Refinements and Applications of Hermite–Hadamard-Type Inequalities Using Hadamard Fractional Integral Operators and GA-Convexity

Abstract

In this paper, several applications of the Hermite–Hadamard inequality for fractional integrals using $GA$-convexity are discussed, including some new refinements and similar extensions, as well as several applications in the Gamma and incomplete Gamma functions.

1. Introduction

We know that an interval $I\subset \mathbb{R}$ is convex if for all $\mu$, $\nu \in I$, we have $\tau \mu +\left(1-\tau \right)\nu \in I$, where $\tau \in \left[0,1\right]$ and a function $\phi :I\to \mathbb{R}$ is convex if for all $\mu$, $\nu \in I$, the inequality

$$\phi \left(\tau \mu +\left(1-\tau \right)\nu \right)\le \tau \phi \left(\mu \right)+\left(1-\tau \right)\phi \left(\nu \right)$$

(1)

holds. A function $\phi :I\to \mathbb{R}$ is concave if the inequality (1) holds in the opposite direction.

Inequalities are an excellent mathematical tool because of their importance in fractional calculus, classical calculus, quantum calculus, stochastic calculus, time-scale calculus, fractal sets, and other topics. Integral inequalities are the essential mathematical tools that connect integrals and inequalities and provide insights into the behavior of functions across specific intervals. We refer the interested reader to the references provided for more information on inequalities [1,2].

Convexity is critical to understanding and solving problems involving fractional integral inequalities. Because of its attributes and definition, its relevance has recently been recognized. The studies [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49] show that convex functions have been generalized in a number of ways and, hence, a vast literature related to the Hermite–Hadamard inequalities [50,51] has been produced using several new novel generalizations of convex functions as well as fractional integral operators. The Hermite–Hadamard inequalities are very useful that provide the upper and lower bounds of the average value of a convex function over an interval. These are mostly used in mathematics to study the properties of convex functions and their applications in optimization and approximation theory. The Hermite–Hadamard integral inequalities are those most commonly found when searching for comprehensive inequalities [50,51]:

$$\phi \left(\frac{\rho +\sigma }{2}\right)\le \frac{1}{\sigma -\rho }{\int }_{\rho }^{\sigma }\phi \left(\mu \right)d\mu \le \frac{\phi \left(\rho \right)+\phi \left(\sigma \right)}{2},$$

(2)

where the function $\phi :I\to \mathbb{R}$ is convex on I and $\phi \in L^1\left(\left[\rho ,\sigma \right]\right)$.

We refer the interested reader a remarkable paper [52] for the Weighted generalization of the inequalities (2) obtained by the famous mathematician Fejér.

2. Literature Review

As well as being helpful for generalization, convexity and convex functions have many other uses. The concept of $GA$-convexity and $GA$-convex functions is one of those generalizations, and is defined as follows:

Definition 1

([53,54]). Let $I\subseteq \left(0,\infty \right)$. A function $\phi :I\to \mathbb{R}$ is considered to be $GA$-convex, if the inequality

$$\phi \left({\mu }^{\tau }{\nu }^{1-\tau }\right)\le \left(1-\tau \right)\phi \left(\mu \right)+\tau \phi \left(\nu \right)$$

(3)

holds for all $\mu ,\nu \in I$ and $\tau \in \left[0,1\right]$. If the inequality (3) holds in the reversed direction, then the function $\phi :I\to \mathbb{R}$ is defined as $GA$-concave.

The following Hermite–Hadamard-type inequalities were proved in [22] by applying $GA$-convexity.

Theorem 1

([22]). Let $\phi :I\subseteq \left(0,\infty \right)\to \mathbb{R}$ be a $GA$-convex function and $\rho ,\sigma \in I$ with $\rho <\sigma$. If $\phi \in L\left(\left[\rho ,\sigma \right]\right)$, then the following inequalities hold:

$$\phi \left(\sqrt{\rho \sigma }\right)\le \frac{1}{ln\sigma -ln\rho }{\int }_{\sigma }^{\rho }\frac{\phi \left(\mu \right)}{\mu }d\mu \le \frac{\phi \left(\rho \right)+\phi \left(\sigma \right)}{2}.$$

(4)

Fractional calculus, which focuses on fractional integration across complex domains, has recently gained prominence and aroused the interest of mathematicians due to its practical applications. The study of fractional integral inequality was influenced by the work of well-known inequalities researchers, such as Ostrowski, Simpson, and Hadamard. Fractional calculus is applied in many fields, including engineering, modeling, economics, mathematical biology, fluid flow, natural phenomena prediction, health-care, and image processing.

The left-sided and right-sided Riemann–Liouville fractional integrals ${\mathcal{I}}_{{\rho }^{+}}^{\kappa }\phi \left(\sigma \right)$ and ${\mathcal{I}}_{{\sigma }^{-}}^{\kappa }\phi \left(\rho \right)$ of order $\kappa >0$ in (5), are defined, respectively, as (see [34,55,56]):

$${\mathcal{I}}_{{\rho }^{+}}^{\kappa }\phi \left(\mu \right):=\frac{1}{\Gamma \left(\kappa \right)}{\int }_{\rho }^{\mu }{\left(\mu -\tau \right)}^{\kappa -1}\phi \left(\tau \right)d\tau ,\rho <\mu$$

and

$${\mathcal{I}}_{{\sigma }^{-}}^{\kappa }\phi \left(\mu \right):=\frac{1}{\Gamma \left(\kappa \right)}{\int }_{\mu }^{\sigma }{\left(\tau -\mu \right)}^{\kappa -1}\phi \left(\tau \right)d\tau ,\mu <\sigma ,$$

where $\Gamma \left(\kappa \right)$ is the Gamma function defined by

$$\Gamma \left(\kappa \right)={\int }_0^{\infty }{e}^{-\tau }{\tau }^{\kappa -1}d\tau \mathrm{and}{\mathcal{I}}_{{\rho }^{+}}^0\phi \left(\mu \right)={\mathcal{I}}_{{\sigma }^{-}}^0\phi \left(\mu \right)=\phi \left(\mu \right).$$

Let $\phi \in L\left(\left[\rho ,\sigma \right]\right)$. The right-hand side and left-hand side Hadamard fractional integrals ${\mathcal{J}}_{{\rho }^{+}}^{\kappa }\phi \left(\sigma \right)$ and ${\mathcal{J}}_{{\sigma }^{-}}^{\kappa }\phi \left(\rho \right)$ of order $\kappa >0$ with $0<\rho <\sigma <\infty$ are defined by (please see the reference [57] for more details on Hadamard fractional integral operators):

$${\mathcal{J}}_{{\rho }^{+}}^{\kappa }\phi \left(\mu \right):=\frac{1}{\Gamma \left(\kappa \right)}{\int }_{\rho }^{\mu }{\left(ln\frac{\mu }{\tau }\right)}^{\kappa -1}\frac{\phi \left(\tau \right)}{\tau }d\tau ,\rho <\mu$$

and

$${\mathcal{J}}_{{\sigma }^{-}}^{\kappa }\phi \left(\mu \right):=\frac{1}{\Gamma \left(\kappa \right)}{\int }_{\mu }^{\sigma }{\left(ln\frac{\tau }{\mu }\right)}^{\kappa -1}\frac{\phi \left(\tau \right)}{\tau }d\tau ,\mu <\sigma ,$$

where $\Gamma \left(\kappa \right)$ is the Gamma function as defined above.

The following fractional integral forms can be used to express Hermite–Hadamard-type inequalities (4) for $GA$-convex functions:

Theorem 2

(Trapezoidal-type fractional inequalities [21]). Let $\phi :I\subseteq \left(0,\infty \right)\to \mathbb{R}$ be a function such that $\phi \in L\left(\left[\rho ,\sigma \right]\right)$, where $\rho ,\sigma \in I$ with $\rho <\sigma$. If φ is a $GA$-convex function on $\left[\rho ,\sigma \right]$, then the following inequalities for fractional integrals hold:

$$\phi \left(\sqrt{\rho \sigma }\right)\le \frac{\Gamma \left(\kappa +1\right)}{2{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }}\left[{\mathcal{J}}_{\rho +}^{\kappa }\phi \left(\sigma \right)+{\mathcal{J}}_{\sigma -}^{\kappa }\phi \left(\rho \right)\right]\le \frac{\phi \left(\rho \right)+\phi \left(\sigma \right)}{2}$$

(5)

with $\kappa >0$.

Theorem 3

(Midpoint-type fractional inequalities [20]). Let $\phi :I\subseteq \left(0,\infty \right)\to \mathbb{R}$ be a function such that $\phi \in L\left(\left[\rho ,\sigma \right]\right)$, where $\rho ,\sigma \in I$ with $\rho <\sigma$. If φ is a $GA$-convex function on $\left[\rho ,\sigma \right]$, then the following inequalities for fractional integrals hold:

$$\phi \left(\sqrt{\rho \sigma }\right)\le \frac{\Gamma \left(\kappa +1\right)}{2^{1-\kappa }{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }}\left[{\mathcal{J}}_{\sqrt{\rho \sigma }+}^{\kappa }\phi \left(\sigma \right)+{\mathcal{J}}_{\sqrt{\rho \sigma }-}^{\kappa }\phi \left(\rho \right)\right]\le \frac{\phi \left(\rho \right)+\phi \left(\sigma \right)}{2}$$

(6)

with $\kappa >0$.

Kunt and İşcan [20] proved the inequalities that provide the bounds for the difference between the terms in the second part of the inequalities (5).

Theorem 4

([20]). Let $\phi :I\subseteq \left(0,\infty \right)\to \mathbb{R}$ be a differentiable mapping on $I°$ such that $\phi \in L\left(\left[\rho ,\sigma \right]\right)$, where $\rho ,\sigma \in I°$ with $\rho <\sigma$ and $\kappa >0$. If $\left|{\phi }^{\prime }\right|$ is $GA$-convex on $\left[\rho ,\sigma \right]$, then the following inequality via fractional integrals holds:

$$\begin{array}{c}\left|\frac{\phi \left(\rho \right)+\phi \left(\sigma \right)}{2}-\frac{\Gamma \left(\kappa +1\right)}{2{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }}\left[{\mathcal{J}}_{\rho +}^{\kappa }\phi \left(\sigma \right)+{\mathcal{J}}_{\sigma -}^{\kappa }\phi \left(\rho \right)\right]\right|\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le \frac{ln\left(\frac{\sigma }{\rho }\right)}{2}\left[C_1\left(\kappa \right)\left|{\phi }^{\prime }\left(\rho \right)\right|+C_1\left(\kappa \right)\left|{\phi }^{\prime }\left(\sigma \right)\right|\right],\end{array}$$

(7)

where

$$C_1\left(\kappa \right)={\int }_0^{\frac{1}{2}}\left[{\left(1-\tau \right)}^{\kappa }-{\tau }^{\kappa }\right]\left[\left(1-\tau \right){\rho }^{1-\tau }{\sigma }^{\tau }+\tau {\rho }^{\tau }{\sigma }^{1-\tau }\right]d\tau$$

and

$$C_2\left(\kappa \right)={\int }_0^{\frac{1}{2}}\left[{\left(1-\tau \right)}^{\kappa }-{\tau }^{\kappa }\right]\left[\tau {\rho }^{1-\tau }{\sigma }^{\tau }+\left(1-\tau \right){\rho }^{\tau }{\sigma }^{1-\tau }\right]d\tau .$$

Theorem 5

([20]). Let $\phi :I\subseteq \left(0,\infty \right)\to \mathbb{R}$ be a differentiable mapping on $I°$ such that $\phi \in L\left(\left[\rho ,\sigma \right]\right)$, where $\rho ,\sigma \in I°$ with $\rho <\sigma$ and $\kappa >0$. If $\left|{\phi }^{\prime }\right|$ is $GA$-convex on $\left[\rho ,\sigma \right]$, then the following inequality via fractional integrals holds:

$$\begin{array}{c}\left|\phi \left(\sqrt{\rho \sigma }\right)-\frac{\Gamma \left(\kappa +1\right)}{2^{1-\kappa }{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }}\left[{\mathcal{J}}_{\sqrt{\rho \sigma }+}^{\kappa }\phi \left(\sigma \right)+{\mathcal{J}}_{\sqrt{\rho \sigma }-}^{\kappa }\phi \left(\rho \right)\right]\right|\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le \frac{ln\left(\frac{\sigma }{\rho }\right)}{2^{1-\kappa }}\left[T_1\left(\kappa \right)\left|{\phi }^{\prime }\left(\rho \right)\right|+T_1\left(\kappa \right)\left|{\phi }^{\prime }\left(\sigma \right)\right|\right],\end{array}$$

(8)

where

$$T_1\left(\kappa \right)={\int }_0^{\frac{1}{2}}{\tau }^{\kappa }\left[\left(1-\tau \right){\rho }^{1-\tau }{\sigma }^{\tau }+\tau {\rho }^{\tau }{\sigma }^{1-\tau }\right]d\tau$$

and

$$T_2\left(\kappa \right)={\int }_0^{\frac{1}{2}}{\tau }^{\kappa }\left[\tau {\rho }^{1-\tau }{\sigma }^{\tau }+\left(1-\tau \right){\rho }^{\tau }{\sigma }^{1-\tau }\right]d\tau .$$

If $\kappa =1$, then one can obtain the following inequalities that give us the bounds for the difference between the terms in the inequalities (4).

Theorem 6

([20]). Let $\phi :I\subseteq \left(0,\infty \right)\to \mathbb{R}$ be a differentiable mapping on $I°$ such that $\phi \in L\left(\left[\rho ,\sigma \right]\right)$, where $\rho ,\sigma \in I°$ with $\rho <\sigma$. If $\left|{\phi }^{\prime }\right|$ is $GA$-convex on $\left[\rho ,\sigma \right]$, then the following inequality via fractional integrals holds:

$$\left|\frac{\phi \left(\rho \right)+\phi \left(\sigma \right)}{2}-\frac{1}{ln\left(\frac{\sigma }{\rho }\right)}{\int }_{\rho }^{\sigma }\frac{\phi \left(\mu \right)}{\mu }d\mu \right|\hspace{1em}\le \hspace{1em}\frac{ln\left(\frac{\sigma }{\rho }\right)}{2}\left[C_1\left(\kappa \right)\left|{\phi }^{\prime }\left(\rho \right)\right|+C_1\left(\kappa \right)\left|{\phi }^{\prime }\left(\sigma \right)\right|\right],$$

(9)

where

$$C_1\left(1\right)={\int }_0^{\frac{1}{2}}\left[1-2\tau \right]\left[\left(1-\tau \right){\rho }^{1-\tau }{\sigma }^{\tau }+\tau {\rho }^{\tau }{\sigma }^{1-\tau }\right]d\tau$$

and

$$C_2\left(1\right)={\int }_0^{\frac{1}{2}}\left[1-2\tau \right]\left[\tau {\rho }^{1-\tau }{\sigma }^{\tau }+\left(1-\tau \right){\rho }^{\tau }{\sigma }^{1-\tau }\right]d\tau .$$

Theorem 7

([20]). Let $\phi :I\subseteq \left(0,\infty \right)\to \mathbb{R}$ be a differentiable mapping on $I°$ such that $\phi \in L\left(\left[\rho ,\sigma \right]\right)$, where $\rho ,\sigma \in I°$ with $\rho <\sigma$. If $\left|{\phi }^{\prime }\right|$ is $GA$-convex on $\left[\rho ,\sigma \right]$, then the following inequality via fractional integrals holds:

$$\left|\phi \left(\sqrt{\rho \sigma }\right)-\frac{1}{ln\left(\frac{\sigma }{\rho }\right)}{\int }_{\rho }^{\sigma }\frac{\phi \left(\mu \right)}{\mu }d\mu \right|\le ln\left(\frac{\sigma }{\rho }\right)\left[T_1\left(1\right)\left|{\phi }^{\prime }\left(\rho \right)\right|+T_1\left(1\right)\left|{\phi }^{\prime }\left(\sigma \right)\right|\right],$$

where

$$T_1\left(1\right)={\int }_0^{\frac{1}{2}}\tau \left[\left(1-\tau \right){\rho }^{1-\tau }{\sigma }^{\tau }+\tau {\rho }^{\tau }{\sigma }^{1-\tau }\right]d\tau$$

and

$$T_2\left(1\right)={\int }_0^{\frac{1}{2}}\tau \left[\tau {\rho }^{1-\tau }{\sigma }^{\tau }+\left(1-\tau \right){\rho }^{\tau }{\sigma }^{1-\tau }\right]d\tau .$$

Motivated by the studies presented in [18,20], we initiated similar studies as described in [18] using one of the novel generalizations of the convexity of functions that is known as the $GA$-convexity of functions. The results presented in this paper are novel since these results make use of the novel generalization of the convexity and an important integral operator of fractional calculus known as the Hadamard fractional integral operator. The results of this manuscript can only provide refinements of Hermite–Hadamard-type integral inequalities that have been proven for $GA$-convex functions using Hadamard fractional integral operators. Section 3 provides the results for the extensions and refinements of the Hermite–Hadamard-type inequalities (4). Section 4 considers the extensions of the fractional Hermite–Hadamard-type inequalities (5) and (6) by using the Hadamard fractional integral operators. Section 5 contains some applications of the bounds of the Gamma function and upper incomplete Gamma functions together with some graphical interpretations to validate the results obtained.

3. Main Results

This section begins with the following result that is an extension of the result (4).

Theorem 8.

Let $\phi :\left[\rho ,\sigma \right]\to \mathbb{R}$ be a $GA$-convex function with $\rho <\sigma$. Then, we have the inequality

$$\begin{array}{c}\phi \left(\sqrt{\rho \sigma }\right)\le \frac{3^{\kappa }-1}{4^{\kappa }}\phi \left(\sqrt{\rho \sigma }\right)+\frac{4^{\kappa }-3^{\kappa }+1}{2\times 4^{\kappa }}\left[\phi \left({\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}\right)+\phi \left({\rho }^{\frac{1}{4}}{\sigma }^{\frac{3}{4}}\right)\right]\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le \frac{\Gamma \left(\kappa +1\right)}{2{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }}\left[{\mathcal{J}}_{\rho +}^{\kappa }\phi \left(\sigma \right)+{\mathcal{J}}_{\sigma -}^{\kappa }\phi \left(\rho \right)\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le \frac{3^{\kappa }-1}{2\times 4^{\kappa }}\left[\phi \left({\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}\right)+\phi \left({\rho }^{\frac{1}{4}}{\sigma }^{\frac{3}{4}}\right)\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{4^{\kappa }-3^{\kappa }+1}{4^{\kappa }}\left[\frac{\phi \left(\rho \right)+\phi \left(\sigma \right)}{2}\right]\le \frac{\phi \left(\rho \right)+\phi \left(\sigma \right)}{2}\end{array}$$

(10)

for $\kappa >0$.

Proof. 

We can derive the following identities from simple computation:

$$\begin{array}{c}\frac{\kappa \Gamma \left(\kappa \right)}{2{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }}\left[{\mathcal{J}}_{\rho +}^{\kappa }\phi \left(\sigma \right)+{\mathcal{J}}_{\sigma -}^{\kappa }\phi \left(\rho \right)\right]\hfill \\ =\frac{\kappa }{2{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }}{\int }_{\rho }^{\sigma }\left[{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa -1}+{\left(ln\frac{\mu }{\rho }\right)}^{\kappa -1}\right]\frac{\phi \left(\mu \right)}{\mu }d\mu \\ \hspace{1em}=\frac{\kappa }{2{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }}{\int }_{\rho }^{{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}}\left[{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa -1}+{\left(ln\frac{\mu }{\rho }\right)}^{\kappa -1}\right]\left[\phi \left(\mu \right)+\phi \left(\rho \sigma {\mu }^{-1}\right)\right]\frac{d\mu }{\mu }\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{\kappa }{2{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }}{\int }_{{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}}^{\sqrt{\rho \sigma }}\left[{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa -1}+{\left(ln\frac{\mu }{\mu }\right)}^{\kappa -1}\right]\left[\phi \left(\mu \right)+\phi \left(\rho \sigma {\mu }^{-1}\right)\right]\frac{d\mu }{\mu }.\end{array}$$

(11)

It is easy to observe that

$$\begin{array}{cc}\hfill & \frac{\kappa }{2{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }}{\int }_{\rho }^{{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}}\left[{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa -1}+{\left(ln\frac{\mu }{\rho }\right)}^{\kappa -1}\right]\frac{d\mu }{\mu }\hfill \\ \hfill & =\frac{1}{2}{\left[ln\left(\frac{\sigma }{\rho }\right)\right]}^{-\kappa }\left\{-{\left[ln\left(\frac{{\sigma }^{3/4}}{{\rho }^{3/4}}\right)\right]}^{\kappa }+{\left[ln\left(\frac{\sigma }{\rho }\right)\right]}^{\kappa }+{\left[ln\left(\frac{\sqrt[4]{\sigma }}{\sqrt[4]{\rho }}\right)\right]}^{\kappa }\right\}\hfill \\ \hfill & =\frac{1}{2}·\frac{4^{\kappa }-3^{\kappa }+1}{4^{\kappa }}.\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & \frac{\kappa }{2{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }}{\int }_{{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}}^{\sqrt{\rho \sigma }}\left[{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa -1}+{\left(ln\frac{\mu }{\rho }\right)}^{\kappa -1}\right]\frac{d\mu }{\mu }\hfill \\ \hfill & =\frac{1}{2}\left\{{\left[ln\left(\frac{{\sigma }^{3/4}}{{\rho }^{3/4}}\right)\right]}^{\kappa }-{\left[ln\left(\frac{\sqrt[4]{\sigma }}{\sqrt[4]{\rho }}\right)\right]}^{\kappa }\right\}{\left[ln\left(\frac{\sigma }{\rho }\right)\right]}^{-\kappa }\hfill \\ \hfill & =\frac{1}{2}·\frac{3^{\kappa }-1}{4^{\kappa }}.\hfill \end{array}$$

Since

$${\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}={\mu }^{\frac{ln\rho -4ln\mu +3ln\sigma }{4ln\rho -8ln\mu +4ln\sigma }}{\left(\rho \sigma {\mu }^{-1}\right)}^{\frac{3ln\rho -4ln\mu +ln\sigma }{4ln\rho -8ln\mu +4ln\sigma }}$$

(12)

and

$${\rho }^{\frac{1}{4}}{\sigma }^{\frac{3}{4}}={\left(\rho \sigma {\mu }^{-1}\right)}^{\frac{ln\rho -4ln\mu +3ln\sigma }{4ln\rho -8ln\mu +4ln\sigma }}{\mu }^{\frac{3ln\rho -4ln\mu +ln\sigma }{4ln\rho -8ln\mu +4ln\sigma }},$$

(13)

where $\mu \in \left[\rho ,{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}\right]$ and $0\le \frac{ln\rho -4ln\mu +3ln\sigma }{4ln\rho -8ln\mu +4ln\sigma },\frac{3ln\rho -4ln\mu +ln\sigma }{4ln\rho -8ln\mu +4ln\sigma }\le 1$.

We can also write the following identities:

$$\sqrt{\rho \sigma }={\mu }^{\frac{1}{2}}{\left(\rho \sigma {\mu }^{-1}\right)}^{\frac{1}{2}},$$

(14)

where $\mu \in \left[{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}},\sqrt{\rho \sigma }\right]$.

Moreover, we also obtain that

$$\begin{array}{cc}\mu \hfill & =exp\left(ln\mu \right)\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill & =exp\left(\frac{ln\sigma -ln\mu }{ln\sigma -ln\rho }ln\rho +\frac{ln\mu -ln\rho }{ln\sigma -ln\rho }ln\sigma \right)\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill & ={\rho }^{\frac{ln\sigma -ln\mu }{ln\sigma -ln\rho }}{\sigma }^{\frac{ln\mu -ln\rho }{ln\sigma -ln\rho }}\hfill \end{array}$$

(17)

and

$$\rho \sigma {\mu }^{-1}={\rho }^{\frac{ln\mu -ln\rho }{ln\sigma -ln\rho }}{\sigma }^{\frac{ln\sigma -ln\mu }{ln\sigma -ln\rho }},$$

(18)

where $\mu \in \left[\rho ,{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}\right]$.

Lastly, we obtain the following identities:

$$\mu ={\left({\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}\right)}^{\frac{ln\rho +3ln\sigma -4ln\mu }{2ln\sigma -2ln\rho }}{\left({\rho }^{\frac{1}{4}}{\sigma }^{\frac{3}{4}}\right)}^{\frac{3ln\rho +ln\sigma -4ln\mu }{2ln\sigma -2ln\rho }}$$

(19)

and

$$\rho \sigma {\mu }^{-1}={\left({\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}\right)}^{\frac{3ln\rho +ln\sigma -4ln\mu }{2ln\sigma -2ln\rho }}{\left({\rho }^{\frac{1}{4}}{\sigma }^{\frac{3}{4}}\right)}^{\frac{ln\rho +3ln\sigma -4ln\mu }{2ln\sigma -2ln\rho }},$$

(20)

where $\mu \in \left[{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}},\sqrt{\rho \sigma }\right]$ with $0\le \frac{ln\rho +3ln\sigma -4ln\mu }{2ln\sigma -2ln\rho },\frac{3ln\rho +ln\sigma -4ln\mu }{2ln\sigma -2ln\rho }\le 1$.

Using the $GA$-convexity of $\phi :\left[\rho ,\sigma \right]\to \mathbb{R}$, we obtain

$$\frac{4^{\kappa }-3^{\kappa }+1}{2\times 4^{\kappa }}\left[\phi \left({\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}\right)+\phi \left({\rho }^{\frac{1}{4}}{\sigma }^{\frac{3}{4}}\right)\right]\ge \frac{4^{\kappa }-3^{\kappa }+1}{4^{\kappa }}\phi \left(\sqrt{\rho \sigma }\right)$$

(21)

and

$$\frac{3^{\kappa }-1}{2\times 4^{\kappa }}\left[\phi \left({\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}\right)+\phi \left({\rho }^{\frac{1}{4}}{\sigma }^{\frac{3}{4}}\right)\right]\le \frac{3^{\kappa }-1}{2\times 4^{\kappa }}\left[\phi \left(\rho \right)+\phi \left(\sigma \right)\right].$$

(22)

Using the aforementioned identities (12)–(14) and the $GA$-convexity of $\phi :\left[\rho ,\sigma \right]\to \mathbb{R}$, we obtain the following inequalities:

$$\begin{array}{c}\frac{4^{\kappa }-3^{\kappa }+1}{2\times 4^{\kappa }}\left[\phi \left({\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}\right)+\phi \left({\rho }^{\frac{1}{4}}{\sigma }^{\frac{3}{4}}\right)\right]=\frac{\kappa }{2{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }}\hfill \\ \hspace{1em}\hspace{1em} \times {\int }_{\rho }^{{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}}\left[{\left(ln\frac{\mu }{\rho }\right)}^{\kappa -1}+{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa -1}\right]\left[\phi \left({\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}\right)+\phi \left({\rho }^{\frac{1}{4}}{\sigma }^{\frac{3}{4}}\right)\right]\frac{d\mu }{\mu }\\ \hspace{1em}\hspace{1em} \le \frac{\kappa }{2{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }}{\int }_{\rho }^{{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}}\left[{\left(ln\frac{\mu }{\rho }\right)}^{\kappa -1}+{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa -1}\right]\\ \hspace{1em}\times \left[\left(\frac{ln\rho -4ln\mu +3ln\sigma }{4ln\rho -8ln\mu +4ln\sigma }\right)\phi \left(\mu \right)+\left(\frac{3ln\rho -4ln\mu +ln\sigma }{4ln\rho -8ln\mu +4ln\sigma }\right)\phi \left(\rho \sigma {\mu }^{-1}\right)\right.\hfill \\ \left.+\left(\frac{3ln\rho -4ln\mu +ln\sigma }{4ln\rho -8ln\mu +4ln\sigma }\right)\phi \left(\mu \right)+\left(\frac{ln\rho -4ln\mu +3ln\sigma }{4ln\rho -8ln\mu +4ln\sigma }\right)\phi \left(\rho \sigma {\mu }^{-1}\right)\right]\frac{d\mu }{\mu }\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em} =\frac{\kappa }{2{\left[ln\left(\frac{\sigma }{\rho }\right)\right]}^{\kappa }}{\int }_{\rho }^{{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}}\left\{{\left[ln\left(\frac{\mu }{\rho }\right)\right]}^{\kappa -1}+{\left[ln\left(\frac{\sigma }{\mu }\right)\right]}^{\kappa -1}\right\}\hfill \\ \hfill \times \left[\phi \left(\mu \right)+\phi \left(\rho \sigma {\mu }^{-1}\right)\right]\frac{d\mu }{\mu }\end{array}$$

(23)

and

$$\begin{array}{c}\frac{3^{\kappa }-1}{4^{\kappa }}\phi \left(\sqrt{\rho \sigma }\right)=\frac{\kappa }{2{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }}{\int }_{{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}}^{\sqrt{\rho \sigma }}\left[{\left(ln\frac{\mu }{\rho }\right)}^{\kappa -1}+{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa -1}\right]2\phi \left(\sqrt{\rho \sigma }\right)\frac{d\mu }{\mu }\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\le \frac{\kappa }{2{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }}{\int }_{{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}}^{\sqrt{\rho \sigma }}\left[{\left(ln\frac{\mu }{\rho }\right)}^{\kappa -1}+{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa -1}\right]\left[\phi \left(\mu \right)+\phi \left(\rho \sigma {\mu }^{-1}\right)\right]\frac{d\mu }{\mu }.\end{array}$$

(24)

Adding (23) and (24), we obtain

$$\begin{array}{c}\frac{3^{\kappa }-1}{4^{\kappa }}\phi \left(\sqrt{\rho \sigma }\right)+\frac{4^{\kappa }-3^{\kappa }+1}{2\times 4^{\kappa }}\left[\phi \left({\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}\right)+\phi \left({\rho }^{\frac{1}{4}}{\sigma }^{\frac{3}{4}}\right)\right]\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le \frac{\kappa \Gamma \left(\kappa \right)}{2{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }}\left\{{\mathcal{J}}_{\rho +}^{\kappa }\phi \left(\sigma \right)+{\mathcal{J}}_{\sigma -}^{\kappa }\phi \left(\rho \right)\right\}.\end{array}$$

(25)

Applying (15) and (18) and using the convexity of $\phi :\left[\rho ,\sigma \right]\to \mathbb{R}$ give

$$\begin{array}{c}\frac{\kappa }{2{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }}{\int }_{\rho }^{{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}}\left[{\left(ln\frac{\mu }{\rho }\right)}^{\kappa -1}+{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa -1}\right]\left[\phi \left(\mu \right)+\phi \left(\rho \sigma {\mu }^{-1}\right)\right]\frac{d\mu }{\mu }\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\le \frac{\kappa }{2{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }}{\int }_{\rho }^{{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}}\left[{\left(ln\frac{\mu }{\rho }\right)}^{\kappa -1}+{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa -1}\right]\\ \times \left[\left(\frac{ln\sigma -ln\mu }{ln\sigma -ln\rho }\right)\phi \left(\rho \right)+\left(\frac{ln\mu -ln\rho }{ln\sigma -ln\rho }\right)\phi \left(\sigma \right)\right.\\ \left.+\left(\frac{ln\mu -ln\rho }{ln\sigma -ln\rho }\right)\phi \left(\rho \right)+\left(\frac{ln\sigma -ln\mu }{ln\sigma -ln\rho }\right)\phi \left(\sigma \right)\right]\frac{d\mu }{\mu }\\ \hspace{1em} \le \frac{\kappa \left[\phi \left(\rho \right)+\phi \left(\sigma \right)\right]}{2{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }}{\int }_{\rho }^{{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}}\left[{\left(ln\frac{\mu }{\rho }\right)}^{\kappa -1}+{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa -1}\right]\frac{d\mu }{\mu }\\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\frac{4^{\kappa }-3^{\kappa }+1}{2\times 4^{\kappa }}\left[\phi \left(\rho \right)+\phi \left(\sigma \right)\right].\end{array}$$

(26)

Applying (19) and (20), together with the convexity of $\phi :\left[\rho ,\sigma \right]\to \mathbb{R}$, yield

$$\begin{array}{c}\frac{\kappa }{2{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }}{\int }_{\rho }^{{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}}\left[{\left(ln\frac{\mu }{\rho }\right)}^{\kappa -1}+{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa -1}\right]\left[\phi \left(\mu \right)+\phi \left(\rho \sigma {\mu }^{-1}\right)\right]\frac{d\mu }{\mu }\hfill \\ \hspace{1em}\le \frac{\kappa }{2{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }}{\int }_{\rho }^{{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}}\left[{\left(ln\frac{\mu }{\rho }\right)}^{\kappa -1}+{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa -1}\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \times \left[\left(\frac{ln\rho +3ln\sigma -4ln\mu }{2ln\sigma -2ln\rho }\right)\phi \left({\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}\right)\right.\hfill \\ \hspace{1em}\hspace{1em}+\left(\frac{4ln\mu -ln\rho -3ln\sigma }{2ln\sigma -2ln\rho }\right)\phi \left({\rho }^{\frac{1}{4}}{\sigma }^{\frac{3}{4}}\right)+\left(\frac{4ln\mu -ln\rho -3ln\sigma }{2ln\sigma -2ln\rho }\right)\phi \left({\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}\right)\hfill \\ \left.\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\left(\frac{ln\rho +3ln\sigma -4ln\mu }{2ln\sigma -2ln\rho }\right)\phi \left({\rho }^{\frac{1}{4}}{\sigma }^{\frac{3}{4}}\right)\right]\frac{d\mu }{\mu }\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\frac{\kappa }{2{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }}{\int }_{\rho }^{{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}}\left[{\left(ln\frac{\mu }{\rho }\right)}^{\kappa -1}+{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa -1}\right]\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \times \left[\phi \left({\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}\right)+\phi \left({\rho }^{\frac{1}{4}}{\sigma }^{\frac{3}{4}}\right)\right]\frac{d\mu }{\mu }\le \frac{3^{\kappa }-1}{2\times 4^{\kappa }}\left[\phi \left({\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}\right)+\phi \left({\rho }^{\frac{1}{4}}{\sigma }^{\frac{3}{4}}\right)\right].\end{array}$$

(27)

Adding (26) and (27), we obtain

$$\begin{array}{c}\frac{\kappa \Gamma \left(\kappa \right)}{2{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }}\left\{{\mathcal{J}}_{\rho +}^{\kappa }\phi \left(\sigma \right)+{\mathcal{J}}_{\sigma -}^{\kappa }\phi \left(\rho \right)\right\}\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le \frac{3^{\kappa }-1}{2\times 4^{\kappa }}\left[\phi \left({\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}\right)+\phi \left({\rho }^{\frac{1}{4}}{\sigma }^{\frac{3}{4}}\right)\right]+\frac{4^{\kappa }-3^{\kappa }+1}{2\times 4^{\kappa }}\left[\phi \left(\rho \right)+\phi \left(\sigma \right)\right].\end{array}$$

(28)

Combining (25) and (28), we obtain the desired result. □

Remark 1.

In Theorem 8, the inequality (10) refines the Hermite–Hadamard-type inequality (5).

Corollary 1.

In Theorem 8, let $\kappa =1$, then we have the inequality

$$\begin{array}{c}\phi \left(\sqrt{\rho \sigma }\right)\le \frac{1}{2}\phi \left(\sqrt{\rho \sigma }\right)+\frac{1}{4}\left[\phi \left({\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}\right)+\phi \left({\rho }^{\frac{1}{4}}{\sigma }^{\frac{3}{4}}\right)\right]\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \le \frac{1}{ln\left(\frac{\sigma }{\rho }\right)}{\int }_{\rho }^{\sigma }\frac{\phi \left(\mu \right)}{\mu }d\mu \le \frac{1}{4}\left[\phi \left({\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}\right)+\phi \left({\rho }^{\frac{1}{4}}{\sigma }^{\frac{3}{4}}\right)\right]\\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{1}{2}\left[\frac{\phi \left(\rho \right)+\phi \left(\sigma \right)}{2}\right]\le \frac{\phi \left(\rho \right)+\phi \left(\sigma \right)}{2}.\end{array}$$

(29)

The inequality provides a refinement of the inequality (4).

4. Extended Inequalities for Fractional Integrals Using Harmonic Convexity

In this section, we establish some results which are extensions of some existing results.

Theorem 9.

Let $\phi :I\subseteq \left(0,\infty \right)\to \mathbb{R}$ be an $L^1\left(\left[\rho ,\sigma \right]\right)$ function with ${\phi }^{\prime }\in L^1\left(\left[\rho ,\sigma \right]\right)$ for ρ, $\sigma \in I°$. If $\phi :I\to \mathbb{R}$ is differentiable on $I°$ and $\left|{\phi }^{\prime }\right|$ is $GA$-convex on $\left[\rho ,\sigma \right]$, then the following inequality holds for $\kappa >0$:

$$\begin{array}{c}\left|\frac{\Gamma \left(\kappa +1\right)}{2{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }}\left[{\mathcal{J}}_{\rho +}^{\kappa }\phi \left(\sigma \right)+{\mathcal{J}}_{\sigma -}^{\kappa }\phi \left(\rho \right)\right]-\right.\hfill \\ \left.\hspace{1em}\hspace{1em}\hspace{1em} \left\{\frac{3^{\kappa }-1}{2\times 4^{\kappa }}\left[\phi \left({\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}\right)+\phi \left({\rho }^{\frac{1}{4}}{\sigma }^{\frac{3}{4}}\right)\right]+\frac{4^{\kappa }-3^{\kappa }+1}{2\times 4^{\kappa }}\left[\phi \left(\rho \right)+\phi \left(\sigma \right)\right]\right\}\right|\\ \hspace{1em} \le ln\left(\frac{\sigma }{\rho }\right)\left[\left|{\phi }^{\prime }\left(\rho \right)\right|+\left|{\phi }^{\prime }\left(\sigma \right)\right|\right]\left[\frac{4^{\kappa +1}+\kappa \times \left(1-3^{\kappa }\right)+2\times 3^{\kappa }-2^{\kappa +2}}{4^{\kappa +1}\left(\kappa +1\right)}\right].\end{array}$$

(30)

Proof. 

Let $h:\left[\rho ,\sigma \right]\to \mathbb{R}$ be defined as

$$h\left(\mu \right)=\left\{\begin{array}{cc}{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa }-{\left(ln\frac{\mu }{\rho }\right)}^{\kappa }-\frac{3^{\kappa }-1}{4^{\kappa }}{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa },& \mu \in \left[\rho ,{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}\right),\\ & \\ {\left(ln\frac{\sigma }{\mu }\right)}^{\kappa }-{\left(ln\frac{\mu }{\rho }\right)}^{\kappa },& \mu \in \left[{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}},{\rho }^{\frac{1}{4}}{\sigma }^{\frac{3}{4}}\right),\\ & \\ {\left(ln\frac{\sigma }{\mu }\right)}^{\kappa }-{\left(ln\frac{\mu }{\rho }\right)}^{\kappa }+\frac{3^{\kappa }-1}{4^{\kappa }}{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa },& \mu \in \left[{\rho }^{\frac{1}{4}}{\sigma }^{\frac{3}{4}},\sigma \right].\end{array}\right.$$

We obtain the following identities using the integration by parts:

$$\begin{array}{c}\frac{1}{2{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }}{\int }_{\rho }^{\sigma }h\left(\mu \right){\phi }^{\prime }\left(\mu \right)d\mu =\frac{\kappa }{2{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }}{\int }_{\rho }^{\sigma }\left[{\left(ln\frac{\mu }{\rho }\right)}^{\kappa -1}+{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa -1}\right]\frac{\phi \left(\mu \right)}{\mu }d\mu \hfill \\ \hspace{1em}\hspace{1em}-\left[\frac{3^{\kappa }-1}{2\times 4^{\kappa }}\left[\phi \left({\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}\right)+\phi \left({\rho }^{\frac{1}{4}}{\sigma }^{\frac{3}{4}}\right)\right]+\frac{4^{\kappa }-3^{\kappa }+1}{2\times 4^{\kappa }}\left[\phi \left(\rho \right)+\phi \left(\sigma \right)\right]\right]\\ \hspace{1em}\hspace{1em}=\frac{\kappa \Gamma \left(\kappa \right)}{2{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }}\left[{\mathcal{J}}_{\rho +}^{\kappa }\phi \left(\sigma \right)+{\mathcal{J}}_{\sigma -}^{\kappa }\phi \left(\rho \right)\right]\\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\left[\frac{3^{\kappa }-1}{2\times 4^{\kappa }}\left[\phi \left({\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}\right)+\phi \left({\rho }^{\frac{1}{4}}{\sigma }^{\frac{3}{4}}\right)\right]+\frac{4^{\kappa }-3^{\kappa }+1}{2\times 4^{\kappa }}\left[\phi \left(\rho \right)+\phi \left(\sigma \right)\right]\right].\end{array}$$

(31)

We observe that

$$\begin{array}{c}{\int }_{\rho }^{{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}}\left[{\left(ln\frac{\mu }{\rho }\right)}^{\kappa }-{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa }-\frac{3^{\kappa }-1}{4^{\kappa }}{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }\right]\left(\frac{ln\sigma -ln\mu }{ln\sigma -ln\rho }\right)\left|{\phi }^{\prime }\left(\rho \right)\right|\frac{d\mu }{\mu }\hfill \\ \hspace{1em}\hspace{1em}+{\int }_{{\rho }^{\frac{1}{4}}{\sigma }^{\frac{3}{4}}}^{\sigma }\left[{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa }-{\left(ln\frac{\mu }{\rho }\right)}^{\kappa }-\frac{3^{\kappa }-1}{4^{\kappa }}{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }\right]\left(\frac{ln\sigma -ln\mu }{ln\sigma -ln\rho }\right)\left|{\phi }^{\prime }\left(\rho \right)\right|\frac{d\mu }{\mu }\\ \hspace{1em}\hspace{1em}={\int }_{\rho }^{{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}}\left[{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa }-{\left(ln\frac{\mu }{\rho }\right)}^{\kappa }-\frac{3^{\kappa }-1}{4^{\kappa }}{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }\right]\left(\frac{ln\sigma -ln\mu }{ln\sigma -ln\rho }\right)\left|{\phi }^{\prime }\left(\rho \right)\right|\frac{d\mu }{\mu }\\ \hspace{1em}\hspace{1em}+{\int }_{\rho }^{{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}}\left[{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa }-{\left(ln\frac{\mu }{\rho }\right)}^{\kappa }-\frac{3^{\kappa }-1}{4^{\kappa }}{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }\right]\left(\frac{ln\mu -ln\rho }{ln\sigma -ln\rho }\right)\left|{\phi }^{\prime }\left(\rho \right)\right|\frac{d\mu }{\mu }\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\left|{\phi }^{\prime }\left(\rho \right)\right|{\int }_{\rho }^{{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}}\left[{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa }-{\left(ln\frac{\mu }{\rho }\right)}^{\kappa }-\frac{3^{\kappa }-1}{4^{\kappa }}{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }\right]\frac{d\mu }{\mu }\\ \hfill \hspace{1em}\hspace{1em}=\left|{\phi }^{\prime }\left(\rho \right)\right|{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa +1}\left[\frac{4^{\kappa +1}+\kappa \left(1-3^{\kappa }\right)-3^{\kappa }-1}{4^{\kappa +1}(\kappa +1)}\right],\end{array}$$

(32)

$$\begin{array}{c}{\int }_{\rho }^{{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}}\left[{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa }-{\left(ln\frac{\mu }{\rho }\right)}^{\kappa }-\frac{3^{\kappa }-1}{4^{\kappa }}{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }\right]\left(\frac{ln\mu -ln\rho }{ln\sigma -ln\rho }\right)\left|{\phi }^{\prime }\left(\sigma \right)\right|\frac{d\mu }{\mu }\hfill \\ \hspace{1em}\hspace{1em}+{\int }_{{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}}^{\sigma }\left[{\left(ln\frac{\mu }{\rho }\right)}^{\kappa }-{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa }-\frac{3^{\kappa }-1}{4^{\kappa }}{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }\right]\left(\frac{ln\mu -ln\rho }{ln\sigma -ln\rho }\right)\left|{\phi }^{\prime }\left(\sigma \right)\right|\frac{d\mu }{\mu }\\ \hspace{1em}\hspace{1em}={\int }_{\rho }^{{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}}\left[{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa }-{\left(ln\frac{\mu }{\rho }\right)}^{\kappa }-\frac{3^{\kappa }-1}{4^{\kappa }}{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }\right]\left(\frac{ln\mu -ln\rho }{ln\sigma -ln\rho }\right)\left|{\phi }^{\prime }\left(\sigma \right)\right|\frac{d\mu }{\mu }\\ \hspace{1em}\hspace{1em}+{\int }_{\rho }^{{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}}\left[{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa }-{\left(ln\frac{\mu }{\rho }\right)}^{\kappa }-\frac{3^{\kappa }-1}{4^{\kappa }}{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }\right]\left(\frac{ln\sigma -ln\mu }{ln\sigma -ln\rho }\right)\left|{\phi }^{\prime }\left(\sigma \right)\right|\frac{d\mu }{\mu }\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\left|{\phi }^{\prime }\left(\sigma \right)\right|{\int }_{\rho }^{{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}}\left[{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa }-{\left(ln\frac{\mu }{\rho }\right)}^{\kappa }-\frac{3^{\kappa }-1}{4^{\kappa }}{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }\right]\frac{d\mu }{\mu }\\ \hfill \hspace{1em}=\left|{\phi }^{\prime }\left(\sigma \right)\right|{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa +1}\left[\frac{4^{\kappa +1}+\kappa \left(1-3^{\kappa }\right)-3^{\kappa }-1}{4^{\kappa +1}(\kappa +1)}\right],\end{array}$$

(33)

$$\begin{array}{c}{\int }_{{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}}^{\sqrt{\rho \sigma }}\left[{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa }-{\left(ln\frac{\mu }{\rho }\right)}^{\kappa }\right]\left(\frac{ln\sigma -ln\mu }{ln\sigma -ln\rho }\right)\left|{\phi }^{\prime }\left(\rho \right)\right|\frac{d\mu }{\mu }\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\int }_{{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}}^{{\rho }^{\frac{1}{4}}{\sigma }^{\frac{3}{4}}}\left[{\left(ln\frac{\mu }{\rho }\right)}^{\kappa }-{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa }\right]\left(\frac{ln\sigma -ln\mu }{ln\sigma -ln\rho }\right)\left|{\phi }^{\prime }\left(\rho \right)\right|\frac{d\mu }{\mu }\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={\int }_{{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}}^{\sqrt{\rho \sigma }}\left[{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa }-{\left(ln\frac{\mu }{\rho }\right)}^{\kappa }\right]\left(\frac{ln\sigma -ln\mu }{ln\sigma -ln\rho }\right)\left|{\phi }^{\prime }\left(\rho \right)\right|\frac{d\mu }{\mu }\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\int }_{{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}}^{\sqrt{\rho \sigma }}\left[{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa }-{\left(ln\frac{\mu }{\rho }\right)}^{\kappa }\right]\left(\frac{ln\mu -ln\rho }{ln\sigma -ln\rho }\right)\left|{\phi }^{\prime }\left(\rho \right)\right|\frac{d\mu }{\mu }\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\left|{\phi }^{\prime }\left(\rho \right)\right|{\int }_{{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}}^{\sqrt{\rho \sigma }}\left[{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa }-{\left(ln\frac{\mu }{\rho }\right)}^{\kappa }\right]\frac{d\mu }{\mu }\\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\left|{\phi }^{\prime }\left(\rho \right)\right|{\left(\frac{\sigma }{\rho }\right)}^{\kappa +1}\left[\frac{3^{\kappa +1}-2^{\kappa +2}+1}{4^{\kappa +1}(\kappa +1)}\right]\end{array}$$

(34)

and

$$\begin{array}{c}{\int }_{{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}}^{\sqrt{\rho \sigma }}\left[{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa }-{\left(ln\frac{\mu }{\rho }\right)}^{\kappa }\right]\left(\frac{ln\mu -ln\rho }{ln\sigma -ln\rho }\right)\left|{\phi }^{\prime }\left(\sigma \right)\right|\frac{d\mu }{\mu }\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\int }_{\sqrt{\rho \sigma }}^{{\rho }^{\frac{1}{4}}{\sigma }^{\frac{3}{4}}}\left[{\left(ln\frac{\mu }{\rho }\right)}^{\kappa }-{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa }\right]\left(\frac{ln\mu -ln\rho }{ln\sigma -ln\rho }\right)\left|{\phi }^{\prime }\left(\sigma \right)\right|\frac{d\mu }{\mu }\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={\int }_{{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}}^{\sqrt{\rho \sigma }}\left[{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa }-{\left(ln\frac{\mu }{\rho }\right)}^{\kappa }\right]\left(\frac{ln\mu -ln\rho }{ln\sigma -ln\rho }\right)\left|{\phi }^{\prime }\left(\sigma \right)\right|\frac{d\mu }{\mu }\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\int }_{{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}}^{\sqrt{\rho \sigma }}\left[{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa }-{\left(ln\frac{\mu }{\rho }\right)}^{\kappa }\right]\left(\frac{ln\sigma -ln\mu }{ln\sigma -ln\rho }\right)\left|{\phi }^{\prime }\left(\sigma \right)\right|\frac{d\mu }{\mu }\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} =\left|{\phi }^{\prime }\left(\sigma \right)\right|{\int }_{{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}}^{\sqrt{\rho \sigma }}\left[{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa }-{\left(ln\frac{\mu }{\rho }\right)}^{\kappa }\right]\frac{d\mu }{\mu }\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\left|{\phi }^{\prime }\left(\sigma \right)\right|{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa +1}\left[\frac{3^{\kappa +1}-2^{\kappa +2}+1}{4^{\kappa +1}(\kappa +1)}\right].\end{array}$$

(35)

Now, by using the convexity of $\phi :\left[\rho ,\sigma \right]\to \mathbb{R}$, and the identities (32)–(35), we obtain

$$\begin{array}{c}\left|\frac{1}{2{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }}{\int }_{\rho }^{\sigma }h\left(\mu \right){\phi }^{\prime }\left(\mu \right)d\mu \right|\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \le \frac{1}{2{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }}{\int }_{\rho }^{\sigma }\left|h\left(\mu \right)\right|\left|{\phi }^{\prime }\left({\rho }^{\frac{ln\sigma -ln\mu }{ln\sigma -ln\rho }}{\sigma }^{\frac{ln\mu -ln\rho }{ln\sigma -ln\rho }}\right)\right|\frac{d\mu }{\mu }\\ \hspace{1em}\hspace{1em}\le \frac{1}{2{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }}{\int }_{\rho }^{\sigma }\left|h\left(\mu \right)\right|\left[\left(\frac{ln\sigma -ln\mu }{ln\sigma -ln\rho }\right)\left|{\phi }^{\prime }\left(\rho \right)\right|+\left(\frac{ln\mu -ln\rho }{ln\sigma -ln\rho }\right)\left|{\phi }^{\prime }\left(\sigma \right)\right|\right]\frac{d\mu }{\mu }\hfill \\ \hspace{1em}\hspace{1em} =\left(ln\frac{\sigma }{\rho }\right)\left[\frac{3^{\kappa +1}+1-2^{\kappa +2}}{(\kappa +1)4^{\kappa +1}}\right]\left[\left|{\phi }^{\prime }\left(\rho \right)\right|+\left|{\phi }^{\prime }\left(\sigma \right)\right|\right]\\ \hfill \hspace{1em}+\left(ln\frac{\sigma }{\rho }\right)\left[\frac{4^{\kappa +1}+\kappa \left(1-3^{\kappa }\right)-4\times 3^{\kappa }}{4^{\kappa +1}(\kappa +1)}\right]\left[\left|{\phi }^{\prime }\left(\rho \right)\right|+\left|{\phi }^{\prime }\left(\sigma \right)\right|\right].\end{array}$$

(36)

The inequality (36), together with (31), give the desired result. □

Theorem 10.

Let $\phi :I\subseteq \left(0,\infty \right)\to \mathbb{R}$ be an $L^1\left(\left[\rho ,\sigma \right]\right)$ function with ${\phi }^{\prime }\in L^1\left(\left[\rho ,\sigma \right]\right)$ for ρ, $\sigma \in I°$. If $\phi :I\to \mathbb{R}$ is differentiable on $I°$ and $\left|{\phi }^{\prime }\right|$ is $GA$-convex on $\left[\rho ,\sigma \right]$, then the following inequality holds for $\kappa >0$:

$$\begin{array}{c}\left|\frac{\Gamma \left(\kappa +1\right)}{2{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }}\left\{{\mathcal{J}}_{\rho +}^{\kappa }\phi \left(\sigma \right)+{\mathcal{J}}_{\sigma -}^{\kappa }\phi \left(\rho \right)\right\}-\left[\frac{4^{\kappa }-3^{\kappa }+1}{2\times 4^{\kappa }}\right.\right.\hfill \\ \left.\left.\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times \left[\phi \left({\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}\right)+\phi \left({\rho }^{\frac{1}{4}}{\sigma }^{\frac{4}{4}}\right)\right]+\frac{3^{\kappa }-1}{4^{\kappa }}\phi \left(\sqrt{\rho \sigma }\right)\right]\right|\\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le ln\left(\frac{\sigma }{\rho }\right)\left[\frac{3^{\kappa }+4^{\kappa }-1}{4^{\kappa +1}}\right]\left[\left|{\phi }^{\prime }\left(\rho \right)\right|+\left|{\phi }^{\prime }\left(\sigma \right)\right|\right].\end{array}$$

(37)

Proof. 

Let $H:\left[\rho ,\sigma \right]\to \mathbb{R}$ be defined as

$$H\left(\mu \right)=\left\{\begin{array}{cc}{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa }-{\left(ln\frac{\mu }{\rho }\right)}^{\kappa }-{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa },& \mu \in \left[\rho ,{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}\right),\\ & \\ {\left(ln\frac{\sigma }{\mu }\right)}^{\kappa }-{\left(ln\frac{\mu }{\rho }\right)}^{\kappa }-\frac{3^{\kappa }-1}{4^{\kappa }}{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa },& \mu \in \left[{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}},\sqrt{\rho \sigma }\right),\\ & \\ {\left(ln\frac{\sigma }{\mu }\right)}^{\kappa }-{\left(ln\frac{\mu }{\rho }\right)}^{\kappa }+\frac{3^{\kappa }-1}{4^{\kappa }}{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa },& \mu \in \left[\sqrt{\rho \sigma },{\rho }^{\frac{1}{4}}{\sigma }^{\frac{3}{4}}\right)\\ & \\ {\left(ln\frac{\sigma }{\mu }\right)}^{\kappa }-{\left(ln\frac{\mu }{\rho }\right)}^{\kappa }+{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa },& \mu \in \left[{\rho }^{\frac{1}{4}}{\sigma }^{\frac{3}{4}},\sigma \right].\end{array}\right.$$

We obtain the following identities using the integration by parts:

$$\begin{array}{c}\frac{1}{2{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }}{\int }_{\rho }^{\sigma }H\left(\mu \right){\phi }^{\prime }\left(\mu \right)d\mu =\frac{1}{2{\left(ln\frac{\sigma }{\sigma }\right)}^{\kappa }}{\int }_{\rho }^{\sigma }\left[{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa }+{\left(ln\frac{\mu }{\rho }\right)}^{\kappa }\right]\phi \left(\mu \right)\frac{d\mu }{\mu }\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}-\left[\frac{4^{\kappa }-3^{\kappa }+1}{2\times 4^{\kappa }}\left[\phi \left({\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}\right)+\phi \left({\rho }^{\frac{1}{4}}{\sigma }^{\frac{4}{4}}\right)\right]+\frac{3^{\kappa }-1}{4^{\kappa }}\phi \left(\sqrt{\rho \sigma }\right)\right]\\ \hspace{1em}\hspace{1em}=\frac{\kappa \Gamma \left(\kappa \right)}{2{\left(ln\frac{\sigma }{\sigma }\right)}^{\kappa }}\left\{{\mathcal{J}}_{\rho +}^{\kappa }\phi \left(\sigma \right)+{\mathcal{J}}_{\sigma -}^{\kappa }\phi \left(\rho \right)\right\}\\ \hfill \hspace{1em}\hspace{1em}\hspace{1em} -\left[\frac{4^{\kappa }-3^{\kappa }+1}{2\times 4^{\kappa }}\left[\phi \left({\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}\right)+\phi \left({\rho }^{\frac{1}{4}}{\sigma }^{\frac{4}{4}}\right)\right]+\frac{3^{\kappa }-1}{4^{\kappa }}\phi \left(\sqrt{\rho \sigma }\right)\right].\end{array}$$

(38)

In a similar way to that of obtaining (32)–(35) in Theorem 9, we observe that the following identities hold:

$$\begin{array}{c}{\int }_{\rho }^{{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}}\left[{\left(ln\frac{\mu }{\rho }\right)}^{\kappa }-{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa }+{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }\right]\left(\frac{ln\sigma -ln\mu }{ln\sigma -ln\rho }\right)\left|{\phi }^{\prime }\left(\rho \right)\right|\frac{d\mu }{\mu }\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\int }_{{\rho }^{\frac{1}{4}}{\sigma }^{\frac{3}{4}}}^{\sigma }\left[{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa }-{\left(ln\frac{\mu }{\rho }\right)}^{\kappa }+{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }\right]\left(\frac{ln\sigma -ln\mu }{ln\sigma -ln\rho }\right)\left|{\phi }^{\prime }\left(\rho \right)\right|\frac{d\mu }{\mu }\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\left|{\phi }^{\prime }\left(\rho \right)\right|{\int }_{\rho }^{{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}}\left[{\left(ln\frac{\mu }{\rho }\right)}^{\kappa }-{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa }+{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }\right]\frac{d\mu }{\mu }\\ \hfill =\left|{\phi }^{\prime }\left(\rho \right)\right|{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa +1}\left[\frac{\left(\kappa -3\right)\times 4^{\kappa }+3^{\kappa +1}+1}{4^{\kappa +1}(\kappa +1)}\right],\end{array}$$

(39)

$$\begin{array}{c}{\int }_{\rho }^{{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}}\left[{\left(ln\frac{\mu }{\rho }\right)}^{\kappa }-{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa }+{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }\right]\left(\frac{ln\mu -ln\rho }{ln\sigma -ln\rho }\right)\left|{\phi }^{\prime }\left(\sigma \right)\right|\frac{d\mu }{\mu }\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}+{\int }_{{\rho }^{\frac{1}{4}}{\sigma }^{\frac{3}{4}}}^{\sigma }\left[{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa }-{\left(ln\frac{\mu }{\rho }\right)}^{\kappa }+{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }\right]\left(\frac{ln\mu -ln\rho }{ln\sigma -ln\rho }\right)\left|{\phi }^{\prime }\left(\sigma \right)\right|\frac{d\mu }{\mu }\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\left|{\phi }^{\prime }\left(\sigma \right)\right|{\int }_{\rho }^{{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}}\left[{\left(ln\frac{\mu }{\rho }\right)}^{\kappa }-{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa }+{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }\right]\frac{d\mu }{\mu }\\ \hfill =\left|{\phi }^{\prime }\left(\sigma \right)\right|{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa +1}\left[\frac{\left(\kappa -3\right)\times 4^{\kappa }+3^{\kappa +1}+1}{4^{\kappa +1}(\kappa +1)}\right],\end{array}$$

(40)

$$\begin{array}{c}{\int }_{{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}}^{\sqrt{\rho \sigma }}\left[{\left(ln\frac{\mu }{\rho }\right)}^{\kappa }-{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa }+\frac{3^{\kappa }-1}{4^{\kappa }}{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }\right]\left(\frac{ln\sigma -ln\mu }{ln\sigma -ln\mu }\right)\left|{\phi }^{\prime }\left(\rho \right)\right|\frac{d\mu }{\mu }\hfill \\ \hspace{1em}\hspace{1em} +{\int }_{\sqrt{\rho \sigma }}^{{\rho }^{\frac{1}{4}}{\sigma }^{\frac{3}{4}}}\left[{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa }-{\left(ln\frac{\mu }{\rho }\right)}^{\kappa }+\frac{3^{\kappa }-1}{4^{\kappa }}{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }\right]\left(\frac{ln\sigma -ln\mu }{ln\sigma -ln\mu }\right)\left|{\phi }^{\prime }\left(\rho \right)\right|\frac{d\mu }{\mu }\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\left|{\phi }^{\prime }\left(\rho \right)\right|{\int }_{\sqrt{\rho \sigma }}^{{\rho }^{\frac{1}{4}}{\sigma }^{\frac{3}{4}}}\left[{\left(ln\frac{\mu }{\rho }\right)}^{\kappa }-{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa }+\frac{3^{\kappa }-1}{4^{\kappa }}{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }\right]\frac{d\mu }{\mu }\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\left|{\phi }^{\prime }\left(\rho \right)\right|{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa +1}\left[\frac{2^{\kappa +2}-2-2\times 3^{\kappa }+\left(3^{\kappa }-1\right)\kappa }{4^{\kappa +1}(\kappa +1)}\right],\end{array}$$

(41)

and

$$\begin{array}{c}{\int }_{{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}}^{\sqrt{\rho \sigma }}\left[{\left(ln\frac{\mu }{\rho }\right)}^{\kappa }-{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa }+\frac{3^{\kappa }-1}{4^{\kappa }}{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }\right]\left(\frac{ln\mu -ln\rho }{ln\sigma -ln\rho }\right)\left|{\phi }^{\prime }\left(\sigma \right)\right|\frac{d\mu }{\mu }\hfill \\ \hspace{1em}\hspace{1em} +{\int }_{\sqrt{\rho \sigma }}^{{\rho }^{\frac{1}{4}}{\sigma }^{\frac{3}{4}}}\left[{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa }-{\left(ln\frac{\mu }{\rho }\right)}^{\kappa }+\frac{3^{\kappa }-1}{4^{\kappa }}{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }\right]\left(\frac{ln\mu -ln\rho }{ln\sigma -ln\rho }\right)\left|{\phi }^{\prime }\left(\sigma \right)\right|\frac{d\mu }{\mu }\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\left|{\phi }^{\prime }\left(\sigma \right)\right|{\int }_{{\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}}^{\sqrt{\rho \sigma }}\left[{\left(ln\frac{\mu }{\rho }\right)}^{\kappa }-{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa }+\frac{3^{\kappa }-1}{4^{\kappa }}{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }\right]\frac{d\mu }{\mu }\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\left|{\phi }^{\prime }\left(\sigma \right)\right|{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa +1}\left[\frac{2^{\kappa +2}-2-2\times 3^{\kappa }+\left(3^{\kappa }-1\right)\kappa }{4^{\kappa +1}(\kappa +1)}\right].\end{array}$$

(42)

Now, by using the convexity of $\phi :\left[\rho ,\sigma \right]\to \mathbb{R}$, and the identities (39)–(42), we obtain

$$\begin{array}{c}\left|\frac{1}{2{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }}{\int }_{\rho }^{\sigma }H\left(\mu \right){\phi }^{\prime }\left(\mu \right)\frac{d\mu }{\mu }\right|\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le \frac{1}{2{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }}{\int }_{\rho }^{\sigma }\left|H\left(\mu \right)\right|\left|{\phi }^{\prime }\left({\rho }^{\frac{ln\sigma -ln\mu }{ln\sigma -ln\rho }}{\sigma }^{\frac{ln\mu -ln\rho }{ln\sigma -ln\rho }}\right)\right|\frac{d\mu }{\mu }\\ \hspace{1em} \le \frac{1}{2{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }}{\int }_{\rho }^{\sigma }\left|H\left(\mu \right)\right|\left[\left(\frac{ln\sigma -ln\mu }{ln\sigma -ln\rho }\right)\left|{\phi }^{\prime }\left(\rho \right)\right|+\left(\frac{ln\mu -ln\rho }{ln\sigma -ln\rho }\right)\left|{\phi }^{\prime }\left(\sigma \right)\right|\right]\frac{d\mu }{\mu }\hfill \\ \hspace{1em}=\left(ln\frac{\sigma }{\rho }\right)\left[\frac{\left(\kappa -3\right)\times 4^{\kappa }+3^{\kappa }+1}{4^{\kappa +1}(\kappa +1)}\right]\left[\left|{\phi }^{\prime }\left(\rho \right)\right|+\left|{\phi }^{\prime }\left(\sigma \right)\right|\right]\\ \hfill \hspace{1em}\hspace{1em}+\left(ln\frac{\sigma }{\rho }\right)\left[\frac{2^{\kappa +2}-2-2\times 3^{\kappa }+\left(3^{\kappa }-1\right)\kappa }{4^{\kappa +1}(\kappa +1)}\right]\left[\left|{\phi }^{\prime }\left(\rho \right)\right|+\left|{\phi }^{\prime }\left(\sigma \right)\right|\right].\end{array}$$

(43)

The inequality (43) when combined with (38) yields the desired result. □

Remark 2.

Theorems 9 and 10 are similar extensions of Theorems 2 and 3.

Remark 3.

Let $\kappa =1$ in Theorems 9 and 10. The following inequalities are then obtained:

$$\begin{array}{c}\left|\frac{1}{ln\left(\frac{\sigma }{\rho }\right)}{\int }_{\rho }^{\sigma }\frac{\phi \left(\mu \right)}{\mu }d\mu -\frac{1}{4}\left[\phi \left(\rho \right)+\phi \left({\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}\right)+\phi \left({\rho }^{\frac{1}{4}}{\sigma }^{\frac{3}{4}}\right)+\phi \left(\sigma \right)\right]\right|\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le \frac{3}{8}ln\left(\frac{\sigma }{\rho }\right)\left[\left|{\phi }^{\prime }\left(\rho \right)\right|+\left|{\phi }^{\prime }\left(\sigma \right)\right|\right].\end{array}$$

(44)

and

$$\begin{array}{c}\left|\frac{1}{ln\left(\frac{\sigma }{\rho }\right)}{\int }_{\rho }^{\sigma }\frac{\phi \left(\mu \right)}{\mu }d\mu -\frac{1}{4}\left[\phi \left({\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}\right)+\phi \left({\rho }^{\frac{1}{4}}{\sigma }^{\frac{3}{4}}\right)+2\phi \left(\sqrt{\rho \sigma }\right)\right]\right|\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le \frac{3}{8}ln\left(\frac{\sigma }{\rho }\right)\left[\left|{\phi }^{\prime }\left(\rho \right)\right|+\left|{\phi }^{\prime }\left(\sigma \right)\right|\right].\end{array}$$

(45)

5. Applications of the Results

The Gamma function and the lower and upper incomplete Gamma functions are defined as

$$\Gamma \left(r\right)={\int }_0^{\infty }{\mu }^{r-1}exp\left(-\mu \right)d\mu ,\Re \left(r\right)>0,$$

$$\Gamma \left(r,\tau \right)={\int }_{\tau }^{\infty }{\mu }^{r-1}exp\left(-\mu \right)d\mu ,Re\left(r\right)>0$$

and

$$\gamma \left(r,\tau \right)={\int }_0^{\tau }{\mu }^{r-1}exp\left(-\mu \right)d\mu ,Re\left(r\right)>0.$$

Let $p\ne 0$ be a real number and consider the convex function $g\left(\tau \right)=exp\left(p\tau \right)$, $\tau \in \mathbb{R}$. Then, the function $\phi :\left(0,\infty \right)\to \mathbb{R}$, $\phi \left(\tau \right)=g(ln\tau )=exp(pln\tau )={\tau }^p$ is a $GA$-convex function on $\left(0,\infty \right)$. Observe that for $0<\rho <\sigma$ and $\kappa$ is a positive integer, we have

$$\begin{array}{cc}\hfill & \frac{\Gamma \left(\kappa +1\right)}{2{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }}{\mathcal{J}}_{\sigma -}^{\kappa }\phi \left(\rho \right)=\frac{\kappa }{2{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }}{\int }_{\rho }^{\sigma }{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa -1}\frac{\phi \left(\mu \right)}{\mu }d\mu \hfill \\ \hfill & =\frac{\kappa }{2{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }}{\int }_{\rho }^{\sigma }{\left(ln\frac{\sigma }{\mu }\right)}^{\kappa -1}{\mu }^{p-1}d\mu \hfill \\ \hfill & =\frac{\kappa {\sigma }^p}{2p^{\kappa }{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }}\left[\Gamma \left(\kappa \right)-\Gamma \left(\kappa ,pln\left(\frac{\sigma }{\rho }\right)\right)\right]\hfill \\ \hfill & =\frac{{\sigma }^p}{2p^{\kappa }{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }}\left[\Gamma (\kappa +1)-\kappa \Gamma \left(\kappa ,pln\left(\frac{\sigma }{\rho }\right)\right)\right]\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & \frac{\Gamma \left(\kappa +1\right)}{2{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }}{\mathcal{J}}_{\rho +}^{\kappa }\phi \left(\sigma \right)=\frac{\kappa }{2{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }}{\int }_{\rho }^{\sigma }{\left(ln\frac{\mu }{\rho }\right)}^{\kappa -1}\frac{\phi \left(\mu \right)}{\mu }d\mu \hfill \\ \hfill & =\frac{\kappa }{2{\left(ln\frac{\sigma }{\rho }\right)}^{\kappa }}{\int }_{\rho }^{\sigma }{\left(ln\frac{\mu }{\rho }\right)}^{\kappa -1}{\mu }^{p-1}d\mu \hfill \\ \hfill & =\frac{\kappa {\rho }^p}{2{\left(-pln\left(\frac{\sigma }{\rho }\right)\right)}^{\kappa }}\left[\Gamma \left(\kappa \right)-\Gamma \left(\kappa ,-pln\left(\frac{\sigma }{\rho }\right)\right)\right]\hfill \\ \hfill & =\frac{{\rho }^p}{2{\left(-pln\left(\frac{\sigma }{\rho }\right)\right)}^{\kappa }}\left[\Gamma (\kappa +1)-\kappa \Gamma \left(\kappa ,-pln\left(\frac{\sigma }{\rho }\right)\right)\right].\hfill \end{array}$$

We can now obtain the following estimates regarding the incomplete Gamma and Gamma functions:

Proposition 1.

Let $p\ne 0$, κ be positive integers in Theorem 8, where $p>1$, then the following inequality holds:

$$\begin{array}{c}{\left(\sqrt{\rho \sigma }\right)}^p\le \frac{3^{\kappa }-1}{4^{\kappa }}{\left(\sqrt{\rho \sigma }\right)}^p+\frac{4^{\kappa }-3^{\kappa }+1}{2\times 4^{\kappa }}\left[{\left({\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}\right)}^p+{\left({\rho }^{\frac{1}{4}}{\sigma }^{\frac{3}{4}}\right)}^p\right]\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le \frac{{\rho }^p}{2{\left(-pln\left(\frac{\sigma }{\rho }\right)\right)}^{\kappa }}\left[\Gamma (\kappa +1)-\kappa \Gamma \left(\kappa ,-pln\left(\frac{\sigma }{\rho }\right)\right)\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{{\sigma }^p}{2{\left(pln\left(\frac{\sigma }{\rho }\right)\right)}^{\kappa }}\left[\Gamma (\kappa +1)-\kappa \Gamma \left(\kappa ,pln\left(\frac{\sigma }{\rho }\right)\right)\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le \frac{3^{\kappa }-1}{2\times 4^{\kappa }}\left[{\left({\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}\right)}^p+{\left({\rho }^{\frac{1}{4}}{\sigma }^{\frac{3}{4}}\right)}^p\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{4^{\kappa }-3^{\kappa }+1}{4^{\kappa }}\left[\frac{{\rho }^p+{\sigma }^p}{2}\right]\le \frac{{\rho }^p+{\sigma }^p}{2}\end{array}$$

(46)

Proof. 

The proof is obvious from Theorem 8 for the function $\phi \left(\mu \right)={\mu }^p$, $\mu \in \left(0,\infty \right),p\ne 0$. The Figure 1 also supports the validity of the inequality (46).

□

Proposition 2.

Let $p\ne 2$ and κ be a positive integer in Theorem 9, then the following inequality holds:

$$\begin{array}{c}\frac{3^{\kappa }-1}{2\times 4^{\kappa }}\left[{\left({\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}\right)}^p+{\left({\rho }^{\frac{1}{4}}{\sigma }^{\frac{3}{4}}\right)}^p\right]+\frac{4^{\kappa }-3^{\kappa }+1}{2\times 4^{\kappa }}\left[{\rho }^p+{\sigma }^p\right]\hfill \\ \hspace{1em}\hspace{1em}-pln\left(\frac{\sigma }{\rho }\right)\left[{\rho }^{p-1}+{\sigma }^{p-1}\right]\left[\frac{4\times \left(4^{\kappa }+2^{\kappa }\right)+\left(\kappa +1\right)\left(1-3^{\kappa }\right)}{4^{\kappa +1}\left(\kappa +1\right)}\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le \frac{{\rho }^p}{2{\left(-pln\left(\frac{\sigma }{\rho }\right)\right)}^{\kappa }}\left[\Gamma (\kappa +1)-\kappa \Gamma \left(\kappa ,-pln\left(\frac{\sigma }{\rho }\right)\right)\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{{\sigma }^p}{2{\left(pln\left(\frac{\sigma }{\rho }\right)\right)}^{\kappa }}\left[\Gamma (\kappa +1)-\kappa \Gamma \left(\kappa ,pln\left(\frac{\sigma }{\rho }\right)\right)\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le \frac{3^{\kappa }-1}{2\times 4^{\kappa }}\left[{\left({\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}\right)}^p+{\left({\rho }^{\frac{1}{4}}{\sigma }^{\frac{3}{4}}\right)}^p\right]+\frac{4^{\kappa }-3^{\kappa }+1}{2\times 4^{\kappa }}\left[{\rho }^p+{\sigma }^p\right]\\ \hfill \hspace{1em}+pln\left(\frac{\sigma }{\rho }\right)\left[{\rho }^{p-1}+{\sigma }^{p-1}\right]\left[\frac{4\times \left(4^{\kappa }+2^{\kappa }\right)+\left(\kappa +1\right)\left(1-3^{\kappa }\right)}{4^{\kappa +1}\left(\kappa +1\right)}\right].\end{array}$$

(47)

Proof. 

The proof follows directly from Theorem 9 using the harmonic convex function $\phi \left(\mu \right)={\mu }^p$, $\mu \in \left(0,\infty \right)$ for $p\ne 0$. The Figure 2 confirms the validity of the above inequality.

□

Proposition 3.

Let $p\ne 2$ and κ in Theorem 10. Then the following inequality holds:

$$\begin{array}{c}\frac{4^{\kappa }-3^{\kappa }+1}{2\times 4^{\kappa }}\left[{\left({\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}\right)}^p+{\left({\rho }^{\frac{1}{4}}{\sigma }^{\frac{4}{4}}\right)}^p\right]+\frac{3^{\kappa }-1}{4^{\kappa }}{\left(\sqrt{\rho \sigma }\right)}^p\hfill \\ \hspace{1em}\hspace{1em}-pln\left(\frac{\sigma }{\rho }\right)\left[{\rho }^{p-1}+{\sigma }^{p-1}\right]\left[\frac{3^{\kappa }+4^{\kappa }-1}{4^{\kappa +1}}\right]\\ \hspace{1em}\hspace{1em}\le \frac{{\rho }^p}{2{\left(-pln\left(\frac{\sigma }{\rho }\right)\right)}^{\kappa }}\left[\Gamma (\kappa +1)-\kappa \Gamma \left(\kappa ,-pln\left(\frac{\sigma }{\rho }\right)\right)\right]\\ \hspace{1em}\hspace{1em}+\frac{{\sigma }^p}{2{\left(pln\left(\frac{\sigma }{\rho }\right)\right)}^{\kappa }}\left[\Gamma (\kappa +1)-\kappa \Gamma \left(\kappa ,pln\left(\frac{\sigma }{\rho }\right)\right)\right]\\ \hspace{1em}\le \frac{4^{\kappa }-3^{\kappa }+1}{2\times 4^{\kappa }}\left[{\left({\rho }^{\frac{3}{4}}{\sigma }^{\frac{1}{4}}\right)}^p+{\left({\rho }^{\frac{1}{4}}{\sigma }^{\frac{4}{4}}\right)}^p\right]+\frac{3^{\kappa }-1}{4^{\kappa }}{\left(\sqrt{\rho \sigma }\right)}^p\\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+pln\left(\frac{\sigma }{\rho }\right)\left[{\rho }^{p-1}+{\sigma }^{p-1}\right]\left[\frac{3^{\kappa }+4^{\kappa }-1}{4^{\kappa +1}}\right].\end{array}$$

(48)

Proof. 

The proof follows directly from Theorem 10 using the harmonic convex function $\phi \left(\mu \right)={\mu }^p$, $\mu \in \left(0,\infty \right)$ for $p\ne 3$. The Figure 3 confirms the validity of the above inequality.

□

6. Concluding Remarks

The mathematical inequalities topic has grown immensely over the past four decades and many papers have been written on this topic presenting novel ideas. In establishing the results for this topic, convexity plays a pivotal role and this has resulted in a variety of generalizations to convexity. As the topic of mathematical inequalities relating to a number of real life phenomena cannot be modeled using ordinary calculus, mathematicians are utilizing fractional calculus and have produced some surprisingly useful results. This study deals with the results of Hermite–Hadamard-type inequalities by applying one of the novel generalizations of convexity, know as the $GA$-convexity (also known as the geometric-arithmetic convexity) and Hadamard fractional integral operators, which are very useful and important fractional integral operators in fractional calculus since classical derivations cannot adequately model the majority of applied issues. In terms of application domains, fractional integral and derivative operators create links between mathematics and other disciplines and offer answers that are very applicable to real-world issues. The results in this paper provide generalizations and refinements of some of the results proved in [10,11,20,21,22]. We believe that the results may motivate mathematicians working in the field of fractional calculus to demonstrate novel results using more generalized forms of convexity together with other fractional integral operators.