circ g_{1}\left( \frac{1}{u_2^{\rho_{1}}},v_2^{\rho_{2}}\right) \ ^{\rho_{2}}I^{\beta}_{1/v_1-}h\circ g_2\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \bigg] \\ &+\ ^{\rho_{1}}I^{\alpha}_{1/u_2+}\left[ f\circ g_{1}\left( \frac{1}{u_1^{\rho_{1}}},v_1^{\rho_{2}}\right)\ ^{\rho_{2}}I^{\beta}_{1/v_2+}h\circ g_2\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right] \end{align*} \begin{align} \label{tt1e1} &+\ ^{\rho_{1}}I^{\alpha}_{1/u_2+}\left[ f\circ g_{1}\left( \frac{1}{u_1^{\rho_{1}}},v_2^{\rho_{2}}\right)\ ^{\rho_{2}}I^{\beta}_{1/v_1-}h\circ g_2\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right]\notag \\ &+\ ^{\rho_{2}}I^{\beta}_{1/v_1-}\left[ f\circ g_{2}\left(u_1^{\rho_{1}}, \frac{1}{v_2^{\rho_{2}}}\right) \ ^{\rho_{1}}I^{\alpha}_{1/u_2+}h\circ g_1\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right]\notag \\ &+ \ ^{\rho_{2}}I^{\beta}_{1/v_1-}\left[ f\circ g_{2}\left(u_2^{\rho_{1}}, \frac{1}{v_2^{\rho_{2}}}\right) \ ^{\rho_{1}}I^{\alpha}_{1/u_1-}h\circ g_1\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right]\notag \\ &+\ ^{\rho_{2}}I^{\beta}_{1/v_2+}\left[ f\circ g_{2}\left(u_1^{\rho_{1}}, \frac{1}{v_1^{\rho_{2}}}\right) \ ^{\rho_{1}}I^{\alpha}_{1/u_2+}h\circ g_1\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right]\notag \\ &+\ ^{\rho_{2}}I^{\beta}_{1/v_2+}\left[ f\circ g_{2}\left(u_2^{\rho_{1}}, \frac{1}{v_1^{\rho_{2}}}\right) \ ^{\rho_{1}}I^{\alpha}_{1/u_1-}h\circ g_1\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right]\notag \\ \end{align} where \(g(x^{\rho_{1}},y^{\rho_{2}})=\left(\frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}} \right) \), \(g_{1}(x^{\rho_{1}},y^{\rho_{2}})=\left(\frac{1}{x^{\rho_{1}}},y^{\rho_{2}} \right)\) and \(g_{2}(x^{\rho_{1}},y^{\rho_{2}})=\left(x^{\rho_{1}},\frac{1}{y^{\rho_{2}}} \right) \), respectively.

Proof. Since \(f\) is co-ordinated harmonically convex on \(\Delta\), then the function \(f_{1/x^{\rho_{1}}}:[v_1^{\rho_{2}},v_2^{\rho_{2}}]\rightarrow \mathbb{R}\), defined by \(f_{1/x^{\rho_{1}}}(y^{\rho_{2}})=f(\frac{1}{x^{\rho_{1}}},y^{\rho_{2}})\) is harmonically convex on \([v_1^{\rho_{2}},v_2^{\rho_{2}}]\) for all \(x^{\rho_{1}}\in \left[ \frac{1}{u_2^{\rho_{1}}},\frac{1}{u_1^{\rho_{1}}}\right] \). Then from (6), we have

Multiplying both sides of (14) by \(\frac{x^{\rho_1-1}\left(x^{\rho_1}-\frac{1}{u_2^{\rho_1}} \right)^{\alpha-1}}{\rho_1^{\alpha-1}\Gamma(\alpha)}\) and \(\frac{x^{\rho_1-1}\left(\frac{1}{u_1^{\rho_1}}-x^{\rho_1} \right)^{\alpha-1}}{\rho_1^{\alpha-1}\Gamma(\alpha)}\), and integrating with respect to \(x\) over \(\left[ \frac{1}{u_2},\frac{1}{u_1}\right] \), respectively, we get and \begin{align*} \frac{\rho_1^{1-\alpha}\rho_2^{1-\beta}}{\Gamma(\alpha)\Gamma(\beta)}&\left[\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(\frac{1}{u_1^{\rho_1}}-x^{\rho_1} \right)^{\alpha-1}\left(y^{\rho_2}-\frac{1}{v_2^{\rho_2}} \right)^{\beta-1} x^{\rho_{1}-1} y^{\rho_{2}-1}f\left(\frac{1}{x^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}} \right)\right. \\ &\times h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx +\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(\frac{1}{u_1^{\rho_1}}-x^{\rho_1} \right)^{\alpha-1}\left(\frac{1}{v_1^{\rho_2}}-y^{\rho_2} \right)^{\beta-1}x^{\rho_{1}-1}y^{\rho_{2}-1} \\ &\times \left.f\left(\frac{1}{x^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}} \right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx\right] \\ \leq &\frac{\rho_1^{1-\alpha}\rho_2^{1-\beta}}{\Gamma(\alpha)\Gamma(\beta)}\left[\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(\frac{1}{u_1^{\rho_1}}-x^{\rho_1} \right)^{\alpha-1}\left(y^{\rho_2}-\frac{1}{v_2^{\rho_2}} \right)^{\beta-1}x^{\rho_1-1} y^{\rho_{2}-1}\right. \\ &\times f\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx +\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(\frac{1}{u_1^{\rho_1}}-x^{\rho_1} \right)^{\alpha-1}\left(\frac{1}{v_1^{\rho_2}}-y^{\rho_2} \right)^{\beta-1} \\ &\left.\times x^{\rho_1-1} y^{\rho_{2}-1}f\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx\right] \end{align*} Using similar arguments for the mapping \(f_{\frac{1}{y^{\rho_2}}}:[u_1^{\rho_1},u_2^{\rho_1}]\rightarrow \mathbb{R}\), \(f_{\frac{1}{y^{\rho_{2}}}}(x^{\rho_{1}})=f(x^{\rho_{1}},\frac{1}{y^{\rho_{2}}})\), we have and By adding the inequalities (15)\(\sim\)(18), we get \begin{align*} ^{\rho_{1}}I^{\alpha}_{1/u_1-}&\left[ f\circ g_{1}\left( \frac{1}{u_2^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right)\ ^{\rho_{2}}I^{\beta}_{1/v_1-}h\circ g_2\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right] \\&+\ ^{\rho_{1}}I^{\alpha}_{1/u_1-}\left[ f\circ g_{1}\left( \frac{1}{u_2^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right) \ ^{\rho_{2}}I^{\beta}_{1/v_2+}h\circ g_2\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right] \\ &+\ ^{\rho_{1}}I^{\alpha}_{1/u_2+}\left[ f\circ g_{1}\left( \frac{1}{u_1^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right)\ ^{\rho_{2}}I^{\beta}_{1/v_1-}h\circ g_2\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right] \\ &+\ ^{\rho_{1}}I^{\alpha}_{1/u_2+}\left[ f\circ g_{1}\left( \frac{1}{u_1^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right)\ ^{\rho_{2}}I^{\beta}_{1/v_2+}h\circ g_2\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right] \\ &+\ ^{\rho_{2}}I^{\beta}_{1/v_1-}\left[ f\circ g_{2}\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}}, \frac{1}{v_2^{\rho_{2}}}\right) \ ^{\rho_{1}}I^{\alpha}_{1/u_1-}h\circ g_1\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right] \\ &+ \ ^{\rho_{2}}I^{\beta}_{1/v_1-}\left[ f\circ g_{2}\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}}, \frac{1}{v_2^{\rho_{2}}}\right) \ ^{\rho_{1}}I^{\alpha}_{1/u_2+}h\circ g_1\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right]\\ &+\ ^{\rho_{2}}I^{\beta}_{1/v_2+}\left[ f\circ g_{2}\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}}, \frac{1}{v_1^{\rho_{2}}}\right) \ ^{\rho_{1}}I^{\alpha}_{1/u_1-}h\circ g_1\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right] \end{align*} \begin{align*} &+\ ^{\rho_{2}}I^{\beta}_{1/v_2+}\left[ f\circ g_{2}\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}}, \frac{1}{v_1^{\rho_{2}}}\right) \ ^{\rho_{1}}I^{\alpha}_{1/u_2+}h\circ g_1\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right] \\ \leq &2\left[\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_1-,1/v_1-}fh\circ g\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) +\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_1-,1/v_2+}fh\circ g\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right)\right. \\ &\left.+\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_2+,1/v_1-}fh\circ g\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) +\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_2+,1/v_2+}fh\circ g\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right] \\ \leq &\ ^{\rho_{1}}I^{\alpha}_{1/u_1-}\left[ f\circ g_{1}\left( \frac{1}{u_2^{\rho_{1}}},v_1^{\rho_{2}}\right)\ ^{\rho_{2}}I^{\beta}_{1/v_2+}h\circ g_2\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right]+\ ^{\rho_{1}}I^{\alpha}_{1/u_1-}\left[ f\circ g_{1}\left( \frac{1}{u_2^{\rho_{1}}},v_2^{\rho_{2}}\right)\right. \\ &\left.\times ^{\rho_{2}}I^{\beta}_{1/v_1-}h\circ g_2\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right]+\ ^{\rho_{1}}I^{\alpha}_{1/u_2+}\left[ f\circ g_{1}\left( \frac{1}{u_1^{\rho_{1}}},v_1^{\rho_{2}}\right)\ ^{\rho_{2}}I^{\beta}_{1/v_2+}h\circ g_2\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right] \\ &+\ ^{\rho_{1}}I^{\alpha}_{1/u_2+}\left[ f\circ g_{1}\left( \frac{1}{u_1^{\rho_{1}}},v_2^{\rho_{2}}\right)\ ^{\rho_{2}}I^{\beta}_{1/v_1-}h\circ g_2\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right]+\ ^{\rho_{2}}I^{\beta}_{1/v_1-}\left[ f\circ g_{2}\left(u_1^{\rho_{1}}, \frac{1}{v_2^{\rho_{2}}}\right)\right. \\ &\left.\times ^{\rho_{1}}I^{\alpha}_{1/u_2+}h\circ g_1\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right]+ \ ^{\rho_{2}}I^{\beta}_{1/v_1-}\left[ f\circ g_{2}\left(u_2^{\rho_{1}}, \frac{1}{v_2^{\rho_{2}}}\right) \ ^{\rho_{1}}I^{\alpha}_{1/u_1-}h\circ g_1\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right] \\ &+\ ^{\rho_{2}}I^{\beta}_{1/v_2+}\left[ f\circ g_{2}\left(u_1^{\rho_{1}}, \frac{1}{v_1^{\rho_{2}}}\right) \ ^{\rho_{1}}I^{\alpha}_{1/u_2+}h\circ g_1\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right]+\ ^{\rho_{2}}I^{\beta}_{1/v_2+}\left[ f\circ g_{2}\left(u_2^{\rho_{1}}, \frac{1}{v_1^{\rho_{2}}}\right)\right. \\ &\left.\times ^{\rho_{1}}I^{\alpha}_{1/u_1-}h\circ g_1\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right]. \end{align*} This completes the second and third inequality of (13). Now, using the first inequality of (6), we find Adding (19) and (20) and using the fact that \(h\) is symmetric, we get \begin{align*} \begin{split} f\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right)&\left[\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_1-,1/v_1-}h\circ g\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) +\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_1-,1/v_2+}h\circ g\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right)\right. \\ &\left.+\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_2+,1/v_1-}h\circ g\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) +\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_2+,1/v_2+}h\circ g\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right] \\ \leq& \ ^{\rho_{1}}I^{\alpha}_{1/u_1-}\left[ f\circ g_{1}\left( \frac{1}{u_2^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right)\ ^{\rho_{2}}I^{\beta}_{1/v_1-}h\circ g_2\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right] \\ &+\ ^{\rho_{1}}I^{\alpha}_{1/u_1-}\left[ f\circ g_{1}\left( \frac{1}{u_2^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right) \ ^{\rho_{2}}I^{\beta}_{1/v_2+}h\circ g_2\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right] \\ &+\ ^{\rho_{1}}I^{\alpha}_{1/u_2+}\left[ f\circ g_{1}\left( \frac{1}{u_1^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right)\ ^{\rho_{2}}I^{\beta}_{1/v_1-}h\circ g_2\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right] \\ &+\ ^{\rho_{1}}I^{\alpha}_{1/u_2+}\left[ f\circ g_{1}\left( \frac{1}{u_1^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right)\ ^{\rho_{2}}I^{\beta}_{1/v_2+}h\circ g_2\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right] \\ &+\ ^{\rho_{2}}I^{\beta}_{1/v_1-}\left[ f\circ g_{2}\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}}, \frac{1}{v_2^{\rho_{2}}}\right) \ ^{\rho_{1}}I^{\alpha}_{1/u_1-}h\circ g_1\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right] \\ &+ \ ^{\rho_{2}}I^{\beta}_{1/v_1-}\left[ f\circ g_{2}\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}}, \frac{1}{v_2^{\rho_{2}}}\right) \ ^{\rho_{1}}I^{\alpha}_{1/u_2+}h\circ g_1\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right] \\ &+\ ^{\rho_{2}}I^{\beta}_{1/v_2+}\left[ f\circ g_{2}\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}}, \frac{1}{v_1^{\rho_{2}}}\right) \ ^{\rho_{1}}I^{\alpha}_{1/u_1-}h\circ g_1\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right] \\ &+\ ^{\rho_{2}}I^{\beta}_{1/v_2+}\left[ f\circ g_{2}\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}}, \frac{1}{v_1^{\rho_{2}}}\right) \ ^{\rho_{1}}I^{\alpha}_{1/u_2+}h\circ g_1\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right]. \end{split} \end{align*} This completes the first inequality of (13). Now, to achieve the last inequality of (13), applying the second inequality of (6) as: and By adding the inequalities (21)\(\sim\)(24), we get the last inequality of (13).

Remark 3.

• 1)   From Theorems 9 and 10, we can get new Hermite-Hadamard-Fejér type inequalities for co-ordinated harmonically convex functions via Riemann-Liouville fractional integral by taking \(\rho_1=\rho_2=1\).
• 2)   From Theorems 9 and 10, we can get new Hermite-Hadamard-Fejér type inequalities for co-ordinated harmonically convex functions via classical integral by taking \(\rho_1=\rho_2=1\) and \(\alpha=\beta=1\).

4. Conclusion

In this paper, firstly we established the Hermite-Hadamard-Fejér type inequalities for harmonically convex function in one dimension which is further used to establish the Hermite-Hadamard-Fejér type inequalities for harmonically convex function via Katugampola fractional integral. The results provided in our paper are the generalizations of some earlier results.

Acknowledgments

This research is supported by National University of Science and Technology(NUST), Islamabad, Pakistan.

Author 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.