Weighted Midpoint Hermite-Hadamard-Fejér Type Inequalities in Fractional Calculus for Harmonically Convex Functions

Abstract

In this paper, we establish a new version of Hermite-Hadamard-Fejér type inequality for harmonically convex functions in the form of weighted fractional integral. Secondly, an integral identity and some weighted midpoint fractional Hermite-Hadamard-Fejér type integral inequalities for harmonically convex functions by involving a positive weighted symmetric functions have been obtained. As shown, all of the resulting inequalities generalize several well-known inequalities, including classical and Riemann–Liouville fractional integral inequalities.

1. Introduction

The theory of convex functions is an essential tool in various fields of pure and applied sciences. There is also a close connection between the theory of convex functions, the theory of inequalities, and fractional differential equations. At the same time, fractional differential equations are one of the most studied fields of mathematics due to their application in the real world. Many inequalities are proved for convex functions but, the most known from the related literature is Hermite-Hadamard inequality.

A function $\mathcal{F}:[{\theta }_1,{\theta }_2]\subset \mathbb{R}\to \mathbb{R}$ on an interval of real line is said to be convex, if for all ${\theta }_1,{\theta }_2\in I$ and $\tau \in [0,1]$, then

$$\mathcal{F}(\tau {\theta }_1+(1-\tau ){\theta }_2)\le \tau \mathcal{F}\left({\theta }_1\right)+(1-\tau )\mathcal{F}\left({\theta }_2\right).$$

(1)

The Hermite-Hadamard integral inequality is a well-known inequality in the subject of convex functional analysis. It has an interesting geometric representation with numerous important applications. The extraordinary inequality states that if $\mathcal{F}:I\to \mathbb{R}$ is a convex mapping on the interval I of real numbers and ${\theta }_1,{\theta }_2\in I$ with ${\theta }_1<{\theta }_2,$ then

$$\mathcal{F}\left(\frac{{\theta }_1+{\theta }_2}{2}\right)\le \frac{1}{{\theta }_2-{\theta }_1}\underset{{\theta }_1}{\overset{{\theta }_2}{\int }}\mathcal{F}\left(\tau \right)d\tau \le \frac{\mathcal{F}\left({\theta }_1\right)+\mathcal{F}\left({\theta }_2\right)}{2}.$$

(2)

Inequality (2) was introduced by C. Hermite [1] and investigated by J. Hadamard [2] in 1893. Both inequalities hold in the inverted direction if $\mathcal{F}$ is concave. Many mathematicians have paid considerable attention to Hermite-Hadamard inequality due to its quality and integrity in mathematical inequality. For significant developments, modifications, and consequences regarding the Hermite-Hadamard uniqueness property and general convex function definitions, for essential details, the interested reader would like to refer to [3,4,5,6,7] and references therein. Fractional calculus and applications have application areas in many different fields such as physics, chemistry and engineering, and mathematics. Applying arithmetic in classical analysis in the fractional analysis is very important in obtaining more realistic results in solving many problems. Many real dynamical systems are better characterized using non-integer order dynamic models based on fractional computation. While integer orders are a model that is not suitable for nature in classical analysis, fractional computation in which arbitrary orders are examined enables us to obtain more realistic approaches. Regarding some papers dealing with fractional integral inequalities via different types of fractional integral operators, we refer readers to [8,9,10,11,12,13,14,15,16].

Furthermore, Sarikaya et al. [17] generalized and reformed the Hermite-Hadamard integral inequality (2) in forms of Riemann–Liouville fractional integrals in 2013.

$$\mathcal{F}\left(\frac{{\theta }_1+{\theta }_2}{2}\right)\le \frac{\Gamma (v+1)}{2{({\theta }_2-{\theta }_1)}^v}\left[{\mathcal{I}}_{{\theta }_1^{+}}^v\mathcal{F}\left({\theta }_2\right)+{\mathcal{I}}_{{\theta }_2^{-}}^v\mathcal{F}\left({\theta }_1\right)\right]\le \frac{\mathcal{F}\left({\theta }_1\right)+\mathcal{F}\left({\theta }_2\right)}{2},$$

(3)

where the function $\mathcal{F}:[{\theta }_1,{\theta }_2]\to \mathbb{R}$ with $0\le {\theta }_1<{\theta }_2$ and $\mathcal{F}\in L^1[{\theta }_1,{\theta }_2]$. Therefore, ${\mathcal{I}}_{{\theta }_1^{+}}^v$ is left-sided Riemann–Liouville fractional integrals and ${\mathcal{I}}_{{\theta }_2^{-}}^v$ is the right-sided Riemann–Liouville fractional integrals with order $v>0$, defined by [18]

$$\begin{array}{c}{\mathcal{I}}_{{\theta }_1^{+}}^v\mathcal{F}\left(\tau \right)=\frac{1}{\Gamma \left(v\right)}{\int }_{{\theta }_1}^{\tau }{(\tau -\kappa )}^{v-1}\mathcal{F}\left(\kappa \right)d\kappa ,\tau >{\theta }_1,\\ {\mathcal{I}}_{{\theta }_2^{-}}^v\mathcal{F}\left(\tau \right)=\frac{1}{\Gamma \left(v\right)}{\int }_{\tau }^{{\theta }_2}{(\kappa -\tau )}^{v-1}\mathcal{F}\left(\kappa \right)d\kappa ,\tau <{\theta }_2.\end{array}$$

(4)

respectively. Here, $\Gamma \left(v\right)$ is the Gamma function and ${\mathcal{I}}_{{\theta }_1^{+}}^0\mathcal{F}\left(\tau \right)={\mathcal{I}}_{{\theta }_2^{-}}^0\mathcal{F}\left(\tau \right)=\mathcal{F}\left(\tau \right).$

Due to the use of the interval’s ends ${\theta }_1,{\theta }_2$, the inequality (3) is called endpoint Hermite-Hadamard inequality.

The midpoint Hermite-Hadamard inequality was discovered by Sarikaya and Yildirim [19] after expending the essential area of the integral inequalities in (2) and (3)

$$\mathcal{F}\left(\frac{{\theta }_1+{\theta }_2}{2}\right)\le \frac{2^{v-1}\Gamma (v+1)}{{({\theta }_2-{\theta }_1)}^v}\left[{\mathcal{I}}_{{\left(\frac{{\theta }_1+{\theta }_2}{2}\right)}^{+}}^v\mathcal{F}\left({\theta }_2\right)+{\mathcal{I}}_{{\left(\frac{{\theta }_1+{\theta }_2}{2}\right)}^{-}}^v\mathcal{F}\left({\theta }_1\right)\right]\le \frac{\mathcal{F}\left({\theta }_1\right)+\mathcal{F}\left({\theta }_2\right)}{2},$$

(5)

where the function $\mathcal{F}:[{\theta }_1,{\theta }_2]\to \mathbb{R}$ is convex and continuous. In [20], I. G. Macdonald gave the following definition.

Definition 1.

Suppose that a function $\mathcal{G}:[{\theta }_1,{\theta }_2]\to [0,\infty )$ and it is symmetric with respect to $\frac{{\theta }_1+{\theta }_2}{2}$ if

$$\mathcal{G}({\theta }_1+{\theta }_2-\kappa )=\mathcal{G}\left(\kappa \right),\mathrm{for}\mathrm{all}\kappa \in [{\theta }_1,{\theta }_2].$$

Fejér proposed the following generalization of Hadamard inequality in 1906 (see [21]):

Theorem 1.

Let $\mathcal{F}:[{\theta }_1,{\theta }_2]\to \mathbb{R}$ be a convex function such that ${\theta }_1<{\theta }_2$. Furthermore, let $\mathcal{G}:[{\theta }_1,{\theta }_2]\to \mathbb{R}$ be a positive, integrable and symmetric to $\frac{{\theta }_1+{\theta }_2}{2}$. Then the following inequality holds:

$$\mathcal{F}\left(\frac{{\theta }_1+{\theta }_2}{2}\right){\int }_{{\theta }_1}^{{\theta }_2}\mathcal{G}\left(\tau \right)d\tau \le {\int }_{{\theta }_1}^{{\theta }_2}\mathcal{F}\left(\tau \right)\mathcal{G}\left(\tau \right)d\tau \le \frac{\mathcal{F}\left({\theta }_1\right)+\mathcal{F}\left({\theta }_2\right)}{2}{\int }_{{\theta }_1}^{{\theta }_2}\mathcal{G}\left(\tau \right)d\tau .$$

(6)

The inequality (6) is well-known as the Fejér-Hadamard inequality in the literature. In the concept of Riemann–Liouville fractional integrals, İ. İşcan [22] discovered the endpoint version of (6), which is also the extension of (3). As a result, the final inequalities are shown as follows:

$$\begin{array}{cc}\mathcal{F}\left(\frac{{\theta }_1+{\theta }_2}{2}\right)\left[{\mathcal{I}}_{{\theta }_1^{+}}^v\mathcal{F}\left({\theta }_2\right)+{\mathcal{I}}_{{\theta }_2^{-}}^v\left(\mathcal{F}\right)\left({\theta }_1\right)\right]\hfill & \le \left[{\mathcal{I}}_{{\theta }_1^{+}}^v\left(\mathcal{F}\mathcal{G}\right)\left({\theta }_2\right)+{\mathcal{I}}_{{\theta }_2^{-}}^v\left(\mathcal{F}\mathcal{G}\right)\left({\theta }_1\right)\right]\hfill \\ \hfill & \le \frac{\mathcal{F}\left({\theta }_1\right)+\mathcal{F}\left({\theta }_2\right)}{2}\left[{\mathcal{I}}_{{\theta }_1^{+}}^v\mathcal{F}\left({\theta }_2\right)+{\mathcal{I}}_{{\theta }_2^{-}}^v\mathcal{F}\left({\theta }_1\right)\right],\hfill \end{array}$$

(7)

where $\mathcal{F}$ is convex and continuous, $\mathcal{G}$ is symmetric and belongs to $L^1[{\theta }_1,{\theta }_2],$ (see Definition 1).

In [23], İ. İşcan gave definition of harmonically convex functions and established following Hermite-Hadamard type inequality for harmonically convex functions as follows:

Definition 2.

Let $I\subset \mathbb{R}\setminus \left\{0\right\}$ be an interval of nonzero real numbers. Then a function $\mathcal{F}:I\to \mathbb{R}$ is said to be harmonically convex if

$$\mathcal{F}\left(\frac{{\theta }_1{\theta }_2}{{\theta }_1\tau +(1-\tau ){\theta }_2}\right)\le \tau \mathcal{F}\left({\theta }_2\right)+(1-\tau )\mathcal{F}\left({\theta }_1\right),$$

holds for all ${\theta }_1,{\theta }_2\in I$ and $\tau \in [0,1]$.

In [24], Latif et al. gave the following definition.

Definition 3.

A function $\mathcal{F}:[{\theta }_1,{\theta }_2]\subseteq \mathbb{R}\setminus \left\{0\right\}\to \mathbb{R}$ is said to be harmonically symmetric with respect to $\frac{2{\theta }_1{\theta }_2}{{\theta }_1+{\theta }_2},$ if

$$\mathcal{F}\left(\frac{1}{\kappa }\right)=\mathcal{F}\left(\frac{1}{\frac{1}{{\theta }_1}+\frac{1}{{\theta }_2}-\kappa }\right),$$

$\kappa \in [{\theta }_1,{\theta }_2].$

İşcan et al. published Hermite-inequality Hadamard’s in fractional integral type for harmonically convex functions In [22], as follows:

Theorem 2.

Let $\mathcal{F}:I\subset \mathbb{R}\setminus \left\{0\right\}\to \mathbb{R}$ be a harmonically convex function and ${\theta }_1,{\theta }_2\in I$ with ${\theta }_1<{\theta }_2$. If $\mathcal{F}\in L^1[{\theta }_1,{\theta }_2],$ then the following inequalities holds:

$$\mathcal{F}\left(\frac{2{\theta }_1{\theta }_2}{{\theta }_1+{\theta }_2}\right)\le \frac{{\theta }_1{\theta }_2}{{\theta }_2-{\theta }_1}{\int }_{{\theta }_1}^{{\theta }_2}\frac{\mathcal{F}\left(\tau \right)}{{\tau }^2}d\tau \le \frac{\mathcal{F}\left({\theta }_1\right)+\mathcal{F}\left({\theta }_2\right)}{2}.$$

(8)

Hermite-Hadamard inequalities for harmonically convex functions were introduced in fractional integral form in [25] as follows:

Theorem 3.

Consider a function $\mathcal{F}:I\subset (0,\infty )\to \mathbb{R}$ such that $\mathcal{F}\in L^1[{\theta }_1,{\theta }_2]$, where ${\theta }_1,{\theta }_2\in I$ with ${\theta }_1<{\theta }_2$. If $\mathcal{F}$ is a harmonically convex function on $[{\theta }_1,{\theta }_2]$, then

$$\begin{array}{cc}\mathcal{F}\left(\frac{2{\theta }_1{\theta }_2}{{\theta }_1+{\theta }_2}\right)\hfill & \le \frac{\Gamma (v+1)}{2}{\left(\frac{{\theta }_1{\theta }_2}{{\theta }_1+{\theta }_2}\right)}^v\left[{\mathcal{I}}_{\frac{1}{{\theta }_1}-}^v\left(\mathcal{F}\circ \mathcal{H}\right)\left(\frac{1}{{\theta }_2}\right){\mathcal{I}}_{\frac{1}{{\theta }_2}+}^v\left(\mathcal{F}\circ \mathcal{H}\right)\left(\frac{1}{{\theta }_1}\right)\right]\hfill \\ \hfill & \le \frac{\mathcal{F}\left({\theta }_1\right)+\mathcal{F}\left({\theta }_2\right)}{2},\hfill \end{array}$$

(9)

with $v>0$ and $\mathcal{H}\left(\kappa \right)=\frac{1}{\kappa }$ where $\kappa =\left[\frac{1}{{\theta }_2},\frac{1}{{\theta }_1}\right].$

For harmonically convex functions, in [17] Chan et al. stated the Hermite-Hadamard-Fejér inequality as follows:

Theorem 4.

Suppose that harmonically convex function $\mathcal{F}:I\subseteq \mathbb{R}\setminus \left\{0\right\}\to \mathbb{R}$. If $\mathcal{F}\in L^1[{\theta }_1,{\theta }_2]$ and $\mathcal{G}:[{\theta }_1,{\theta }_2]\subseteq \mathbb{R}\setminus \left\{0\right\}\to \mathbb{R}$ is positive, an integrable, and harmonically symmetric with respect to $\frac{2{\theta }_1{\theta }_2}{{\theta }_1+{\theta }_2},$ then

$$\mathcal{F}\left(\frac{2{\theta }_1{\theta }_2}{{\theta }_1+{\theta }_2}\right){\int }_{{\theta }_1}^{{\theta }_2}\frac{\mathcal{G}\left(\tau \right)}{{\tau }^2}d\tau \le {\int }_{{\theta }_1}^{{\theta }_2}\frac{\mathcal{F}\left(\tau \right)\mathcal{G}\left(\tau \right)}{{\tau }^2}d\tau \le \frac{\mathcal{F}\left({\theta }_1\right)+\mathcal{F}\left({\theta }_2\right)}{2}{\int }_{{\theta }_1}^{{\theta }_2}\frac{\mathcal{G}\left(\tau \right)}{{\tau }^2}d\tau ,$$

(10)

and ${\theta }_1,{\theta }_2\in I$ with ${\theta }_1<{\theta }_2.$

In [26], İşcan et al. proved Hermite-Hadamard-Fejér type inequalities for harmonically convex functions through fractional integrals:

Theorem 5.

Let $\mathcal{F}:[{\theta }_1,{\theta }_2]\to \mathbb{R}$ be a harmonically convex function and ${\theta }_1,{\theta }_2\in I$ with ${\theta }_1<{\theta }_2$. If $\mathcal{F}\in L^1[{\theta }_1,{\theta }_2]$ and $\mathcal{G}:[{\theta }_1,{\theta }_2]\to \mathbb{R}$ is positive, an integrable, and harmonically symmetric with respect to $\frac{2{\theta }_1{\theta }_2}{{\theta }_1+{\theta }_2},$ then

$$\begin{array}{c}\mathcal{F}\left(\frac{2{\theta }_1{\theta }_2}{{\theta }_1+{\theta }_2}\right)\left[{\mathcal{I}}_{\frac{1}{{\theta }_1}-}^v\left(\mathcal{G}\circ \mathcal{H}\right)\left(\frac{1}{{\theta }_2}\right){\mathcal{I}}_{\frac{1}{{\theta }_2}+}^v\left(\mathcal{G}\circ \mathcal{H}\right)\left(\frac{1}{{\theta }_1}\right)\right]\hfill \\ \le \left[{\mathcal{I}}_{\frac{1}{{\theta }_1}-}^v\left(\mathcal{F}\mathcal{G}\circ \mathcal{H}\right)\left(\frac{1}{{\theta }_2}\right){\mathcal{I}}_{\frac{1}{{\theta }_2}+}^v\left(\mathcal{F}\mathcal{G}\circ \mathcal{H}\right)\left(\frac{1}{{\theta }_1}\right)\right]\hfill \\ \le \frac{\mathcal{F}\left({\theta }_1\right)+\mathcal{F}\left({\theta }_2\right)}{2}\left[{\mathcal{I}}_{\frac{1}{{\theta }_1}-}^v\left(\mathcal{G}\circ \mathcal{H}\right)\left(\frac{1}{{\theta }_2}\right){\mathcal{I}}_{\frac{1}{{\theta }_2}+}^v\left(\mathcal{G}\circ \mathcal{H}\right)\left(\frac{1}{{\theta }_1}\right)\right],\hfill \end{array}$$

(11)

with $v>0$ and $\mathcal{H}\left(\kappa \right)=\frac{1}{\kappa },\kappa \in \left[\frac{1}{{\theta }_2},\frac{1}{{\theta }_1}\right].$

In [27], Fahad et al. presented weighted fractional integrals as follows:

Definition 4.

Let $({\theta }_1,{\theta }_2)\le \mathbb{R}$ and $\sigma \left(\tau \right)$ be an increasing positive and monotone function on the interval $({\theta }_1,{\theta }_2]$ with a continuous derivative ${\sigma }^{\prime }\left(\tau \right)$ on the open interval $({\theta }_1,{\theta }_2)$. Then the weighted fractional integrals on the left and right sides of a function $\mathcal{F}$ according to another function $\sigma \left(\tau \right)$ on $[{\theta }_1,{\theta }_2]$ are shown as below:

$$\begin{array}{cc}& \left({}_w{\mathcal{I}}_{{\theta }_1^{+}}^{v,\sigma }\mathcal{F}\right)\left(\tau \right)={\displaystyle \frac{{\left[w\left(\tau \right)\right]}^{-1}}{\Gamma \left(v\right)}}{\int }_{{\theta }_1}^{\tau }{\sigma }^{\prime }\left(\kappa \right){\left(\sigma \left(\tau \right)-\sigma \left(\kappa \right)\right)}^{v-1}\mathcal{F}\left(\kappa \right)w\left(\kappa \right)d\kappa ,\\ & \left({}_w{\mathcal{I}}_{{\theta }_2^{-}}^{v,\sigma }\mathcal{F}\right)\left(\tau \right)={\displaystyle \frac{{\left[w\left(\tau \right)\right]}^{-1}}{\Gamma \left(v\right)}}{\int }_{\tau }^{{\theta }_2}{\sigma }^{\prime }\left(\kappa \right){\left(\sigma \left(\kappa \right)-\sigma \left(\tau \right)\right)}^{v-1}\mathcal{F}\left(\kappa \right)w\left(\kappa \right)d\kappa ,\end{array}$$

(12)

with $v>0$ and for ${\left[w\left(\tau \right)\right]}^{-1}:=\frac{1}{w\left(\tau \right)}$ with $w\left(\tau \right)\ne 0.$

Remark 1.

Using Definition 4, we have

 (i)

Equation (12) can be restated in the following form: If $\sigma \left(\tau \right)=\tau$ with $w\left(\tau \right)=1$, then the weighted fractional integrals reduce to the classical Riemann–Liouville fractional integrals.

 (ii)

Putting $w\left(\tau \right)=1$, so we obtain the fractional integrals of $\mathcal{F}$ with regard to the function $\sigma \left(\tau \right),$ for more details see [28,29]:

$$\begin{array}{cc}& \left({\mathcal{I}}_{{\theta }_1^{+}}^{v,\sigma }\mathcal{F}\right)\left(\tau \right)=\frac{1}{\Gamma \left(v\right)}{\int }_{{\theta }_1}^{\tau }{\sigma }^{\prime }\left(\kappa \right){\left(\sigma \left(\tau \right)-\sigma \left(\kappa \right)\right)}^{v-1}\mathcal{F}\left(\kappa \right)w\left(\kappa \right)d\kappa ,\\ & \left({\mathcal{I}}_{{\theta }_2^{-}}^{v,\sigma }\mathcal{F}\right)\left(\tau \right)=\frac{1}{\Gamma \left(v\right)}{\int }_{\tau }^{{\theta }_2}{\sigma }^{\prime }\left(\kappa \right){\left(\sigma \left(\kappa \right)-\sigma \left(\tau \right)\right)}^{v-1}\mathcal{F}\left(\kappa \right)w\left(\kappa \right)d\kappa ,\end{array}$$

(13)

with$v>0.$

We recall the following special functions which are known as Beta and hypergeometric function

$$\beta ({\gamma }_1,{\gamma }_2)=\frac{\Gamma \left({\gamma }_1\right)\Gamma \left({\gamma }_2\right)}{\Gamma ({\gamma }_1+{\gamma }_2)}={\int }_0^1{\tau }^{{\gamma }_1-1}{(1-\tau )}^{{\gamma }_2-1}d\tau ,{\gamma }_1,{\gamma }_2>0.$$

$$\begin{array}{c}{}_2{F}_1({\gamma }_1,{\gamma }_2;{\gamma }_3,\zeta )=\frac{1}{\beta ({\gamma }_2,{\gamma }_3-{\gamma }_2)}{\int }_0^1{\tau }^{{\gamma }_2-1}{(1-\tau )}^{{\gamma }_3-{\gamma }_2-1}{(1-\zeta \tau )}^{-{\gamma }_1}d\tau ,\\ {\gamma }_3>{\gamma }_2>0,\left|\zeta \right|<1,\end{array}$$

respectively, (see [18]).

The polygamma function of order m is a meromorphic function on the complex numbers $\mathbb{C}$ defined as the $(m+1)$th derivative of the logarithm of the gamma function:

$${\psi }^{\left(m\right)}\left(\zeta \right):=\frac{d^m}{d{\zeta }^m}\psi \left(\zeta \right)=\frac{d^{m+1}}{d{\zeta }^{m+1}}ln\Gamma \left(\zeta \right).$$

Thus

$${\psi }^{\left(0\right)}\left(\zeta \right)=\psi \left(\zeta \right)=\frac{{\Gamma }^{{}^{\prime }}\left(\zeta \right)}{\Gamma \left(\zeta \right)}$$

holds where $\psi \left(\zeta \right)$ is the digamma function and $\Gamma \left(\zeta \right)$ is the gamma function.

When $m>0$ and $Re>0$, the integral representation of polygamma is given by

$$\begin{array}{cc}\hfill {\psi }^{\left(m\right)}\left(\zeta \right)& ={\left(-1\right)}^{m+1}{\int }_0^{\infty }\frac{{\tau }^m{e}^{-\zeta \tau }}{1-e^{-\tau }}d\tau \hfill \\ \hfill & =-{\int }_0^1\frac{{\tau }^{\zeta -1}}{1-\tau }{\left(ln\tau \right)}^{m}d\tau .\hfill \end{array}$$

The generalized hypergeometric function is given by a hypergeometric series, i.e., a series for which the ratio of successive terms can be written as

$$\frac{c_{k+1}}{c_k}=\frac{(k+{\theta }_1)(k+{\theta }_2)...(k+{\theta }_p)}{(k+{\pi }_1)(k+{\pi }_2)...(k+{\pi }_q)(k+1)}.$$

The resulting generalized hypergeometric function is written

$$\begin{array}{cc}\hfill \sum _{k=0}^{\infty }{c}_k{x}^k& {=}_p{F}_q\left[\begin{array}{cc}\begin{array}{c}{\theta }_1,{\theta }_2...,{\theta }_p\\ {\pi }_1,{\pi }_2...,{\pi }_p\end{array}& ;x\end{array}\right]\hfill \\ \hfill & {=}_p{F}_q\left({\theta }_1,{\theta }_2...,{\theta }_p;{\pi }_1,{\pi }_2...,{\pi }_p;x\right)\hfill \\ \hfill & =\sum _{k=0}^{\infty }\frac{{\left({\theta }_1\right)}_k{\left({\theta }_2\right)}_k...{\left({\theta }_p\right)}_k{x}^k}{{\left({\pi }_1\right)}_k{\left({\pi }_2\right)}_k...{\left({\pi }_p\right)}_{k}k!},\hfill \end{array}$$

where $x\in \mathbb{C},$ $p\le q,$ ${\theta }_i,{\pi }_j\in \mathbb{C},$ ${\pi }_j\ne 0,-1,-2,\cdots ,i=1,2,\cdots ,p,j=1,2,\cdots ,q$ and ${\left(\theta \right)}_k$ is the Pochhammer symbol or rising factorial defined by

$${\left(\theta \right)}_k=\frac{\Gamma \left(\theta +k\right)}{\Gamma \left(\theta \right)}=\theta \left(\theta +1\right)\left(\theta +2\right)...\left(\theta +k+1\right).$$

In this article, we will use fractional weighted integrals (12) with nonnegative symmetric weighted functions in the kernel, necessary and auxiliary lemmas to study Hermite-Hadamard-Fejér type inequalities in Section 2. We shall prove our key results in Section 3, which will include new midpoint fractional Hermite-Hadamard-Fejér type integral inequalities as well as some related results. The conclusion will be presented in Section 4.

2. Auxiliary Results

Lemma 1.

If $\mathcal{F}:[{\theta }_1,{\theta }_2]\to \mathbb{R}$ is integrable and harmonically symmetric with respect to $\frac{2{\theta }_1{\theta }_2}{{\theta }_1+{\theta }_2}$, then

 (i)

for each $\kappa \in [0,1]$, we have

$$w\left(\frac{2{\theta }_1{\theta }_2}{\kappa {\theta }_2+\left(2-\kappa \right){\theta }_1}\right)=w\left(\frac{2{\theta }_1{\theta }_2}{\kappa {\theta }_1+\left(2-\kappa \right){\theta }_2}\right).$$

(14)

 (ii)

For $v>0$, we have

$$\begin{array}{c}\hfill \left({\mathcal{I}}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{+}}^{v,\sigma }\left(w\circ \mathcal{H}\circ \sigma \right)\right)\left({\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)\right)=\left({}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{-}}{\mathcal{I}}^{v,\sigma }\left(w\circ \mathcal{H}\circ \sigma \right)\right)\left({\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)\right)\\ \hfill =\frac{1}{2}\left[\left({\mathcal{I}}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{+}}^{v,\sigma }\left(w\circ \mathcal{H}\circ \sigma \right)\right)\left({\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)\right)\right.\\ \hfill \left.+\left({}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{-}}{\mathcal{I}}^{v,\sigma }\left(w\circ \mathcal{H}\circ \sigma \right)\right)\left({\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)\right)\right],\end{array}$$

(15)

and $\mathcal{H}\left(\tau \right)=\frac{1}{\tau },\tau \in \left[\frac{1}{{\theta }_2},\frac{1}{{\theta }_1}\right].$

Proof.

(i)

Suppose that $\tau =\frac{2{\theta }_1{\theta }_2}{\kappa {\theta }_2+\left(2-\kappa \right){\theta }_1}$ with $\tau \in \left[{\theta }_1,{\theta }_2\right]$ and $\kappa \in [0,1]$ such that $\frac{1}{\frac{1}{{\theta }_1}+\frac{1}{{\theta }_2}-\frac{1}{\tau }}=\frac{2{\theta }_1{\theta }_2}{\kappa {\theta }_1+\left(2-\kappa \right){\theta }_2}$. Then, using the hypotheses and Definition 3, we can obtain (14).

(ii)

w has symmetry characteristic, then

$$\left(w\circ \mathcal{H}\circ \sigma \right)\left(\kappa \right)=w\left(\frac{1}{\sigma \left(\kappa \right)}\right)=w\left(\frac{1}{\frac{1}{{\theta }_1}+\frac{1}{{\theta }_2}-\sigma \left(\kappa \right)}\right),\forall \kappa \in \left[{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right),{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)\right].$$

Hence, from above and setting $\frac{1}{\sigma \left(\tau \right)}=\frac{1}{\frac{1}{{\theta }_1}+\frac{1}{{\theta }_2}-\sigma \left(\kappa \right)}$, it follows that

$$\begin{array}{cc}\hfill & \left({}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{-}}{\mathcal{I}}^{v,\sigma }\left(w\circ \mathcal{H}\circ \sigma \right)\right)\left({\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)\right)\hfill \\ \hfill & =\frac{1}{\Gamma \left(v\right)}{\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}{\left(\sigma \left(\tau \right)-\frac{1}{{\theta }_2}\right)}^{v-1}\left(w\circ \mathcal{H}\circ \sigma \right)\left(\tau \right){\sigma }^{\prime }\left(\tau \right)d\tau \hfill \\ \hfill & =\frac{1}{\Gamma \left(v\right)}{\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}{\left(\sigma \left(\tau \right)-\frac{1}{{\theta }_2}\right)}^{v-1}w\left(\frac{1}{\frac{1}{{\theta }_1}+\frac{1}{{\theta }_2}-\sigma \left(\tau \right)}\right){\sigma }^{\prime }\left(\tau \right)d\tau \hfill \\ \hfill & =\frac{1}{\Gamma \left(v\right)}{\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}^{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}{\left(\frac{1}{{\theta }_1}-\sigma \left(\kappa \right)\right)}^{v-1}w\left(\frac{1}{\sigma \left(\kappa \right)}\right)\left(-{\sigma }^{\prime }\left(\kappa \right)\right)d\kappa \hfill \\ \hfill & =\frac{1}{\Gamma \left(v\right)}{\int }_{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}{\left(\frac{1}{{\theta }_1}-\sigma \left(\tau \right)\right)}^{v-1}\left(w\circ \mathcal{H}\circ \sigma \right)\left(\kappa \right){\sigma }^{\prime }\left(\kappa \right)d\kappa \hfill \\ \hfill & =\left({\mathcal{I}}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{+}}^{v,\sigma }\left(w\circ \mathcal{H}\circ \sigma \right)\right)\left({\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)\right),\hfill \end{array}$$

which brings the needed equality (15).

□

Theorem 6.

Let $0<{\theta }_1<{\theta }_2$, let $\mathcal{F}:[{\theta }_1,{\theta }_2]\to \mathbb{R}$ be an $L^1$ harmonically convex function and $w:[{\theta }_1,{\theta }_2]\to \mathbb{R}$ is nonnegative, an integrable, and symmetric weighted function with respect to $\frac{2{\theta }_1{\theta }_2}{{\theta }_1+{\theta }_2}$. If σ is is an increasing and positive function from $[{\theta }_1,{\theta }_2)$ onto itself such that its derivative ${\sigma }^{{}^{\prime }}\left(\tau \right)$ is continuous on $({\theta }_1,{\theta }_2)$ for $v>0$, then

$$\begin{array}{c}\mathcal{F}\left(\frac{2{\theta }_1{\theta }_2}{{\theta }_1+{\theta }_2}\right)\left[\left({\mathcal{I}}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{+}}^{v,\sigma }\left(w\circ \mathcal{H}\circ \sigma \right)\right)\left({\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)\right)+\left({}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{-}}{\mathcal{I}}^{v,\sigma }\left(w\circ \mathcal{H}\circ \sigma \right)\right)\right.\hfill \\ \left.\times \left({\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)\right)\right]\le w\left(\frac{1}{{\theta }_1}\right)\left({}_{w\circ \mathcal{H}\circ \sigma }{\mathcal{I}}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{+}}^{v,\sigma }\left(\mathcal{F}\circ \mathcal{H}\circ \sigma \right)\right)\left({\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)\right)\hfill \\ +w\left(\frac{1}{{\theta }_2}\right)\left({}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{-}}{\mathcal{I}}_{w\circ \mathcal{H}\circ \sigma }^{v,\sigma }\left(\mathcal{F}\circ \mathcal{H}\circ \sigma \right)\right)\left({\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)\right)\le \frac{\mathcal{F}\left({\theta }_1\right)+\mathcal{F}\left({\theta }_2\right)}{2}\hfill \\ \times \left[\left({\mathcal{I}}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{+}}^{v,\sigma }\left(w\circ \mathcal{H}\circ \sigma \right)\right)\left({\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)\right)+\left({}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{-}}{\mathcal{I}}^{v,\sigma }\left(w\circ \mathcal{H}\circ \sigma \right)\right)\left({\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)\right)\right].\hfill \end{array}$$

(16)

Proof. 

Since $\mathcal{F}$ is a harmonically convex function on $[{\theta }_1,{\theta }_2]$, we write

$$\mathcal{F}\left(\frac{2{\pi }_1{\pi }_2}{{\pi }_1+{\pi }_2}\right)\le \frac{\mathcal{F}\left({\pi }_1\right)+\mathcal{F}\left({\pi }_2\right)}{2},\mathrm{for}\mathrm{all}{\pi }_1,{\pi }_2\in [{\theta }_1,{\theta }_2].$$

Therefore, for ${\pi }_1=\frac{2{\theta }_1{\theta }_2}{\kappa {\theta }_1+\left(2-\kappa \right){\theta }_2}$ and ${\pi }_2=\frac{2{\theta }_1{\theta }_2}{\kappa {\theta }_2+\left(2-\kappa \right){\theta }_1}$, $\kappa \in [0,1],$ it follows

$$2\mathcal{F}\left(\frac{2{\theta }_1{\theta }_2}{{\theta }_1+{\theta }_2}\right)\le \mathcal{F}\left(\frac{2{\theta }_1{\theta }_2}{\kappa {\theta }_1+\left(2-\kappa \right){\theta }_2}\right)+\mathcal{F}\left(\frac{2{\theta }_1{\theta }_2}{\kappa {\theta }_2+\left(2-\kappa \right){\theta }_1}\right).$$

(17)

Multiplying both sides of (17) by ${\kappa }^{v-1}w\left(\frac{2{\theta }_1{\theta }_2}{\kappa {\theta }_1+\left(2-\kappa \right){\theta }_2}\right)$ and integrating the resulting inequality with respect to $\kappa$ over $[0,1]$, we obtain

$$\begin{array}{c}2\mathcal{F}\left({\displaystyle \frac{2{\theta }_1{\theta }_2}{{\theta }_1+{\theta }_2}}\right){\int }_0^1{\kappa }^{v-1}w\left({\displaystyle \frac{2{\theta }_1{\theta }_2}{\kappa {\theta }_1+\left(2-\kappa \right){\theta }_2}}\right)d\kappa \hfill \\ \le {\int }_0^1{\kappa }^{v-1}\mathcal{F}\left({\displaystyle \frac{2{\theta }_1{\theta }_2}{\kappa {\theta }_1+\left(2-\kappa \right){\theta }_2}}\right)w\left({\displaystyle \frac{2{\theta }_1{\theta }_2}{\kappa {\theta }_1+\left(2-\kappa \right){\theta }_2}}\right)d\kappa \hfill \\ +{\int }_0^1{\kappa }^{v-1}\mathcal{F}\left({\displaystyle \frac{2{\theta }_1{\theta }_2}{\kappa {\theta }_2+\left(2-\kappa \right){\theta }_1}}\right)w\left({\displaystyle \frac{2{\theta }_1{\theta }_2}{\kappa {\theta }_1+\left(2-\kappa \right){\theta }_2}}\right)d\kappa .\hfill \end{array}$$

(18)

From the left-hand side of the inequality in (18), we use (15) to obtain

$$\begin{array}{cc}\hfill & \frac{1}{2}{\left(\frac{2{\theta }_1{\theta }_2}{{\theta }_2-\theta {}_1}\right)}^v\Gamma \left(v\right)\left[\left({\mathcal{I}}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{+}}^{v,\sigma }\left(w\circ \mathcal{H}\circ \sigma \right)\right)\left({\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)\right)\right.\hfill \\ \hfill & \left.+\left({}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{-}}{\mathcal{I}}^{v,\sigma }\left(w\circ \mathcal{H}\circ \sigma \right)\right)\left({\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)\right)\right]\hfill \\ \hfill & ={\left(\frac{2{\theta }_1{\theta }_2}{{\theta }_2-{\theta }_1}\right)}^v\Gamma \left(v\right)\left({\mathcal{I}}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{+}}^{v,\sigma }\left(w\circ \mathcal{H}\circ \sigma \right)\right)\left({\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)\right)\hfill \\ \hfill & ={\left(\frac{2{\theta }_1{\theta }_2}{{\theta }_2-{\theta }_1}\right)}^v{\int }_{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}{\left(\frac{1}{{\theta }_1}-\sigma \left(\tau \right)\right)}^{v-1}\left(w\circ \mathcal{H}\circ \sigma \right)\left(\tau \right){\sigma }^{\prime }\left(\tau \right)d\tau \hfill \\ \hfill & ={\int }_{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}{\left(\frac{2{\theta }_1{\theta }_2\left(\frac{1}{{\theta }_1}-\sigma \left(\tau \right)\right)}{{\theta }_2-{\theta }_1}\right)}^{v-1}\left(w\circ \mathcal{H}\circ \sigma \right)\left(\tau \right){\sigma }^{\prime }\left(\tau \right)\frac{2{\theta }_1{\theta }_{2}d\tau }{{\theta }_2-{\theta }_1}\hfill \\ \hfill & ={\int }_0^1{\kappa }^{v-1}w\left(\frac{2{\theta }_1{\theta }_2}{\kappa {\theta }_1+\left(2-\kappa \right){\theta }_2}\right)d\kappa .\hfill \end{array}$$

It follows that

$$\begin{array}{c}2\mathcal{F}\left(\frac{2{\theta }_1{\theta }_2}{{\theta }_1+{\theta }_2}\right){\int }_0^1{\kappa }^{v-1}w\left(\frac{2{\theta }_1{\theta }_2}{\kappa {\theta }_1+\left(2-\kappa \right){\theta }_2}\right)d\kappa ={\left(\frac{2{\theta }_1{\theta }_2}{{\theta }_2-\theta {}_1}\right)}^v\Gamma \left(v\right)\mathcal{F}\left(\frac{2{\theta }_1{\theta }_2}{{\theta }_1+{\theta }_2}\right)\hfill \\ \times \left[\left({\mathcal{I}}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{+}}^{v,\sigma }\left(w\circ \mathcal{H}\circ \sigma \right)\right)\left({\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)\right)+\left({}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{-}}{\mathcal{I}}^{v,\sigma }\left(w\circ \mathcal{H}\circ \sigma \right)\right)\left({\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)\right)\right].\hfill \end{array}$$

(19)

We can demonstrate that by calculating the weighted fractional operators,

$$\begin{array}{cc}\hfill & w\left(\frac{1}{{\theta }_1}\right)\left({\mathcal{I}}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{+}}^{v,\sigma }\left(w\circ \mathcal{H}\circ \sigma \right)\right)\left({\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)\right)+w\left(\frac{1}{{\theta }_2}\right)\left({}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{-}}{\mathcal{I}}^{v,\sigma }\left(w\circ \mathcal{H}\circ \sigma \right)\right)\left({\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)\right)\hfill \\ \hfill & =w\left(\frac{1}{{\theta }_1}\right)\frac{{\left(w\circ \mathcal{H}\circ \sigma \right)}^{-1}{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}{\Gamma \left(v\right)}{\int }_{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}{\left(\frac{1}{{\theta }_1}-\sigma \left(\tau \right)\right)}^{v-1}\left(\mathcal{F}\circ \mathcal{H}\circ \sigma \right)\left(\tau \right)\left(w\circ \mathcal{H}\circ \sigma \right)\left(\tau \right){\sigma }^{\prime }\left(\tau \right)d\tau \hfill \\ \hfill & +w\left(\frac{1}{{\theta }_2}\right)\frac{{\left(w\circ \mathcal{H}\circ \sigma \right)}^{-1}{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}{\Gamma \left(v\right)}{\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}{\left(\sigma \left(\tau \right)-\frac{1}{{\theta }_2}\right)}^{v-1}\left(\mathcal{F}\circ \mathcal{H}\circ \sigma \right)\left(\tau \right)\left(w\circ \mathcal{H}\circ \sigma \right)\left(\tau \right){\sigma }^{\prime }\left(\tau \right)d\tau \hfill \\ \hfill & =\frac{{\left({\theta }_2-{\theta }_1\right)}^v}{{\left(2{\theta }_1{\theta }_2\right)}^v\Gamma \left(v\right)}{\int }_{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}{\left(\frac{2{\theta }_1{\theta }_2\left(\frac{1}{{\theta }_1}-\sigma \left(\tau \right)\right)}{{\theta }_2-{\theta }_1}\right)}^{v-1}\left(\mathcal{F}\circ \mathcal{H}\circ \sigma \right)\left(\tau \right)\left(w\circ \mathcal{H}\circ \sigma \right)\left(\tau \right){\sigma }^{\prime }\left(\tau \right)\frac{2{\theta }_1{\theta }_{2}d\tau }{{\theta }_2-{\theta }_1}\hfill \\ \hfill & +\frac{{\left({\theta }_2-{\theta }_1\right)}^v}{{\left(2{\theta }_1{\theta }_2\right)}^v\Gamma \left(v\right)}{\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}{\left(\frac{2{\theta }_1{\theta }_2\left(\sigma \left(\tau \right)-\frac{1}{{\theta }_2}\right)}{{\theta }_2-{\theta }_1}\right)}^{v-1}\left(\mathcal{F}\circ \mathcal{H}\circ \sigma \right)\left(\tau \right)\left(w\circ \mathcal{H}\circ \sigma \right)\left(\tau \right){\sigma }^{\prime }\left(\tau \right)\frac{2{\theta }_1{\theta }_{2}d\tau }{{\theta }_2-{\theta }_1},\hfill \end{array}$$

so,

$${\left[\left(w\circ \mathcal{H}\circ \sigma \right){\sigma }^{-1}\left(\frac{1}{y}\right)\right]}^{-1}={\displaystyle \frac{1}{\left(w\circ \mathcal{H}\circ \sigma \right){\sigma }^{-1}\left(\frac{1}{y}\right)}}={\displaystyle \frac{1}{w\left(y\right)}},\mathrm{for}y=\left[\frac{1}{{\theta }_1},\frac{1}{{\theta }_2}\right].$$

Setting ${\gamma }_1=\frac{2{\theta }_1{\theta }_2\left(\frac{1}{{\theta }_1}-\sigma \left(\tau \right)\right)}{{\theta }_2-{\theta }_1}$ and ${\gamma }_2=\frac{2{\theta }_1{\theta }_2\left(\sigma \left(\tau \right)-\frac{1}{{\theta }_2}\right)}{{\theta }_2-{\theta }_1}$, one can deduce that

$$\begin{array}{cc}\hfill & w\left(\frac{1}{{\theta }_1}\right)\left({\mathcal{I}}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{+}}^{v,\sigma }\left(w\circ \mathcal{H}\circ \sigma \right)\right)\left({\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)\right)\hfill \\ & +w\left(\frac{1}{{\theta }_2}\right)\left({}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{-}}{\mathcal{I}}^{v,\sigma }\left(w\circ \mathcal{H}\circ \sigma \right)\right)\left({\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)\right)\hfill \\ \hfill & =\frac{{\left({\theta }_2-{\theta }_1\right)}^v}{{\left(2{\theta }_1{\theta }_2\right)}^v\Gamma \left(v\right)}\left[{\int }_0^1\gamma {{}_1}^{v-1}\mathcal{F}\left(\frac{2{\theta }_1{\theta }_2}{\gamma {}_1{\theta }_1+\left(2-\gamma {}_1\right){\theta }_2}\right)w\left(\frac{2{\theta }_1{\theta }_2}{\gamma {}_1{\theta }_1+\left(2-\gamma {}_1\right){\theta }_2}\right)d\gamma {}_1\right.\hfill \\ \hfill & \left.+{\int }_0^1\gamma {{}_2}^{v-1}\mathcal{F}\left(\frac{2{\theta }_1{\theta }_2}{\gamma {}_2{\theta }_2+\left(2-\gamma {}_2\right){\theta }_1}\right)w\left(\frac{2{\theta }_1{\theta }_2}{\gamma {}_2{\theta }_2+\left(2-\gamma {}_2\right){\theta }_1}\right)d\gamma {}_2\right]\hfill \\ \hfill & =\frac{{\left({\theta }_2-{\theta }_1\right)}^v}{{\left(2{\theta }_1{\theta }_2\right)}^v\Gamma \left(v\right)}\left[{\int }_0^1{\kappa }^{v-1}\mathcal{F}\left(\frac{2{\theta }_1{\theta }_2}{\kappa {\theta }_1+\left(2-\kappa \right){\theta }_2}\right)w\left(\frac{2{\theta }_1{\theta }_2}{\kappa {\theta }_1+\left(2-\kappa \right){\theta }_2}\right)d\kappa \right.\hfill \\ \hfill & \left.+{\int }_0^1{\kappa }^{v-1}\mathcal{F}\left(\frac{2{\theta }_1{\theta }_2}{\kappa {\theta }_2+\left(2-\kappa \right){\theta }_1}\right)w\left(\frac{2{\theta }_1{\theta }_2}{\kappa {\theta }_1+\left(2-\kappa \right){\theta }_2}\right)d\kappa \right].\hfill \end{array}$$

As a consequence,

$$\begin{array}{c}{\int }_0^1{\kappa }^{v-1}\mathcal{F}\left(\frac{2{\theta }_1{\theta }_2}{\kappa {\theta }_1+\left(2-\kappa \right){\theta }_2}\right)w\left(\frac{2{\theta }_1{\theta }_2}{\kappa {\theta }_1+\left(2-\kappa \right){\theta }_2}\right)d\kappa \hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\int }_0^1{\kappa }^{v-1}\mathcal{F}\left(\frac{2{\theta }_1{\theta }_2}{\kappa {\theta }_2+\left(2-\kappa \right){\theta }_1}\right)w\left(\frac{2{\theta }_1{\theta }_2}{\kappa {\theta }_1+\left(2-\kappa \right){\theta }_2}\right)d\kappa \\ =\frac{{\left(2{\theta }_1{\theta }_2\right)}^v\Gamma \left(v\right)}{{\left({\theta }_2-{\theta }_1\right)}^v}\left[w\left(\frac{1}{{\theta }_2}\right)\left({}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{-}}{\mathcal{I}}_{w\circ \mathcal{H}\circ \sigma }^{v,\sigma }\left(\mathcal{F}\circ \mathcal{H}\circ \sigma \right)\right)\left({\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)\right)\right.\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left.+w\left(\frac{1}{{\theta }_1}\right)\left({}_{w\circ \mathcal{H}\circ \sigma }{\mathcal{I}}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{+}}^{v,\sigma }\left(\mathcal{F}\circ \mathcal{H}\circ \sigma \right)\right)\left({\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)\right)\right].\end{array}$$

(20)

When we use (19) and (20) in (18), we obtain the following result

$$\begin{array}{c}\mathcal{F}\left(\frac{2{\theta }_1{\theta }_2}{{\theta }_1+{\theta }_2}\right)\left[\left({\mathcal{I}}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{+}}^{v,\sigma }\left(w\circ \mathcal{H}\circ \sigma \right)\right)\left({\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)\right)\right.\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\left.+\left({}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{-}}{\mathcal{I}}^{v,\sigma }\left(w\circ \mathcal{H}\circ \sigma \right)\right)\left({\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)\right)\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\le w\left(\frac{1}{{\theta }_1}\right)\left({\mathcal{I}}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{+}}^{v,\sigma }\left(w\circ \mathcal{H}\circ \sigma \right)\right)\left({\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)\right)\\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+w\left(\frac{1}{{\theta }_2}\right)\left({}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{-}}{\mathcal{I}}^{v,\sigma }\left(w\circ \mathcal{H}\circ \sigma \right)\right)\left({\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)\right).\end{array}$$

(21)

As a result, left inequality of (18) has been proven.

The second inequality of (18) can be proved using the harmonically convex function of $\mathcal{F}$.

$$\mathcal{F}\left(\frac{2{\theta }_1{\theta }_2}{\kappa {\theta }_1+\left(2-\kappa \right){\theta }_2}\right)+\mathcal{F}\left(\frac{2{\theta }_1{\theta }_2}{\kappa {\theta }_2+\left(2-\kappa \right){\theta }_1}\right)\le \mathcal{F}\left({\theta }_1\right)+\mathcal{F}\left({\theta }_2\right).$$

(22)

Multiplying both sides of (23) by ${\kappa }^{v-1}w\left(\frac{2{\theta }_1{\theta }_2}{\kappa {\theta }_1+\left(2-\kappa \right){\theta }_2}\right)$ and we obtain by integrating the resulting inequality in terms of $\kappa$ on $[0,1]$.

$$\begin{array}{c}{\int }_0^1{\kappa }^{v-1}\mathcal{F}\left(\frac{2{\theta }_1{\theta }_2}{\kappa {\theta }_1+\left(2-\kappa \right){\theta }_2}\right)w\left(\frac{2{\theta }_1{\theta }_2}{\kappa {\theta }_1+\left(2-\kappa \right){\theta }_2}\right)d\kappa \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\int }_0^1{\kappa }^{v-1}\mathcal{F}\left(\frac{2{\theta }_1{\theta }_2}{\kappa {\theta }_2+\left(2-\kappa \right){\theta }_1}\right)w\left(\frac{2{\theta }_1{\theta }_2}{\kappa {\theta }_1+\left(2-\kappa \right){\theta }_2}\right)d\kappa \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le \left[\mathcal{F}\left({\theta }_1\right)+\mathcal{F}\left({\theta }_2\right)\right]{\int }_0^1{\kappa }^{v-1}w\left(\frac{2{\theta }_1{\theta }_2}{\kappa {\theta }_1+\left(2-\kappa \right){\theta }_2}\right)d\kappa .\end{array}$$

(23)

Then, using (14) and (20) in (24), we get

$$\begin{array}{c}w\left(\frac{1}{{\theta }_1}\right)\left({}_{w\circ \mathcal{H}\circ \sigma }{\mathcal{I}}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{+}}^{v,\sigma }\left(\mathcal{F}\circ \mathcal{H}\circ \sigma \right)\right)\left({\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}+w\left(\frac{1}{{\theta }_2}\right)\left({}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{-}}{\mathcal{I}}_{w\circ \mathcal{H}\circ \sigma }^{v,\sigma }\left(\mathcal{F}\circ \mathcal{H}\circ \sigma \right)\right)\left({\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\le \frac{\mathcal{F}\left({\theta }_1\right)+\mathcal{F}\left({\theta }_2\right)}{2}\left[\left({\mathcal{I}}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{+}}^{v,\sigma }\left(w\circ \mathcal{H}\circ \sigma \right)\right)\left({\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)\right)\right.\\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left.+\left({}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{-}}{\mathcal{I}}^{v,\sigma }\left(w\circ \mathcal{H}\circ \sigma \right)\right)\left({\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)\right)\right].\end{array}$$

(24)

This ends our proof. □

Remark 2.

We can derive the following special results from Theorem 6:

 (i)

If $\sigma \left(\tau \right)=\tau$, then inequality (16) yields

$$\begin{array}{c}\mathcal{F}\left(\frac{2{\theta }_1{\theta }_2}{{\theta }_1+{\theta }_2}\right)\left[\left({\mathcal{I}}_{{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{+}}^v\left(w\circ \mathcal{H}\right)\right)\left(\frac{1}{{\theta }_1}\right)+\left({}_{{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{-}}{\mathcal{I}}^v\left(w\circ \mathcal{H}\right)\right)\left(\frac{1}{{\theta }_2}\right)\right]\hfill \\ \le w\left(\frac{1}{{\theta }_1}\right)\left({}_{w\circ \mathcal{H}}{\mathcal{I}}_{{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{+}}^v\left(\mathcal{F}\circ \mathcal{H}\right)\right)\left(\frac{1}{{\theta }_1}\right)+w\left(\frac{1}{{\theta }_2}\right)\left({}_{{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{-}}{\mathcal{I}}_{w\circ \mathcal{H}}^v\left(\mathcal{F}\circ \mathcal{H}\right)\right)\left(\frac{1}{{\theta }_2}\right)\hfill \\ \le \frac{\mathcal{F}\left({\theta }_1\right)+\mathcal{F}\left({\theta }_2\right)}{2}\left[\left({\mathcal{I}}_{{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{+}}^v\left(w\circ \mathcal{H}\right)\right)\left(\frac{1}{{\theta }_1}\right)+\left({}_{{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{-}}{\mathcal{I}}^v\left(w\circ \mathcal{H}\right)\right)\left(\frac{1}{{\theta }_2}\right)\right].\hfill \end{array}$$

(25)

where

$$\begin{array}{cc}& \left({}_w{\mathcal{I}}_{{\theta }_1^{+}}^v\mathcal{F}\right)\left(\tau \right)=\frac{{\left[w\left(\tau \right)\right]}^{-1}}{\Gamma \left(v\right)}{\int }_{{\theta }_1}^{\tau }{\left(\tau -\kappa \right)}^{v-1}\mathcal{F}\left(\kappa \right)w\left(\kappa \right)d\kappa ,\\ & \left({}_w{\mathcal{I}}_{{\theta }_2^{-}}^v\mathcal{F}\right)\left(\tau \right)=\frac{{\left[w\left(\tau \right)\right]}^{-1}}{\Gamma \left(v\right)}{\int }_{\tau }^{{\theta }_2}{\left(\kappa -\tau \right)}^{v-1}\mathcal{F}\left(\kappa \right)w\left(\kappa \right)d\kappa ,v>0.\end{array}$$

(26)

 (ii)

If $\sigma \left(\tau \right)=\tau$ and $v=1$, then inequality (16) is identical to inequality in (10).

 (iii)

If $\sigma \left(\tau \right)=\tau$ and $w=1$, then inequality (16) is identical to inequality in (3).

 (iv)

If $\sigma \left(\tau \right)=\tau$ and $v=1$, then inequality (16) is identical to inequality in (8).

Lemma 2.

Let $0<{\theta }_1<{\theta }_2$, let $\mathcal{F}:[{\theta }_1,{\theta }_2]\to \mathbb{R}$ be a continuous with a derivative ${\mathcal{F}}^{\prime }\in L^1[{\theta }_1,{\theta }_2]$ such that $\mathcal{F}\left(\tau \right)=\mathcal{F}\left(\frac{1}{{\theta }_1}\right)+{\int }_{\frac{1}{{\theta }_1}}^{\tau }{\mathcal{F}}^{\prime }\left(\kappa \right)d\kappa$ and let $w:[{\theta }_1,{\theta }_2]\to \mathbb{R}$ is nonnegative, an integrable, positive, and symmetric weighted function with respect to $\frac{2{\theta }_1{\theta }_2}{{\theta }_1+{\theta }_2}$. If σ is an increasing and positive function from $[{\theta }_1,{\theta }_2)$ onto itself such that its derivative ${\sigma }^{{}^{\prime }}\left(\tau \right)$ is continuous on $({\theta }_1,{\theta }_2)$, for $v>0,$ then, we have

$$\begin{array}{c}\mathcal{F}\left(\frac{2{\theta }_1{\theta }_2}{{\theta }_1+{\theta }_2}\right)\left[\left({\mathcal{I}}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{+}}^{v,\sigma }\left(w\circ \mathcal{H}\circ \sigma \right)\right)\left({\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)\right)+\left({}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{-}}{\mathcal{I}}^{v,\sigma }\left(w\circ \mathcal{H}\circ \sigma \right)\right)\right.\hfill \\ \left.\left({\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)\right)\right]-\left[w\left(\frac{1}{{\theta }_1}\right)\left({}_{w\circ \mathcal{H}\circ \sigma }{\mathcal{I}}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{+}}^{v,\sigma }\left(\mathcal{F}\circ \mathcal{H}\circ \sigma \right)\right)\left({\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)\right)\right.\\ \left.+w\left(\frac{1}{{\theta }_2}\right)\left({}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{-}}{\mathcal{I}}_{w\circ \mathcal{H}\circ \sigma }^{v,\sigma }\left(\mathcal{F}\circ \mathcal{H}\circ \sigma \right)\right)\left({\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)\right)\right]\\ \hspace{1em}\hspace{1em}=\frac{1}{\Gamma \left(v\right)}{\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}\left[{\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{\kappa }{\sigma }^{\prime }\left(\tau \right){\left(\sigma \left(\tau \right)-\frac{1}{{\theta }_2}\right)}^{v-1}\left(w\circ \mathcal{H}\circ \sigma \right)\left(\tau \right)d\tau \right]\left({\mathcal{F}}^{\prime }\circ \mathcal{H}\circ \sigma \right)\left(\kappa \right){\sigma }^{\prime }\left(\kappa \right)d\kappa \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\frac{1}{\Gamma \left(v\right)}{\int }_{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}\left[{\int }_{\kappa }^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}{\sigma }^{\prime }\left(\tau \right){\left(\frac{1}{{\theta }_1}-\sigma \left(\tau \right)\right)}^{v-1}\left(w\circ \mathcal{H}\circ \sigma \right)\left(\tau \right)d\tau \right]\left({\mathcal{F}}^{\prime }\circ \mathcal{H}\circ \sigma \right)\left(\kappa \right){\sigma }^{\prime }\left(\kappa \right)d\kappa \end{array}$$

Proof. 

Let us set

$$\begin{array}{cc}\hfill & \frac{1}{\Gamma \left(v\right)}{\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}\left[{\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{\kappa }{\sigma }^{\prime }\left(\tau \right){\left(\sigma \left(\tau \right)-\frac{1}{{\theta }_2}\right)}^{v-1}\left(w\circ \mathcal{H}\circ \sigma \right)\left(\tau \right)d\tau \right]\left({\mathcal{F}}^{\prime }\circ \mathcal{H}\circ \sigma \right)\left(\kappa \right){\sigma }^{\prime }\left(\kappa \right)d\kappa \hfill \\ \hfill & -\frac{1}{\Gamma \left(v\right)}{\int }_{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}\left[{\int }_{\kappa }^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}{\sigma }^{\prime }\left(\tau \right){\left(\frac{1}{{\theta }_1}-\sigma \left(\tau \right)\right)}^{v-1}\left(w\circ \mathcal{H}\circ \sigma \right)\left(\tau \right)d\tau \right]\left({\mathcal{F}}^{\prime }\circ \mathcal{H}\circ \sigma \right)\left(\kappa \right){\sigma }^{\prime }\left(\kappa \right)d\kappa \hfill \end{array}$$

$$\begin{array}{cc}\hfill & =\frac{1}{\Gamma \left(v\right)}{\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}\left[{\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{\kappa }{\sigma }^{\prime }\left(\tau \right){\left(\sigma \left(\tau \right)-\frac{1}{{\theta }_2}\right)}^{v-1}\left(w\circ \mathcal{H}\circ \sigma \right)\left(\tau \right)d\tau \right]\left({\mathcal{F}}^{\prime }\circ \mathcal{H}\circ \sigma \right)\left(\kappa \right){\sigma }^{\prime }\left(\kappa \right)d\kappa \hfill \\ \hfill & +\frac{-1}{\Gamma \left(v\right)}{\int }_{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}\left[{\int }_{\kappa }^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}{\sigma }^{\prime }\left(\tau \right){\left(\frac{1}{{\theta }_1}-\sigma \left(\tau \right)\right)}^{v-1}\left(w\circ \mathcal{H}\circ \sigma \right)\left(\tau \right)d\tau \right]\left({\mathcal{F}}^{\prime }\circ \mathcal{H}\circ \sigma \right)\left(\kappa \right){\sigma }^{\prime }\left(\kappa \right)d\kappa \hfill \\ \hfill & :={\mathbb{Z}}_1+{\mathbb{Z}}_2.\hfill \end{array}$$

Taking integrating by parts, applying Lemma 1, (12) and (13), we get

$$\begin{array}{cc}\hfill & {\mathbb{Z}}_1={\left.\frac{1}{\Gamma \left(v\right)}\left[{\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{\kappa }{\sigma }^{\prime }\left(\tau \right){\left(\sigma \left(\tau \right)-\frac{1}{{\theta }_2}\right)}^{v-1}\left(w\circ \mathcal{H}\circ \sigma \right)\left(\tau \right)d\tau \right]\left(\mathcal{F}\circ \mathcal{H}\circ \sigma \right)\left(\kappa \right)\right|}_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}\hfill \\ \hfill & -\frac{1}{\Gamma \left(v\right)}{\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}{\sigma }^{\prime }\left(\kappa \right){\left(\sigma \left(\tau \right)-\frac{1}{{\theta }_2}\right)}^{v-1}\left(w\circ \mathcal{H}\circ \sigma \right)\left(\kappa \right)\left(\mathcal{F}\circ \mathcal{H}\circ \sigma \right)\left(\kappa \right)d\kappa \hfill \\ \hfill & =\left(\frac{1}{\Gamma \left(v\right)}{\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}{\sigma }^{\prime }\left(\tau \right){\left(\sigma \left(\tau \right)-\frac{1}{{\theta }_2}\right)}^{v-1}\left(w\circ \mathcal{H}\circ \sigma \right)\left(\tau \right)d\tau \right)\mathcal{F}\left(\frac{2{\theta }_1{\theta }_2}{{\theta }_1+{\theta }_2}\right)-w\left(\frac{1}{{\theta }_2}\right)\hfill \\ \hfill & \times \frac{{\left(w\circ \mathcal{H}\circ \sigma \right)}^{-1}{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}{\Gamma \left(v\right)}{\int }_{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}{\sigma }^{\prime }\left(\kappa \right){\left(\sigma \left(\tau \right)-\frac{1}{{\theta }_2}\right)}^{v-1}\left(w\circ \mathcal{H}\circ \sigma \right)\left(\kappa \right)\left(\mathcal{F}\circ \mathcal{H}\circ \sigma \right)\left(\kappa \right)d\kappa \hfill \\ \hfill & =\mathcal{F}\left(\frac{2{\theta }_1{\theta }_2}{{\theta }_1+{\theta }_2}\right)\left({}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{-}}{\mathcal{I}}^{v,\sigma }\left(w\circ \mathcal{H}\circ \sigma \right)\right)\left({\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)\right)\hfill \\ \hfill & -w\left(\frac{1}{{\theta }_2}\right)\left({}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{-}}{\mathcal{I}}^{v,\sigma }\left(w\circ \mathcal{H}\circ \sigma \right)\right)\left({\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)\right)\hfill \\ \hfill & =\frac{1}{2}\mathcal{F}\left(\frac{2{\theta }_1{\theta }_2}{{\theta }_1+{\theta }_2}\right)\left[\left({\mathcal{I}}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{+}}^{v,\sigma }\left(w\circ \mathcal{H}\circ \sigma \right)\right)\left({\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)\right)\right.\hfill \\ \hfill & \left.+\left({}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{-}}{\mathcal{I}}^{v,\sigma }\left(w\circ \mathcal{H}\circ \sigma \right)\right)\left({\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)\right)\right]\hfill \\ \hfill & -w\left(\frac{1}{{\theta }_2}\right)\left({}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{-}}{\mathcal{I}}^{v,\sigma }\left(w\circ \mathcal{H}\circ \sigma \right)\right)\left({\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)\right).\hfill \end{array}$$

Similarly, we get

$$\begin{array}{cc}\hfill & {\mathbb{Z}}_2={\left.\frac{-1}{\Gamma \left(v\right)}\left[{\int }_{\kappa }^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}{\sigma }^{\prime }\left(\tau \right){\left(\frac{1}{{\theta }_1}-\sigma \left(\tau \right)\right)}^{v-1}\left(w\circ \mathcal{H}\circ \sigma \right)\left(\tau \right)d\tau \right]\left(\mathcal{F}\circ \mathcal{H}\circ \sigma \right)\left(\kappa \right)\right|}_{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}\hfill \\ \hfill & -\frac{1}{\Gamma \left(v\right)}{\int }_{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}{\sigma }^{\prime }\left(\kappa \right){\left(\frac{1}{{\theta }_1}-\sigma \left(\tau \right)\right)}^{v-1}\left(w\circ \mathcal{H}\circ \sigma \right)\left(\kappa \right)\left(\mathcal{F}\circ \mathcal{H}\circ \sigma \right)\left(\kappa \right)d\kappa \hfill \\ & =\left(\frac{1}{\Gamma \left(v\right)}{\int }_{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}{\sigma }^{\prime }\left(\tau \right){\left(\frac{1}{{\theta }_1}-\sigma \left(\tau \right)\right)}^{v-1}\left(w\circ \mathcal{H}\circ \sigma \right)\left(\tau \right)d\tau \right)\mathcal{F}\left(\frac{2{\theta }_1{\theta }_2}{{\theta }_1+{\theta }_2}\right)\hfill \\ & -w\left(\frac{1}{{\theta }_1}\right)\frac{{\left(w\circ \mathcal{H}\circ \sigma \right)}^{-1}{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}{\Gamma \left(v\right)}\hfill \\ \hfill & \times {\int }_{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}{\sigma }^{\prime }\left(\kappa \right){\left(\frac{1}{{\theta }_1}-\sigma \left(\tau \right)\right)}^{v-1}\left(w\circ \mathcal{H}\circ \sigma \right)\left(\kappa \right)\left(\mathcal{F}\circ \mathcal{H}\circ \sigma \right)\left(\kappa \right)d\kappa \hfill \end{array}$$

$$\begin{array}{cc}\hfill & =\mathcal{F}\left(\frac{2{\theta }_1{\theta }_2}{{\theta }_1+{\theta }_2}\right)\left({\mathcal{I}}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{+}}^{v,\sigma }\left(w\circ \mathcal{H}\circ \sigma \right)\right)\left({\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)\right)\hfill \\ \hfill & -w\left(\frac{1}{{\theta }_1}\right)\left({}_{w\circ \mathcal{H}\circ \sigma }{\mathcal{I}}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{+}}^{v,\sigma }\left(\mathcal{F}\circ \mathcal{H}\circ \sigma \right)\right)\left({\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)\right)\hfill \\ \hfill & =\frac{1}{2}\mathcal{F}\left(\frac{2{\theta }_1{\theta }_2}{{\theta }_1+{\theta }_2}\right)\left[\left({\mathcal{I}}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{+}}^{v,\sigma }\left(w\circ \mathcal{H}\circ \sigma \right)\right)\left({\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)\right)\right.\hfill \\ \hfill & \left.+\left({}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{-}}{\mathcal{I}}^{v,\sigma }\left(w\circ \mathcal{H}\circ \sigma \right)\right)\left({\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)\right)\right]\hfill \\ \hfill & -w\left(\frac{1}{{\theta }_1}\right)\left({}_{w\circ \mathcal{H}\circ \sigma }{\mathcal{I}}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{+}}^{v,\sigma }\left(\mathcal{F}\circ \mathcal{H}\circ \sigma \right)\right)\left({\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)\right).\hfill \end{array}$$

As a result, we can conclude:

$$\begin{array}{c}{\mathbb{Z}}_1+{\mathbb{Z}}_2:=\mathcal{F}\left(\frac{2{\theta }_1{\theta }_2}{{\theta }_1+{\theta }_2}\right)\left[\left({\mathcal{I}}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{+}}^{v,\sigma }\left(w\circ \mathcal{H}\circ \sigma \right)\right)\left({\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)\right)\right.\hfill \\ \hspace{1em}\hspace{1em}\left.+\left({}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{-}}{\mathcal{I}}^{v,\sigma }\left(w\circ \mathcal{H}\circ \sigma \right)\right)\left({\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)\right)\right]\\ \hspace{1em}\hspace{1em}-\left[w\left(\frac{1}{{\theta }_1}\right)\left({}_{w\circ \mathcal{H}\circ \sigma }{\mathcal{I}}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{+}}^{v,\sigma }\left(\mathcal{F}\circ \mathcal{H}\circ \sigma \right)\right)\left({\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)\right)\right.\\ \hfill \hspace{1em}\hspace{1em}\left.+w\left(\frac{1}{{\theta }_2}\right)\left({}_{{\sigma }^{-1}{\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{-}}{\mathcal{I}}_{w\circ \mathcal{H}\circ \sigma }^{v,\sigma }\left(\mathcal{F}\circ \mathcal{H}\circ \sigma \right)\right)\left({\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)\right)\right],\end{array}$$

(27)

we achieve the intended result. □

3. Main Results

We can conclude the following Hermite-Hadamard-Fejér inequalities with the help of Lemma 2.

Theorem 7.

Suppose that all the conditions of Lemma 2 and $|{\mathcal{F}}^{\prime }|$ is harmonically convex on $[{\theta }_1,{\theta }_2]$ and σ is an increasing and positive function from $[{\theta }_1,{\theta }_2)$ onto itself such that its derivative ${\sigma }^{{}^{\prime }}\left(\tau \right)$ is continuous on $({\theta }_1,{\theta }_2)$, for $v>0,$ then we have

$$\begin{array}{c}\left|{\mathbb{Z}}_1+{\mathbb{Z}}_2\right|\le \frac{{\left|\right|w\left|\right|}_{\left[\frac{1}{{\theta }_2},\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right],\infty }}{({\theta }_1-{\theta }_2)v\Gamma \left(v\right)}\left[{\mathbb{W}}_1\left({\theta }_1,{\theta }_2\right)\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_1\right)\right|+{\mathbb{W}}_2\left({\theta }_1,{\theta }_2\right)\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_2\right)\right|\right]\hfill \\ +\frac{{\left|\right|w\left|\right|}_{\left[\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2},\frac{1}{{\theta }_1}\right],\infty }}{({\theta }_1-{\theta }_2)v\Gamma \left(v\right)}\left[{\mathbb{W}}_3\left({\theta }_1,{\theta }_2\right)\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_1\right)\right|+{\mathbb{W}}_4\left({\theta }_1,{\theta }_2\right)\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_2\right)\right|\right]\\ \le \frac{{\left|\right|w\left|\right|}_{\left[\frac{1}{{\theta }_1},\frac{1}{{\theta }_2}\right],\infty }}{({\theta }_1-{\theta }_2)v\Gamma \left(v\right)}\left\{\left({\mathbb{W}}_1\left({\theta }_1,{\theta }_2\right)+{\mathbb{W}}_3\left({\theta }_1,{\theta }_2\right)\right)\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_1\right)\right|\right.\\ \hfill \left.+\left({\mathbb{W}}_2\left({\theta }_1,{\theta }_2\right)+{\mathbb{W}}_4\left({\theta }_1,{\theta }_2\right)\right)\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_2\right)\right|\right\},\end{array}$$

(28)

where ${\mathbb{W}}_1\left({\theta }_1,{\theta }_2\right),$${\mathbb{W}}_2\left({\theta }_1,{\theta }_2\right),$${\mathbb{W}}_3\left({\theta }_1,{\theta }_2\right),$ and ${\mathbb{W}}_4\left({\theta }_1,{\theta }_2\right)$ are defined as follows:

$$\begin{array}{cc}\hfill {\mathbb{W}}_1\left({\theta }_1,{\theta }_2\right)=& \frac{2^{-v}{\left({\theta }_1{\theta }_2\right)}^{1-v}{\left({\theta }_1+{\theta }_2\right)}^v{}_2{F}_1\left(-v,-v;1-v;\frac{2{\theta }_1}{{\theta }_1+{\theta }_2}\right)}{v}\hfill \\ \hfill & +\frac{2^{-v-1}({\theta }_1-{\theta }_2){\left(\frac{1}{{\theta }_1}-\frac{1}{{\theta }_2}\right)}^v}{{\theta }_2(v+1)}-\pi {\theta }_1{\left(\frac{1}{{\theta }_2}\right)}^{v-1}csc\left(\pi v\right),\hfill \end{array}$$

$${\mathbb{W}}_2\left({\theta }_1,{\theta }_2\right)\frac{2^{-v-1}{\left({\theta }_1{\theta }_2\right)}^{-v}\left(\begin{array}{c}v\left(2^{v+1}(v+1)\pi {\theta }_2{\theta }_1^{v+2}csc\left(\pi v\right)+{({\theta }_2-{\theta }_1)}^{v+1}\right)\\ -2(v+1){\theta }_1^2{\theta }_2{({\theta }_1+{\theta }_2)}^v{}_2{F}_1\left(-v,-v;1-v;\frac{2{\theta }_1}{{\theta }_1+{\theta }_2}\right)\end{array}\right)}{{\theta }_{1}v(v+1)},$$

$${\mathbb{W}}_3\left({\theta }_1,{\theta }_2\right)=\frac{2^{-v-1}({\theta }_2-{\theta }_1){\left(\frac{1}{{\theta }_1}-\frac{1}{{\theta }_2}\right)}^v\left({\theta }_1{\theta }_2{}_2{F}_1\left(1,v+1;v+2;\frac{{\theta }_2-{\theta }_1}{2{\theta }_2}\right)-1\right)}{{\theta }_2\left(v+1\right)}$$

and

$${\mathbb{W}}_4\left({\theta }_1,{\theta }_2\right)\frac{2^{-v-1}({\theta }_1-{\theta }_2){\left(\frac{1}{{\theta }_1}-\frac{1}{{\theta }_2}\right)}^v\left({\theta }_1^2{}_2{F}_1\left(1,v+1;v+2;\frac{{\theta }_2-{\theta }_1}{2{\theta }_2}\right)-1\right)}{{\theta }_1(v+1)}.$$

Proof. 

Using the Lemma 2 as well as properties of the modulus and the harmonically convex function of $|{\mathcal{F}}^{\prime }|$, we get

$$\begin{array}{c}\left|{\mathbb{Z}}_1+{\mathbb{Z}}_2\right|=\left|\frac{1}{\Gamma \left(v\right)}{\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}\left[{\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{\kappa }{\sigma }^{\prime }\left(\tau \right){\left(\sigma \left(\tau \right)-\frac{1}{{\theta }_2}\right)}^{v-1}\left(w\circ \mathcal{H}\circ \sigma \right)\left(\tau \right)d\tau \right]\left({\mathcal{F}}^{\prime }\circ \mathcal{H}\circ \sigma \right)\left(\kappa \right){\sigma }^{\prime }\left(\kappa \right)d\kappa \right.\hfill \\ \left.-\frac{1}{\Gamma \left(v\right)}{\int }_{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}\left[{\int }_{\kappa }^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}{\sigma }^{\prime }\left(\tau \right){\left(\frac{1}{{\theta }_1}-\sigma \left(\tau \right)\right)}^{v-1}\left(w\circ \mathcal{H}\circ \sigma \right)\left(\tau \right)d\tau \right]\left({\mathcal{F}}^{\prime }\circ \mathcal{H}\circ \sigma \right)\left(\kappa \right){\sigma }^{\prime }\left(\kappa \right)d\kappa \right|\\ \le \frac{1}{\Gamma \left(v\right)}{\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}\left|{\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{\kappa }{\sigma }^{\prime }\left(\tau \right){\left(\sigma \left(\tau \right)-\frac{1}{{\theta }_2}\right)}^{v-1}\left(w\circ \mathcal{H}\circ \sigma \right)\left(\tau \right)d\tau \right|\left|\left({\mathcal{F}}^{\prime }\circ \mathcal{H}\circ \sigma \right)\left(\kappa \right)\right|{\sigma }^{\prime }\left(\kappa \right)d\kappa \\ \hfill +\frac{1}{\Gamma \left(v\right)}{\int }_{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}\left|{\int }_{\kappa }^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}{\sigma }^{\prime }\left(\tau \right){\left(\frac{1}{{\theta }_1}-\sigma \left(\tau \right)\right)}^{v-1}\left(w\circ \mathcal{H}\circ \sigma \right)\left(\tau \right)d\tau \right|\left|\left({\mathcal{F}}^{\prime }\circ \mathcal{H}\circ \sigma \right)\left(\kappa \right)\right|{\sigma }^{\prime }\left(\kappa \right)d\kappa .\end{array}$$

(29)

Since $\left|{\mathcal{F}}^{{}^{\prime }}\right|$ is harmonic-convex on $\left[{\theta }_1,{\theta }_2\right]$, where $\kappa \in \left[{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right),{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)\right]$

$$\begin{array}{c}\left|({\mathcal{F}}^{\prime }\circ \mathcal{H}\circ \sigma )\left(\kappa \right)\right|=\left|{\mathcal{F}}^{\prime }\left(\left(\frac{{\theta }_1({\theta }_2-\sigma \left(\kappa \right))}{\sigma \left(\kappa \right)({\theta }_2-{\theta }_1)}\right)\frac{1}{{\theta }_1}+\left(\frac{{\theta }_2(\sigma \left(\kappa \right)-{\theta }_1)}{\sigma \left(\kappa \right)({\theta }_2-{\theta }_1)}\right)\frac{1}{{\theta }_2}\right)\right|\hfill \\ \hfill \le \frac{{\theta }_1({\theta }_2-\sigma \left(\kappa \right))}{\sigma \left(\kappa \right)({\theta }_2-{\theta }_1)}\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_1\right)\right|+\frac{{\theta }_2(\sigma \left(\kappa \right)-{\theta }_1)}{\sigma \left(\kappa \right)({\theta }_2-{\theta }_1)}\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_2\right)\right|.\end{array}$$

(30)

Consequently, we obtain

$$\begin{array}{c}\left|{\mathbb{Z}}_1+{\mathbb{Z}}_2\right|\le \frac{{\left|\right|w\left|\right|}_{\left[\frac{1}{{\theta }_2},\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right],\infty }}{({\theta }_2-{\theta }_1)\Gamma \left(v\right)}{\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}\left|{\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{\kappa }{\sigma }^{\prime }\left(\tau \right){\left(\sigma \left(\tau \right)-\frac{1}{{\theta }_2}\right)}^{v-1}d\tau \right|\hfill \\ \times \left[\frac{{\theta }_1({\theta }_2-\sigma \left(\kappa \right))}{\sigma \left(\kappa \right)}\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_1\right)\right|+\frac{{\theta }_2(\sigma \left(\kappa \right)-{\theta }_1)}{\sigma \left(\kappa \right)}\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_2\right)\right|\right]{\sigma }^{\prime }\left(\kappa \right)d\kappa \\ +\frac{{\left|\right|w\left|\right|}_{\left[\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2},\frac{1}{{\theta }_1}\right],\infty }}{({\theta }_2-{\theta }_1)\Gamma \left(v\right)}{\int }_{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}\left|{\int }_{\kappa }^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}{\sigma }^{\prime }\left(\tau \right){\left(\frac{1}{{\theta }_1}-\sigma \left(\tau \right)\right)}^{v-1}d\tau \right|\\ \hfill \times \left[\frac{{\theta }_1({\theta }_2-\sigma \left(\kappa \right))}{\sigma \left(\kappa \right)}\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_1\right)\right|+\frac{{\theta }_2(\sigma \left(\kappa \right)-{\theta }_1)}{\sigma \left(\kappa \right)}\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_2\right)\right|\right]{\sigma }^{\prime }\left(\kappa \right)d\kappa ,\end{array}$$

(31)

where

$${\int }_{\kappa }^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}{\sigma }^{\prime }\left(\tau \right){\left(\frac{1}{{\theta }_1}-\sigma \left(\tau \right)\right)}^{v-1}d\tau =\frac{{\left(\frac{1}{{\theta }_1}-\sigma \left(\kappa \right)\right)}^v}{v}$$

and

$${\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{\kappa }{\sigma }^{\prime }\left(\tau \right){\left(\sigma \left(\tau \right)-\frac{1}{{\theta }_2}\right)}^{v-1}d\tau =\frac{{\left(\sigma \left(\kappa \right)-\frac{1}{{\theta }_2}\right)}^v}{v}.$$

Using the above calculations, we obtain the following integral

$$\begin{array}{cc}\hfill & \left|{\mathbb{Z}}_1+{\mathbb{Z}}_2\right|\le \frac{{\left|\right|w\left|\right|}_{\left[\frac{1}{{\theta }_2},\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right],\infty }}{({\theta }_2-{\theta }_1)v\Gamma \left(v\right)}{\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}\left[\frac{{\theta }_1{\left(\sigma \left(\kappa \right)-\frac{1}{{\theta }_2}\right)}^v({\theta }_2-\sigma \left(\kappa \right))}{\sigma \left(\kappa \right)}\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_1\right)\right|\right.\hfill \\ & \left.+\frac{{\theta }_2{\left(\sigma \left(\kappa \right)-\frac{1}{{\theta }_2}\right)}^v(\sigma \left(\kappa \right)-{\theta }_1)}{\sigma \left(\kappa \right)}\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_2\right)\right|\right]{\sigma }^{\prime }\left(\kappa \right)d\kappa \hfill \\ \hfill & +\frac{{\left|\right|w\left|\right|}_{\left[\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2},\frac{1}{{\theta }_1}\right],\infty }}{({\theta }_2-{\theta }_1)v\Gamma \left(v\right)}{\int }_{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}\left[\frac{{\theta }_1{\left(\frac{1}{{\theta }_1}-\sigma \left(\kappa \right)\right)}^v({\theta }_2-\sigma \left(\kappa \right))}{\sigma \left(\kappa \right)}\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_1\right)\right|\right.\hfill \\ \hfill & \left.+\frac{{\theta }_2{\left(\frac{1}{{\theta }_2}-\sigma \left(\kappa \right)\right)}^v(\sigma \left(\kappa \right)-{\theta }_1)}{\sigma \left(\kappa \right)}\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_2\right)\right|\right]{\sigma }^{\prime }\left(\kappa \right)d\kappa \hfill \end{array}$$

(32)

We can obtain the necessary result (28) by doing basic integral calculations based on inequality (32). □

Theorem 8.

Suppose that all the conditions of Lemma 2 and$|{\mathcal{F}}^{\prime }{|}^q$is harmonically convex on$[{\theta }_1,{\theta }_2]$ with $q\ge 1$ and σ is an increasing and positive function from $[{\theta }_1,{\theta }_2)$ onto itself such that its derivative ${\sigma }^{{}^{\prime }}\left(\tau \right)$ is continuous on $({\theta }_1,{\theta }_2)$, for $v>0$, then we have

$$\begin{array}{c}\left|{\mathbb{Z}}_1+{\mathbb{Z}}_2\right|\le {\left(\frac{{\left({\theta }_2-{\theta }_1\right)}^{v+1}}{{\left(2{\theta }_1{\theta }_2\right)}^{v+1}v(v+1)}\right)}^{1-\frac{1}{q}}\hfill \\ \times \left[\frac{{\left|\right|w\left|\right|}_{\left[\frac{1}{{\theta }_2},\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right],\infty }}{({\theta }_1-{\theta }_2)\Gamma \left(v\right)}{\left({\mathbb{W}}_1\left({\theta }_1,{\theta }_2\right){\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_1\right)\right|}^q+{\mathbb{W}}_2\left({\theta }_1,{\theta }_2\right){\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_2\right)\right|}^q\right)}^{\frac{1}{q}}\right.\\ \left.+\frac{{\left|\right|w\left|\right|}_{\left[\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2},\frac{1}{{\theta }_1}\right],\infty }}{({\theta }_1-{\theta }_2)\Gamma \left(v\right)}{\left({\mathbb{W}}_3\left({\theta }_1,{\theta }_2\right){\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_1\right)\right|}^q+{\mathbb{W}}_4\left({\theta }_1,{\theta }_2\right){\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_2\right)\right|}^q\right)}^{\frac{1}{q}}\right]\\ \le {\left(\frac{{\left({\theta }_2-{\theta }_1\right)}^{v+1}}{{\left(2{\theta }_1{\theta }_2\right)}^{v+1}v(v+1)}\right)}^{1-\frac{1}{q}}\left(\frac{{\left|\right|w\left|\right|}_{\left[\frac{1}{{\theta }_1},\frac{1}{{\theta }_2}\right],\infty }}{({\theta }_1-{\theta }_2)\Gamma \left(v\right)}\right)\left\{{\left(\left({\mathbb{W}}_1\left({\theta }_1,{\theta }_2\right)+{\mathbb{W}}_3\left({\theta }_1,{\theta }_2\right)\right){\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_1\right)\right|}^q\right)}^{\frac{1}{q}}\right.\\ \hfill \left.+{\left(\left({\mathbb{W}}_2\left({\theta }_1,{\theta }_2\right)+{\mathbb{W}}_4\left({\theta }_1,{\theta }_2\right)\right){\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_2\right)\right|}^q\right)}^{\frac{1}{q}}\right\},\end{array}$$

(33)

where ${\mathbb{W}}_1\left({\theta }_1,{\theta }_2\right),$ ${\mathbb{W}}_2\left({\theta }_1,{\theta }_2\right),$ ${\mathbb{W}}_3\left({\theta }_1,{\theta }_2\right),$ and ${\mathbb{W}}_4\left({\theta }_1,{\theta }_2\right)$ are defined in Theorem 7.

Proof. 

Using the Lemma 2 as well as properties of power mean inequality and the harmonically convex function of $|{\mathcal{F}}^{\prime }{|}^q$, we get

$$\begin{array}{c}\left|{\mathbb{Z}}_1+{\mathbb{Z}}_2\right|\hfill \\ \le \frac{1}{\Gamma \left(v\right)}{\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{{\theta }_2+{\theta }_1}{2{\theta }_1{\theta }_2}\right)}\left|{\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{\kappa }{\sigma }^{\prime }\left(\tau \right){\left(\sigma \left(\tau \right)-\frac{1}{{\theta }_2}\right)}^{v-1}\left(w\circ \mathcal{H}\circ \sigma \right)\left(\tau \right)d\tau \right|\left|\left({\mathcal{F}}^{\prime }\circ \mathcal{H}\circ \sigma \right)\left(\kappa \right)\right|{\sigma }^{\prime }\left(\kappa \right)d\kappa \\ +\frac{1}{\Gamma \left(v\right)}{\int }_{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}\left|{\int }_{\kappa }^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}{\sigma }^{\prime }\left(\tau \right){\left(\frac{1}{{\theta }_1}-\sigma \left(\tau \right)\right)}^{v-1}\left(w\circ \mathcal{H}\circ \sigma \right)\left(\tau \right)d\tau \right|\left|\left({\mathcal{F}}^{\prime }\circ \mathcal{H}\circ \sigma \right)\left(\kappa \right)\right|{\sigma }^{\prime }\left(\kappa \right)d\kappa \\ \le \frac{1}{\Gamma \left(v\right)}{\left({\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}\left|{\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{\kappa }{\sigma }^{\prime }\left(\tau \right){\left(\sigma \left(\tau \right)-\frac{1}{{\theta }_2}\right)}^{v-1}\left(w\circ \mathcal{H}\circ \sigma \right)\left(\tau \right)d\tau \right|{\sigma }^{\prime }\left(\kappa \right)d\kappa \right)}^{1-\frac{1}{q}}\\ \times {\left({\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}\left|{\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{\kappa }{\sigma }^{\prime }\left(\tau \right){\left(\sigma \left(\tau \right)-\frac{1}{{\theta }_2}\right)}^{v-1}\left(w\circ \mathcal{H}\circ \sigma \right)\left(\tau \right)d\tau \right|{\left|\left({\mathcal{F}}^{\prime }\circ \mathcal{H}\circ \sigma \right)\left(\kappa \right)\right|}^q{\sigma }^{\prime }\left(\kappa \right)d\kappa \right)}^{\frac{1}{q}}\\ +\frac{1}{\Gamma \left(v\right)}{\left({\int }_{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}\left|{\int }_{\kappa }^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}{\sigma }^{\prime }\left(\tau \right){\left(\frac{1}{{\theta }_1}-\sigma \left(\tau \right)\right)}^{v-1}\left(w\circ \mathcal{H}\circ \sigma \right)\left(\tau \right)d\tau \right|{\sigma }^{\prime }\left(\kappa \right)d\kappa \right)}^{1-\frac{1}{q}}\\ \hfill \times {\left({\int }_{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}\left|{\int }_{\kappa }^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}{\sigma }^{\prime }\left(\tau \right){\left(\frac{1}{{\theta }_1}-\sigma \left(\tau \right)\right)}^{v-1}\left(w\circ \mathcal{H}\circ \sigma \right)\left(\tau \right)d\tau \right|{\left|\left({\mathcal{F}}^{\prime }\circ \mathcal{H}\circ \sigma \right)\left(\kappa \right)\right|}^q{\sigma }^{\prime }\left(\kappa \right)d\kappa \right)}^{\frac{1}{q}}.\end{array}$$

(34)

Since $\left|{\mathcal{F}}^{{}^{\prime }}\right|$ is harmonic-convex on $\left[{\theta }_1,{\theta }_2\right]$, where $\kappa \in \left[{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right),{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)\right]$

$$\begin{array}{c}\left|({\mathcal{F}}^{\prime }\circ \mathcal{H}\circ \sigma )\left(\kappa \right)\right|=\left|{\mathcal{F}}^{\prime }\left(\left(\frac{{\theta }_1({\theta }_2-\sigma \left(\kappa \right))}{\sigma \left(\kappa \right)({\theta }_2-{\theta }_1)}\right)\frac{1}{{\theta }_1}+\left(\frac{{\theta }_2(\sigma \left(\kappa \right)-{\theta }_1)}{\sigma \left(\kappa \right)({\theta }_2-{\theta }_1)}\right)\frac{1}{{\theta }_2}\right)\right|\hfill \\ \hfill \le \frac{{\theta }_1({\theta }_2-\sigma \left(\kappa \right))}{\sigma \left(\kappa \right)({\theta }_2-{\theta }_1)}\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_1\right)\right|+\frac{{\theta }_2(\sigma \left(\kappa \right)-{\theta }_1)}{\sigma \left(\kappa \right)({\theta }_{21}-{\theta }_1)}\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_2\right)\right|.\end{array}$$

(35)

As a result, we get

$$\begin{array}{c}\left|{\mathbb{Z}}_1+{\mathbb{Z}}_2\right|\le \frac{{\left|\right|w\left|\right|}_{\left[\frac{1}{{\theta }_2},\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right],\infty }}{\Gamma \left(v\right)}{\left({\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}\left|{\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{\kappa }{\sigma }^{\prime }\left(\tau \right){\left(\sigma \left(\tau \right)-\frac{1}{{\theta }_2}\right)}^{v-1}d\tau \right|{\sigma }^{\prime }\left(\kappa \right)d\kappa \right)}^{1-\frac{1}{q}}\hfill \\ \times {\left({\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}\left|{\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{\kappa }{\sigma }^{\prime }\left(\tau \right){\left(\sigma \left(\tau \right)-\frac{1}{{\theta }_2}\right)}^{v-1}d\tau \right|{\left|\left({\mathcal{F}}^{\prime }\circ \mathcal{H}\circ \sigma \right)\left(\kappa \right)\right|}^q{\sigma }^{\prime }\left(\kappa \right)d\kappa \right)}^{\frac{1}{q}}\hfill \\ +\frac{{\left|\right|w\left|\right|}_{\left[\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2},\frac{1}{{\theta }_1}\right],\infty }}{\Gamma \left(v\right)}{\left({\int }_{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}\left|{\int }_{\kappa }^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}{\sigma }^{\prime }\left(\tau \right){\left(\frac{1}{{\theta }_1}-\sigma \left(\tau \right)\right)}^{v-1}d\tau \right|{\sigma }^{\prime }\left(\kappa \right)d\kappa \right)}^{1-\frac{1}{q}}\hfill \\ \times {\left({\int }_{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}\left|{\int }_{\kappa }^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}{\sigma }^{\prime }\left(\tau \right){\left(\frac{1}{{\theta }_1}-\sigma \left(\tau \right)\right)}^{v-1}d\tau \right|{\left|\left({\mathcal{F}}^{\prime }\circ \mathcal{H}\circ \sigma \right)\left(\kappa \right)\right|}^q{\sigma }^{\prime }\left(\kappa \right)d\kappa \right)}^{\frac{1}{q}}\hfill \\ \le \frac{{\left|\right|w\left|\right|}_{\left[\frac{1}{{\theta }_2},\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right],\infty }}{({\theta }_1-{\theta }_2)\Gamma \left(v\right)}{\left({\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}\left|{\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{\kappa }{\sigma }^{\prime }\left(\tau \right){\left(\sigma \left(\tau \right)-\frac{1}{{\theta }_2}\right)}^{v-1}d\tau \right|{\sigma }^{\prime }\left(\kappa \right)d\kappa \right)}^{1-\frac{1}{q}}\hfill \\ \times \left({\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}\left|{\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{\kappa }{\sigma }^{\prime }\left(\tau \right){\left(\sigma \left(\tau \right)-\frac{1}{{\theta }_2}\right)}^{v-1}d\tau \right|\left[\frac{{\theta }_1({\theta }_2-\sigma \left(\kappa \right))}{\sigma \left(\kappa \right)}{\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_1\right)\right|}^q\right.\right.\hfill \\ {\left.\left.+\frac{{\theta }_2(\sigma \left(\kappa \right)-{\theta }_1)}{\sigma \left(\kappa \right)}{\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_2\right)\right|}^q\right]{\sigma }^{\prime }\left(\kappa \right)d\kappa \right)}^{\frac{1}{q}}\hfill \\ +\frac{{\left|\right|w\left|\right|}_{\left[\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2},\frac{1}{{\theta }_1}\right],\infty }}{({\theta }_1-{\theta }_2)\Gamma \left(v\right)}{\left({\int }_{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}\left|{\int }_{\kappa }^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}{\sigma }^{\prime }\left(\tau \right){\left(\frac{1}{{\theta }_1}-\sigma \left(\tau \right)\right)}^{v-1}d\tau \right|{\sigma }^{\prime }\left(\kappa \right)d\kappa \right)}^{1-\frac{1}{q}}\hfill \\ \times \left({\int }_{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}\left|{\int }_{\kappa }^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}{\sigma }^{\prime }\left(\tau \right){\left(\frac{1}{{\theta }_1}-\sigma \left(\tau \right)\right)}^{v-1}d\tau \right|\left[\frac{{\theta }_1({\theta }_2-\sigma \left(\kappa \right))}{\sigma \left(\kappa \right)}{\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_1\right)\right|}^q\right.\right.\hfill \\ {\left.\left.+\frac{{\theta }_2(\sigma \left(\kappa \right)-{\theta }_1)}{\sigma \left(\kappa \right)}{\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_2\right)\right|}^q\right]{\sigma }^{\prime }\left(\kappa \right)d\kappa \right)}^{\frac{1}{q}},\hfill \end{array}$$

(36)

where it is obvious that

$$\begin{array}{c}{\int }_{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}\left|{\int }_{\kappa }^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}{\sigma }^{\prime }\left(\tau \right){\left(\frac{1}{{\theta }_1}-\sigma \left(\tau \right)\right)}^{v-1}d\tau \right|{\sigma }^{\prime }\left(\kappa \right)d\kappa \hfill \\ \hfill ={\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}\left|{\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{\kappa }{\sigma }^{\prime }\left(\tau \right){\left(\sigma \left(\tau \right)-\frac{1}{{\theta }_2}\right)}^{v-1}d\tau \right|{\sigma }^{\prime }\left(\kappa \right)d\kappa =\frac{{\left({\theta }_2-{\theta }_1\right)}^{v+1}}{{\left(2{\theta }_1{\theta }_2\right)}^{v+1}v(v+1)}.\end{array}$$

(37)

We can obtain the following integral after inserting (37) into (36)

$$\begin{array}{cc}\hfill & \left|{\mathbb{Z}}_1+{\mathbb{Z}}_2\right|\hfill \\ & \le \frac{{\left|\right|w\left|\right|}_{\left[\frac{1}{{\theta }_2},\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right],\infty }}{({\theta }_1-{\theta }_2)\Gamma \left(v\right)}{\left(\frac{{\left({\theta }_2-{\theta }_1\right)}^{v+1}}{{\left(2{\theta }_1{\theta }_2\right)}^{v+1}v(v+1)}\right)}^{1-\frac{1}{q}}\hfill \\ & \times \left({\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}\left|{\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{\kappa }{\sigma }^{\prime }\left(\tau \right){\left(\sigma \left(\tau \right)-\frac{1}{{\theta }_2}\right)}^{v-1}d\tau \right|\left[\frac{{\theta }_1({\theta }_2-\sigma \left(\kappa \right))}{\sigma \left(\kappa \right)}{\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_1\right)\right|}^q\right.\right.\hfill \\ & {\left.\left.+\frac{{\theta }_2(\sigma \left(\kappa \right)-{\theta }_1)}{\sigma \left(\kappa \right)}{\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_2\right)\right|}^q\right]{\sigma }^{\prime }\left(\kappa \right)d\kappa \right)}^{\frac{1}{q}}\hfill \\ & +\frac{{\left|\right|w\left|\right|}_{\left[\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2},\frac{1}{{\theta }_1}\right],\infty }}{({\theta }_2-{\theta }_1)\Gamma \left(v\right)}{\left(\frac{{\left({\theta }_2-{\theta }_1\right)}^{v+1}}{{\left(2{\theta }_1{\theta }_2\right)}^{v+1}v(v+1)}\right)}^{1-\frac{1}{q}}\hfill \\ & \times \left({\int }_{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}\left|{\int }_{\kappa }^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}{\sigma }^{\prime }\left(\tau \right){\left(\frac{1}{{\theta }_1}-\sigma \left(\tau \right)\right)}^{v-1}d\tau \right|\left[\frac{{\theta }_1({\theta }_2-\sigma \left(\kappa \right))}{\sigma \left(\kappa \right)}{\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_1\right)\right|}^q\right.\right.\hfill \\ & {\left.\left.+\frac{{\theta }_2(\sigma \left(\kappa \right)-{\theta }_1)}{\sigma \left(\kappa \right)}{\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_2\right)\right|}^q\right]{\sigma }^{\prime }\left(\kappa \right)d\kappa \right)}^{\frac{1}{q}}.\hfill \end{array}$$

(38)

We can obtain the necessary result (33) by doing basic integral calculations based on inequality (38). □

Theorem 9.

Suppose that all the conditions of Lemma 2 and $|{\mathcal{F}}^{\prime }{|}^q$ is harmonically convex on $[{\theta }_1,{\theta }_2]$ with $q>1$ and σ is an increasing and positive function from $[{\theta }_1,{\theta }_2)$ onto itself such that its derivative ${\sigma }^{{}^{\prime }}\left(\tau \right)$ is continuous on $({\theta }_1,{\theta }_2)$, for $v>0$, then we have

$$\begin{array}{c}\left|{\mathbb{Z}}_1+{\mathbb{Z}}_2\right|\le {\left(\frac{{\left({\theta }_2-{\theta }_1\right)}^{pv+1}}{{\left(2{\theta }_1{\theta }_2\right)}^{pv+1}pv(pv+1)}\right)}^{\frac{1}{p}}\hfill \\ \times \left[\frac{{\left|\right|w\left|\right|}_{\left[\frac{1}{{\theta }_2},\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right],\infty }}{({\theta }_2-{\theta }_1)\Gamma \left(v\right)}{\left({\mathbb{X}}_1\left({\theta }_1,{\theta }_2;\sigma \left(\kappa \right)\right){\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_1\right)\right|}^q+{\mathbb{X}}_2\left({\theta }_1,{\theta }_2;\sigma \left(\kappa \right)\right){\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_2\right)\right|}^q\right)}^{\frac{1}{q}}\right.\\ \hfill +\left.\frac{{\left|\right|w\left|\right|}_{\left[\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2},\frac{1}{{\theta }_1}\right],\infty }}{({\theta }_2-{\theta }_1)\Gamma \left(v\right)}{\left({\mathbb{X}}_3\left({\theta }_1,{\theta }_2;\sigma \left(\kappa \right)\right){\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_1\right)\right|}^q+{\mathbb{X}}_2\left({\theta }_1,{\theta }_2;\sigma \left(\kappa \right)\right){\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_2\right)\right|}^q\right)}^{\frac{1}{q}}\right]\\ \le {\left(\frac{{\left({\theta }_2-{\theta }_1\right)}^{pv+1}}{{\left(2{\theta }_1{\theta }_2\right)}^{pv+1}pv(pv+1)}\right)}^{\frac{1}{p}}\left(\frac{{\left|\right|w\left|\right|}_{\left[\frac{1}{{\theta }_1},\frac{1}{{\theta }_2}\right],\infty }}{({\theta }_2-{\theta }_1)\Gamma \left(v\right)}\right)\left\{{\left(\left({\mathbb{X}}_1\left({\theta }_1,{\theta }_2;\sigma \left(\kappa \right)\right)+{\mathbb{X}}_3\left({\theta }_1,{\theta }_2;\sigma \left(\kappa \right)\right)\right){\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_1\right)\right|}^q\right)}^{\frac{1}{q}}\right.\hfill \\ \hfill +\left.{\left(\left({\mathbb{X}}_2\left({\theta }_1,{\theta }_2;\sigma \left(\kappa \right)\right){\mathbb{X}}_4\left({\theta }_1,{\theta }_2;\sigma \left(\kappa \right)\right)\right){\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_2\right)\right|}^q\right)}^{\frac{1}{q}}\right\},\end{array}$$

(39)

where ${\mathbb{X}}_1\left({\theta }_1,{\theta }_2;\sigma \left(\kappa \right)\right),$ ${\mathbb{X}}_2\left({\theta }_1,{\theta }_2;\sigma \left(\kappa \right)\right),$ ${\mathbb{X}}_3\left({\theta }_1,{\theta }_2;\sigma \left(\kappa \right)\right),$ and ${\mathbb{X}}_4\left({\theta }_1,{\theta }_2;\sigma \left(\kappa \right)\right),$ are defined as follows:

$${\mathbb{X}}_1\left({\theta }_1,{\theta }_2;\sigma \left(\kappa \right)\right)=\frac{1}{2}\left(\frac{{\theta }_1}{{\theta }_2}-1\right)+{\theta }_1{\theta }_2\left(ln\left(\frac{{\theta }_1+{\theta }_2}{{\theta }_1{\theta }_2}\right)-ln\left(\frac{2}{{\theta }_2}\right)\right),$$

$${\mathbb{X}}_2\left({\theta }_1,{\theta }_2;\sigma \left(\kappa \right)\right)=\frac{1}{2}\left(\frac{{\theta }_2}{{\theta }_1}-2{\theta }_1{\theta }_2\left(ln\left(\frac{{\theta }_1+{\theta }_2}{{\theta }_1{\theta }_2}\right)-ln\left(\frac{2}{{\theta }_2}\right)\right)-1\right),$$

$${\mathbb{X}}_3\left({\theta }_1,{\theta }_2;\sigma \left(\kappa \right)\right)=\frac{1}{2}\left(\frac{{\theta }_1}{{\theta }_2}-1\right)+{\theta }_1{\theta }_2\left(-ln\left(\frac{{\theta }_1+{\theta }_2}{{\theta }_1{\theta }_2}\right)+ln\left(\frac{1}{{\theta }_1}\right)+ln2\right)$$

and

$${\mathbb{X}}_4\left({\theta }_1,{\theta }_2;\sigma \left(\kappa \right)\right)=\frac{1}{2}\left(\frac{{\theta }_2}{{\theta }_1}-2{\theta }_1{\theta }_2\left(-ln\left(\frac{{\theta }_1+{\theta }_2}{{\theta }_1{\theta }_2}\right)-ln{\theta }_1+ln2\right)-1\right).$$

Proof. 

Using the Lemma 2 as well as properties of well-known Hölder’s inequality and the harmonically convex function of $|{\mathcal{F}}^{\prime }{|}^q$, we have

$$\begin{array}{c}\left|{\mathbb{Z}}_1+{\mathbb{Z}}_2\right|\le \frac{1}{\Gamma \left(v\right)}{\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}\left|{\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{\kappa }{\sigma }^{\prime }\left(\tau \right){\left(\sigma \left(\tau \right)-\frac{1}{{\theta }_2}\right)}^{v-1}\left(w\circ \mathcal{H}\circ \sigma \right)\left(\tau \right)d\tau \right|\left|\left({\mathcal{F}}^{\prime }\circ \mathcal{H}\circ \sigma \right)\left(\kappa \right)\right|{\sigma }^{\prime }\left(\kappa \right)d\kappa \hfill \\ +\frac{1}{\Gamma \left(v\right)}{\int }_{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}\left|{\int }_{\kappa }^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}{\sigma }^{\prime }\left(\tau \right){\left(\frac{1}{{\theta }_1}-\sigma \left(\tau \right)\right)}^{v-1}\left(w\circ \mathcal{H}\circ \sigma \right)\left(\tau \right)d\tau \right|\left|\left({\mathcal{F}}^{\prime }\circ \mathcal{H}\circ \sigma \right)\left(\kappa \right)\right|{\sigma }^{\prime }\left(\kappa \right)d\kappa \hfill \\ \le \frac{1}{\Gamma \left(v\right)}{\left({\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}{\left|{\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{\kappa }{\sigma }^{\prime }\left(\tau \right){\left(\sigma \left(\tau \right)-\frac{1}{{\theta }_2}\right)}^{v-1}\left(w\circ \mathcal{H}\circ \sigma \right)\left(\tau \right)d\tau \right|}^p{\sigma }^{\prime }\left(\kappa \right)d\kappa \right)}^{\frac{1}{p}}\hfill \\ \times {\left({\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}{\left|\left({\mathcal{F}}^{\prime }\circ \mathcal{H}\circ \sigma \right)\left(\kappa \right)\right|}^q{\sigma }^{\prime }\left(\kappa \right)d\kappa \right)}^{\frac{1}{q}}\hfill \\ +\frac{1}{\Gamma \left(v\right)}{\left({\int }_{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}{\left|{\int }_{\kappa }^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}{\sigma }^{\prime }\left(\tau \right){\left(\frac{1}{{\theta }_1}-\sigma \left(\tau \right)\right)}^{v-1}\left(w\circ \mathcal{H}\circ \sigma \right)\left(\tau \right)d\tau \right|}^p{\sigma }^{\prime }\left(\kappa \right)d\kappa \right)}^{\frac{1}{p}}\hfill \\ \times {\left({\int }_{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}{\left|\left({\mathcal{F}}^{\prime }\circ \mathcal{H}\circ \sigma \right)\left(\kappa \right)\right|}^p{\sigma }^{\prime }\left(\kappa \right)d\kappa \right)}^{\frac{1}{p}}.\hfill \end{array}$$

(40)

Since $\left|{\mathcal{F}}^{{}^{\prime }}\right|$ is harmonic-convex on $\left[{\theta }_1,{\theta }_2\right]$, where $\kappa \in \left[{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right),{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)\right]$

$$\begin{array}{c}\left|({\mathcal{F}}^{\prime }\circ \mathcal{H}\circ \sigma )\left(\kappa \right)\right|=\left|{\mathcal{F}}^{\prime }\left(\left(\frac{{\theta }_1({\theta }_2-\sigma \left(\kappa \right))}{\sigma \left(\kappa \right)({\theta }_2-{\theta }_1)}\right)\frac{1}{{\theta }_1}+\left(\frac{{\theta }_2(\sigma \left(\kappa \right)-{\theta }_1)}{\sigma \left(\kappa \right)({\theta }_2-{\theta }_1)}\right)\frac{1}{{\theta }_2}\right)\right|\hfill \\ \hfill \le \frac{{\theta }_1({\theta }_2-\sigma \left(\kappa \right))}{\sigma \left(\kappa \right)({\theta }_2-{\theta }_1)}\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_1\right)\right|+\frac{{\theta }_2(\sigma \left(\kappa \right)-{\theta }_1)}{\sigma \left(\kappa \right)({\theta }_2-{\theta }_1)}\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_2\right)\right|.\end{array}$$

(41)

As a result, we get

$$\begin{array}{c}\left|{\mathbb{Z}}_1+{\mathbb{Z}}_2\right|\le \frac{{\left|\right|w\left|\right|}_{\left[\frac{1}{{\theta }_2},\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right],\infty }}{\Gamma \left(v\right)}{\left({\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}{\left|{\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{\kappa }{\sigma }^{\prime }\left(\tau \right){\left(\sigma \left(\tau \right)-\frac{1}{{\theta }_2}\right)}^{v-1}d\tau \right|}^p{\sigma }^{\prime }\left(\kappa \right)d\kappa \right)}^{\frac{1}{p}}\hfill \\ \times {\left({\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}{\left|\left({\mathcal{F}}^{\prime }\circ \mathcal{H}\circ \sigma \right)\left(\kappa \right)\right|}^q{\sigma }^{\prime }\left(\kappa \right)d\kappa \right)}^{\frac{1}{q}}\\ +\frac{{\left|\right|w\left|\right|}_{\left[\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2},\frac{1}{{\theta }_1}\right],\infty }}{\Gamma \left(v\right)}{\left({\int }_{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}{\left|{\int }_{\kappa }^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}{\sigma }^{\prime }\left(\tau \right){\left(\frac{1}{{\theta }_1}-\sigma \left(\tau \right)\right)}^{v-1}d\tau \right|}^p{\sigma }^{\prime }\left(\kappa \right)d\kappa \right)}^{\frac{1}{p}}\\ \times {\left({\int }_{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}{\left|\left({\mathcal{F}}^{\prime }\circ \mathcal{H}\circ \sigma \right)\left(\kappa \right)\right|}^q{\sigma }^{\prime }\left(\kappa \right)d\kappa \right)}^{\frac{1}{q}}\\ \le \frac{{\left|\right|w\left|\right|}_{\left[\frac{1}{{\theta }_2},\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right],\infty }}{({\theta }_2-{\theta }_1)\Gamma \left(v\right)}{\left({\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}{\left|{\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{\kappa }{\sigma }^{\prime }\left(\tau \right){\left(\sigma \left(\tau \right)-\frac{1}{{\theta }_2}\right)}^{v-1}d\tau \right|}^p{\sigma }^{\prime }\left(\kappa \right)d\kappa \right)}^{\frac{1}{p}}\\ \times {\left({\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}\left[\frac{{\theta }_1({\theta }_2-\sigma \left(\kappa \right))}{\sigma \left(\kappa \right)}{\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_1\right)\right|}^q+\frac{{\theta }_2(\sigma \left(\kappa \right)-{\theta }_1)}{\sigma \left(\kappa \right)}{\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_2\right)\right|}^q\right]{\sigma }^{\prime }\left(\kappa \right)d\kappa \right)}^{\frac{1}{q}}\\ +\frac{{\left|\right|w\left|\right|}_{\left[\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2},\frac{1}{{\theta }_1}\right],\infty }}{({\theta }_2-{\theta }_1)\Gamma \left(v\right)}{\left({\int }_{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}{\left|{\int }_{\kappa }^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}{\sigma }^{\prime }\left(\tau \right){\left(\frac{1}{{\theta }_1}-\sigma \left(\tau \right)\right)}^{v-1}d\tau \right|}^p{\sigma }^{\prime }\left(\kappa \right)d\kappa \right)}^{\frac{1}{p}}\\ \hfill \times {\left({\int }_{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}\left[\frac{{\theta }_1({\theta }_2-\sigma \left(\kappa \right))}{\sigma \left(\kappa \right)}{\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_1\right)\right|}^q+\frac{{\theta }_2(\sigma \left(\kappa \right)-{\theta }_1)}{\sigma \left(\kappa \right)}{\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_2\right)\right|}^q\right]{\sigma }^{\prime }\left(\kappa \right)d\kappa \right)}^{\frac{1}{q}},\end{array}$$

(42)

where it is obvious

$$\begin{array}{c}{\int }_{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}{\left|{\int }_{\kappa }^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}{\sigma }^{\prime }\left(\tau \right){\left(\frac{1}{{\theta }_1}-\sigma \left(\tau \right)\right)}^{v-1}d\tau \right|}^p{\sigma }^{\prime }\left(\kappa \right)d\kappa \hfill \\ \hfill ={\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}{\left|{\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{\kappa }{\sigma }^{\prime }\left(\tau \right){\left(\sigma \left(\tau \right)-\frac{1}{{\theta }_2}\right)}^{v-1}d\tau \right|}^p{\sigma }^{\prime }\left(\kappa \right)d\kappa =\frac{{\left({\theta }_2-{\theta }_1\right)}^{pv+1}}{{\left(2{\theta }_1{\theta }_2\right)}^{pv+1}pv(pv+1)}.\end{array}$$

(43)

We obtain the following integral after inserting (43) into (42):

$$\begin{array}{c}\left|{\mathbb{Z}}_1+{\mathbb{Z}}_2\right|\le \frac{{\left|\right|w\left|\right|}_{\left[\frac{1}{{\theta }_2},\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right],\infty }}{({\theta }_2-{\theta }_1)\Gamma \left(v\right)}{\left(\frac{{\left({\theta }_2-{\theta }_1\right)}^{pv+1}}{{\left(2{\theta }_1{\theta }_2\right)}^{pv+1}pv(pv+1)}\right)}^{\frac{1}{p}}\hfill \\ \times {\left({\int }_{{\sigma }^{-1}\left(\frac{1}{{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}\left[\frac{{\theta }_1({\theta }_2-\sigma \left(\kappa \right))}{\sigma \left(\kappa \right)}{\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_1\right)\right|}^q+\frac{{\theta }_2(\sigma \left(\kappa \right)-{\theta }_1)}{\sigma \left(\kappa \right)}{\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_2\right)\right|}^q\right]{\sigma }^{\prime }\left(\kappa \right)d\kappa \right)}^{\frac{1}{q}}\\ +\frac{{\left|\right|w\left|\right|}_{\left[\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2},\frac{1}{{\theta }_1}\right],\infty }}{({\theta }_2-{\theta }_1)\Gamma \left(v\right)}{\left(\frac{{\left({\theta }_2-{\theta }_1\right)}^{pv+1}}{{\left(2{\theta }_1{\theta }_2\right)}^{pv+1}pv(pv+1)}\right)}^{\frac{1}{p}}\\ \hfill \times {\left({\int }_{{\sigma }^{-1}\left(\frac{{\theta }_1+{\theta }_2}{2{\theta }_1{\theta }_2}\right)}^{{\sigma }^{-1}\left(\frac{1}{{\theta }_1}\right)}\left[\frac{{\theta }_1({\theta }_2-\sigma \left(\kappa \right))}{\sigma \left(\kappa \right)}{\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_1\right)\right|}^q+\frac{{\theta }_2(\sigma \left(\kappa \right)-{\theta }_1)}{\sigma \left(\kappa \right)}{\left|\left({\mathcal{F}}^{{}^{\prime }}\circ \mathcal{H}\right)\left({\theta }_2\right)\right|}^q\right]{\sigma }^{\prime }\left(\kappa \right)d\kappa \right)}^{\frac{1}{q}}.\end{array}$$

(44)

We can obtain the necessary result (39) by doing basic integral calculations based on inequality (44). □

4. Conclusions

In this paper, inequalities of the Hermite-Hadamard-Fejér type for harmonically convex functions in fractional integral forms are given in this study. Using weighted fractional integrals with positive weighted symmetric function kernels, an integral identity and various midpoint fractional Hermite-Hadamard-Fejér type integral inequalities for harmonically convex functions are also found.