Conformable fractional integral inequalities for <em>GG</em>- and <em>GA</em>-convex functions

Yousaf Khurshid; Muhammad Adil Khan; Yu-Ming Chu; Yousaf Khurshid; Muhammad Adil Khan; Yu-Ming Chu

doi:10.3934/math.2020322

AIMS Mathematics

2020, Volume 5, Issue 5: 5012-5030. doi: 10.3934/math.2020322

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Research article

Conformable fractional integral inequalities for GG- and GA-convex functions

1.
Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan
2.
Department of Mathematics, Huzhou University, Huzhou 313000, P. R. China
3.
Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science & Technology, Changsha 410114, P. R. China

Received: 28 March 2020 Accepted: 02 June 2020 Published: 10 June 2020
MSC : 26A33, 26A51, 26D15

In the article, we present several new Hermite-Hadamard type inequalities for GG- and GA-convex functions via the conformable fractional integrals. Our results are the generalizations of some previously known results.

Keywords:

Citation: Yousaf Khurshid, Muhammad Adil Khan, Yu-Ming Chu. Conformable fractional integral inequalities for GG- and GA-convex functions[J]. AIMS Mathematics, 2020, 5(5): 5012-5030. doi: 10.3934/math.2020322

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Abstract

1. Introduction

Let $I\subseteq \mathbb{R}$ be an interval. Then a real-valued function $h: I\rightarrow \mathbb{R}$ is said to be convex (concave) on the interval $I$ if the inequality

$\begin{equation*} h(t \kappa_{1}+(1-t)\kappa_{2})\leq (\geq) t h(\kappa_{1})+(1-t)h(\kappa_{2}) \end{equation*}$

holds for all $\kappa_{1}, \kappa_{2}\in I$ and $t\in [0, 1]$ .

It is well known that convexity (concavity) has wide applications in pure and applied mathematics ^{[1,2,3,4,5,6,7,8,9,10,11,12]}. The well known Hermite-Hadamard inequality ^{[13,14,15,16,17,18,19,20]} for the convex (concave) function $h: I\rightarrow \mathbb{R}$ can be stated as follows:

$\begin{equation*} h\left(\frac{\kappa_{1}+\kappa_{2}}{2}\right)\leq (\geq)\frac{1}{\kappa_{2}-\kappa_{1}}\int_{\kappa_{1}}^{\kappa_{2}}h(x)dx\leq (\geq)\frac{h(\kappa_{1})+h(\kappa_{2})}{2} \end{equation*}$

for all $\kappa_{1}, \kappa_{2}\in I$ with $\kappa_{1}\neq \kappa_{2}$ .

Recently, many generalizations, invariants and extensions have been made for the convexity, for example, harmonic-convexity ^[21,22], exponential-convexity ^[23,24], $s$ -convexity ^[25,26], Schur-convexity ^[27,28,29], strong convexity ^{[30,31,32,33]}, $H_{p, q}$ -convexity ^{[34,35,36,37,38]}, generalized convexity ^[39], $GG$ - and $GA$ -convexities ^[40], preinvexity ^[41] and quasi-convexity ^[42]. In particular, many remarkable inequalities can be found in the literature ^{[43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58]} via the convexity theory.

Niculescu ^[59,60] defined the $GG$ - and $GA$ -convex functions as follows.

Definition 1.1. (See ^[59]) A real-valued function $h: I\rightarrow [0, \infty)$ is said to be $GG$ -convex on the interval $I$ if the inequality

$\begin{equation*} h\left(\kappa_{1}^{t}\kappa_{2}^{1-t}\right)\leq h(\kappa_{1})^{t}h(\kappa_{2})^{1-t} \end{equation*}$

holds for all $\kappa_{1}, \kappa_{2}\in I$ and $t\in[0, 1]$ .

Definition 1.2. (See ^[60]) A real-valued function $h : I\rightarrow [0, \infty)$ is said to be $GA$ -convex if the inequality

$\begin{equation*} h\left(\kappa_{1}^{t}\kappa_{2}^{1-t}\right)\leq t h(\kappa_{1})+(1-t)h(\kappa_{2}) \end{equation*}$

holds for all $\kappa_{1}, \kappa_{2}\in I$ and $t\in[0, 1]$ .

Ardıç et al. ^[61] established several novel inequalities (Theorem 1.1) involving the $GG$ - and $GA$ -convex functions via an identity (Lemma 1.1) for differentiable functions.

Lemma 1.1. (See ^[61]) Let $\kappa_{1}, \kappa_{2}\in (0, \infty)$ with $\kappa_{1} < \kappa_{2}$ and $h:[\kappa_{1}, \kappa_{2}]\rightarrow\mathbb{R}$ be a differentiable function such that $h^{\prime}\in L([\kappa_{1}, \kappa_{2}])$ . Then the identity

$\begin{equation} \kappa_{2}^{2}h(\kappa_{2})-\kappa_{1}^{2}h(\kappa_{1})-2\int^{\kappa_{2}}_{\kappa_{1}}xh(x)dx \end{equation}$

(1.1)

$\begin{equation*} = (\log\kappa_{2}-\log\eta)\int_{0}^{1}(\kappa_{2}^{t}\eta^{1-t})^{3}h^{\prime}(\kappa_{2}^{t}\eta^{1-t})dt +(\log\eta-\log\kappa_{1})\int_{0}^{1}(\eta^{t}\kappa_{1}^{1-t})^{3}h^{\prime}(\eta^{t}\kappa_{1}^{1-t})dt \end{equation*}$

holds for all $\eta\in[\kappa_{1}, \kappa_{2}]$ .

Theorem 1.1. (See ^[61]) Let $\kappa_{1}, \kappa_{2}\in (0, \infty)$ with $\kappa_{1} < \kappa_{2}$ and $h:[\kappa_{1}, \kappa_{2}]\rightarrow\mathbb{R}$ be a differentiable function such that $h^{\prime}\in L([\kappa_{1}, \kappa_{2}])$ . Then the following statements are true:

$(1)$ If $|h^{\prime}(x)|$ is $GG$ -convex on $[\kappa_{1}, \kappa_{2}]$ , then the inequality

$\begin{equation} \left|\kappa_{2}^{2}h(\kappa_{2})-\kappa_{1}^{2}h(\kappa_{1})-2\int^{\kappa_{2}}_{\kappa_{1}}x h(x)dx\right| \end{equation}$

(1.2)

$\begin{equation*} \leq(\log\kappa_{2}-\log\eta)L(\kappa_{2}^{3}|h^{\prime}(\kappa_{2})|, \eta^{3}|h^{\prime}(\eta)|)+(\log\eta-\log\kappa_{1}) L(\eta^{3}|h^{\prime}(\eta)|, \kappa_{1}^{3}|h^{\prime}(\kappa_{1})|) \end{equation*}$

holds for all $\eta\in[\kappa_{1}, \kappa_{2}]$ , where $L(\kappa_{1}, \kappa_{2}) = (\kappa_{2}-\kappa_{1})/(\log\kappa_{2}-\log\kappa_{1})$ is the logarithmic mean of $\kappa_{1}$ and $\kappa_{2}$ .

$(2)$ If $\vartheta, \gamma > 1$ with $1/\vartheta+1/\gamma = 1$ and $|h^{\prime}(x)|^{\gamma}$ is $GG$ -convex on $[\kappa_{1}, \kappa_{2}]$ , then the inequalities

$\begin{equation} \left|\kappa_{2}^{2}h(\kappa_{2})-\kappa_{1}^{2}h(\kappa_{1})-2\int^{\kappa_{2}}_{\kappa_{1}}xh(x)dx\right| \end{equation}$

(1.3)

$\begin{equation*} \leq(\log\kappa_{2}-\log\eta)(L(\kappa_{2}^{3\vartheta}, \eta^{3\vartheta}))^{\frac{1}{\vartheta}} (L(|h'(\kappa_{2})|^{\gamma}, |h^{\prime}(\eta)|^{\gamma}))^{\frac{1}{\gamma}} \end{equation*}$

$\begin{equation*} +(\log\eta-\log\kappa_{1})(L(\eta^{3\vartheta}, \kappa_{1}^{3\vartheta}))^{\frac{1}{\vartheta}} (L(|h{\prime}(\eta)|^{\gamma}, \kappa_{1}^{3}|h{\prime}(\kappa_{1})|^{\gamma}))^{\frac{1}{\gamma}}, \end{equation*}$

$\begin{equation} \left|\kappa_{2}^{2}h(\kappa_{2})-\kappa_{1}^{2}h(\kappa_{1})-2\int^{\kappa_{2}}_{\kappa_{1}}xh(x)dx\right| \end{equation}$

(1.4)

$\begin{equation*} \leq(\log\kappa_{2}-\log\eta)(L(\kappa_{2}^{3\gamma}|h^{\prime}(\kappa_{2})|^{\gamma}, \eta^{3\gamma}|h^{\prime}(\eta)|^{\gamma}))^{\frac{1}{\gamma}} \end{equation*}$

$\begin{equation*} +(\log\eta-\log\kappa_{1})(L(\eta^{3\gamma}|h^{\prime}(\eta)|^{\gamma}, \kappa_{1}^{3\gamma}|h'(\kappa_{1})|^{\gamma}))^{\frac{1}{\gamma}} \end{equation*}$

and

$\begin{equation} \left|\kappa_{2}^{2}h(\kappa_{2})-\kappa_{1}^{2}h(\kappa_{1})-2\int^{\kappa_{2}}_{\kappa_{1}}x h(x)dx\right| \end{equation}$

(1.5)

$\begin{equation*} \leq(\log\kappa_{2}-\log\eta)(L(\kappa_{2}^{3}, \eta^{3}))^{1-\frac{1}{\gamma}} (L(\kappa_{2}^{3}|h^{\prime}(\kappa_{2})|^{\gamma}, \eta^{3}|h{\prime}(\eta)|^{\gamma}))^{\frac{1}{\gamma}} \end{equation*}$

$\begin{equation*} +(\log\eta-\log\kappa_{1})(L(\eta^{3}, \kappa_{1}^{3}))^{1-\frac{1}{\gamma}} (L(\eta^{3}|h^{\prime}(\eta)|^{\gamma}, \kappa_{1}^{3}|h^{\prime}(\kappa_{1})|^{\gamma}))^{\frac{1}{\gamma}} \end{equation*}$

hold for all $\eta\in[\kappa_{1}, \kappa_{2}]$ .

$(3)$ If $|h^{\prime}(x)|$ is $GA$ -convex on $[\kappa_{1}, \kappa_{2}]$ , then we have

$\begin{equation} \left|\kappa_{2}^{2}h(\kappa_{2})-\kappa_{1}^{2}h(\kappa_{1})-2\int^{\kappa_{2}}_{\kappa_{1}}xh(x)dx\right| \end{equation}$

(1.6)

$\begin{equation*} \leq\frac{|h^{\prime}(\kappa_{2})|}{3}\left[\kappa_{2}^{3}-L (\eta^{3}, \kappa_{2}^{3})\right] +\frac{|h^{\prime}(\eta)|}{3}\big[L(\eta^{3}, \kappa_{2}^{3})-L(\kappa_{1}^{3}, \eta^{3})\big]+\frac{|h^{\prime}(\kappa_{1})|}{3} \left[L(\kappa_{1}^{3}, \eta^{3})-\eta^{3}\right] \end{equation*}$

for all $\eta\in[\kappa_{1}, \kappa_{2}]$ .

$(4)$ If $\vartheta, \gamma > 1$ with $1/\vartheta+1/\gamma = 1$ and $|h^{\prime}(x)|^{\gamma}$ is $GA$ -convex on $[\kappa_{1}, \kappa_{2}]$ , then one has

$\begin{equation} \left|\kappa_{2}^{2}h(\kappa_{2})-\kappa_{1}^{2}h(\kappa_{1})-2\int^{\kappa_{2}}_{\kappa_{1}}x h(x)dx\right| \end{equation}$

(1.7)

$\begin{equation*} \leq(\log\kappa_{2}-\log\eta)^{1-\frac{1}{\gamma}}(L(\kappa_{2}^{3}, \eta^{3}))^{1-\frac{1}{\gamma}} \left(\frac{|h^{\prime}(\kappa_{2})|^{\gamma}\left[\kappa_{2}^{3} -L(\eta^{3}, \kappa_{2}^{3})\right]+|h^{\prime}(\eta)|^{\gamma}\left[L(\eta^{3}, \kappa_{2}^{3}) -\eta^{3}\right]}{3}\right)^{\frac{1}{\gamma}} \end{equation*}$

$\begin{equation*} +(\log\eta-\log\kappa_{1})^{1-\frac{1}{\gamma}}(L (\eta^{3}, \kappa_{1}^{3}))^{1-\frac{1}{\gamma}} \left(\frac{|h^{\prime}(\eta)|^{\gamma}\left[\eta^{3} -L(\kappa_{1}^{3}, \eta^{3})\right]+|h^{\prime}(\kappa_{1})|^{\gamma} \left[L(\kappa_{1}^{3}, \eta^{3})-\kappa_{1}^{3}\right]}{3}\right)^{\frac{1}{\gamma}}, \end{equation*}$

$\begin{equation} \left|\kappa_{2}^{2}h(\kappa_{2})-\kappa_{1}^{2}h(\kappa_{1})-2\int^{\kappa_{2}}_{\kappa_{1}}x h(x)dx\right| \end{equation}$

(1.8)

$\begin{equation*} \leq\frac{(\log\kappa_{2}-\log\eta)^{1-\frac{1}{\gamma}}}{\vartheta^{\frac{1}{\gamma}}} \left(L\left(\kappa_{2}^{\frac{3(\gamma-\vartheta)}{\gamma-1}}, \eta^{\frac{3(\gamma-\vartheta)} {\gamma-1}}\right)\right)^{\frac{\gamma-1}{\gamma}}(A_{\gamma}(\kappa_{2}, \eta))^{\frac{1}{\gamma}} \end{equation*}$

$\begin{equation*} +\frac{(\log\eta-\log\kappa_{1})^{1-\frac{1}{\gamma}}}{\vartheta^{\frac{1}{\gamma}}} \left(L\left(\eta^{\frac{3(\gamma-\vartheta)}{\gamma-1}}, \kappa_{1}^{\frac{3(\gamma-\vartheta)} {\gamma-1}}\right)\right)^{\frac{\gamma-1}{\gamma}}(A_{\gamma}(\eta, \kappa_{1}))^{\frac{1}{\gamma}}, \end{equation*}$

$\begin{equation} \left|\kappa_{2}^{2}h(\kappa_{2})-\kappa_{1}^{2}h(\kappa_{1})-2\int^{\kappa_{2}}_{\kappa_{1}}xh(x)dx\right| \end{equation}$

(1.9)

$\begin{equation*} \leq(\log\kappa_{2}-\log\eta)^{1-\frac{1}{\gamma}}\left(L\left(\kappa_{2}^{\frac{3\gamma}{\gamma-1}}, \eta^{\frac{3\gamma}{\gamma-1}}\right)\right)^{1-\frac{1}{\gamma}} \left(\frac{|h^{\prime}(\kappa_{2})|^{\gamma}+|h^{\prime}(\eta)|^{\gamma}}{2}\right)^{\frac{1}{\gamma}} \end{equation*}$

$\begin{equation*} +(\log\eta-\log\kappa_{1})^{1-\frac{1}{\gamma}}\left(L\left(\eta^{\frac{3\gamma}{\gamma-1}}, \kappa_{1}^{\frac{3\gamma}{\gamma-1}}\right)\right)^{1-\frac{1}{\gamma}} \left(\frac{|h^{\prime}(\eta)|^{\gamma}+|h^{\prime}(\kappa_{1})|^{\gamma}}{2}\right)^{\frac{1}{\gamma}}, \end{equation*}$

$\begin{equation} \left|\kappa_{2}^{2}h(\kappa_{2})-\kappa_{1}^{2}h(\kappa_{1})-2\int^{\kappa_{2}}_{\kappa_{1}}x h(x)dx\right| \end{equation}$

(1.10)

$\begin{equation*} \leq\frac{(\log\kappa_{2}-\log\eta)^{1-\frac{1}{\gamma}}}{\gamma^{\frac{1}{\gamma}}}(A_{\gamma}(\kappa_{2}, \eta))^{1/\gamma} +\frac{(\log\eta-\log\kappa_{1})^{1-\frac{1}{\gamma}}}{\gamma^{\frac{1}{\gamma}}}(A_{\gamma}(\eta, \kappa_{1}))^{1/\gamma}, \end{equation*}$

where

$\begin{equation*} A_{\gamma}(\kappa_{2}, \eta) = \frac{|h^{\prime}(\kappa_{2})|^{\gamma} \left[\kappa_{2}^{3\gamma}-L(\eta^{3\gamma}, \kappa_{2}^{3\gamma})\right] +|h^{\prime}(\eta)|^\gamma\left[L(\eta^{3\gamma}, \kappa_{2}^{3\gamma})-\eta^{3\gamma}\right]}{3} \end{equation*}$

and

$\begin{equation*} A_{\gamma}(\eta, \kappa_{1}) = \frac{|h^{\prime}(\eta)|^{\gamma} \left[\eta^{3\gamma}-L (\kappa_{1}^{3\gamma}, \eta^{3\gamma})\right] +|h^{\prime}(\kappa_{1})|^\gamma \left[L(\kappa_{1}^{3\gamma}, \eta^{3\gamma})-\kappa_{1}^{3\gamma}\right]}{3}. \end{equation*}$

The conformable fractional derivative $D_{\alpha}(h)(t)$ ^[62] of order $0 < \alpha\leq1$ at $t > 0$ for a function $h: [0, \infty)\rightarrow \mathbb{R}$ is defined by

$\begin{equation*} D_{\alpha}(h)(t) = \lim\limits_{\epsilon \rightarrow 0}\frac{h(t+\epsilon t^{1-\alpha})-h(t)}{\epsilon}, \end{equation*}$

$h$ is said to be $\alpha$ -fractional differentiable if the conformable fractional derivative $D_{\alpha}(h)(t)$ exists. The conformable fractional derivative at $0$ is defined by $h^{\alpha}(0) = \lim_{t \rightarrow 0^{+}}h^{\alpha}(t)$ . If $h_{1}$ and $h_{2}$ are $\alpha$ -differentiable at $t > 0$ , and $\kappa_{1}, \kappa_{2}, \lambda, c\in\mathbb{R}$ are constants, then the conformable fractional derivative satisfies the following formulas

$\begin{equation*} \frac{d_{\alpha}}{d_{\alpha}t}\left(t^{\lambda}\right) = \lambda t^{\lambda-\alpha}, \quad \frac{d_{\alpha}}{d_{\alpha}t}(c) = 0, \end{equation*}$

$\begin{equation*} \frac{d_{\alpha}}{d_{\alpha}t}\left(\kappa_{1}h_{1}(t)+\kappa_{2}h_{2}(t)\right) = \kappa_{1}\frac{d_{\alpha}}{d_{\alpha}t}(h_{1}(t))+\kappa_{2} \frac{d_{\alpha}}{d_{\alpha}t}(h_{2}(t)), \end{equation*}$

$\begin{equation*} \frac{d_{\alpha}}{d_{\alpha}t}\left(h_{1}(t)h_{2}(t)\right) = h_{1}(t)\frac{d_{\alpha}}{d_{\alpha}t}(h_{2}(t))+h_{2}(t) \frac{d_{\alpha}}{d_{\alpha}t}(h_{1}(t)), \end{equation*}$

$\begin{equation*} \frac{d_{\alpha}}{d_{\alpha}t}\left(\frac{h_{1}(t)}{h_{2}(t)}\right) = \frac {h_{2}(t)\frac{d_{\alpha}}{d_{\alpha}t}(h_{1}(t))-h_{1}(t)\frac{d_{\alpha}}{d_{\alpha}t}(h_{2}(t))}{(h_{2}(t))^{2}} \end{equation*}$

and

$\begin{equation*} \frac{d_{\alpha}}{d_{\alpha}t}\left(h_{1}(h_{2}(t))\right) = h_{1}'(h_{2}(t))\frac{d_{\alpha}}{d_{\alpha}t}(h_{2}(t)) \end{equation*}$

if $h_{1}$ differentiable at $h_{2}(t)$ . Moreover,

$\begin{equation*} \frac{d_{\alpha}}{d_{\alpha}t}\left(h_{1}(t)\right) = t^{1-\alpha} \frac{d}{dt}(h_{1}(t)) \end{equation*}$

if $h_{1}$ is differentiable.

Let $\alpha\in(0, 1]$ and $0\leq \kappa_{1} < \kappa_{2}$ . Then the function $h: [\kappa_{1}, \kappa_{2}]\rightarrow\mathbb{R}$ is said to be $\alpha$ -fractional integrable on $[\kappa_{1}, \kappa_{2}]$ if the integral

$\begin{equation*} \int_{\kappa_{1}}^{\kappa_{2}}h(x)d_{\alpha}x = \int_{\kappa_{1}}^{\kappa_{2}}h(x)x^{\alpha-1}dx \end{equation*}$

exists and is finite. All $\alpha$ -fractional integrable functions on $[\kappa_{1}, \kappa_{2}]$ is denoted by $L_{\alpha}([\kappa_{1}, \kappa_{2}])$ . Note that

$\begin{equation*} I_{\alpha}^{\kappa_{1}}(h_{1})(s) = I_{1}^{\kappa_{1}}(s^{\alpha-1}h_{1}) = \int_{\kappa_{1}}^{s}\frac{h_{1}(x)}{x^{1-\alpha}}dx \end{equation*}$

for all $\alpha\in (0, 1]$ , where the integral is the usual Riemann improper integral.

Recently, the conformable integrals and derivatives have attracted the attention of many researchers. Anderson ^[63] established the conformable integral version of the Hermite-Hadamard inequality as follows:

$\begin{equation*} \frac{\alpha}{\kappa_{2}^{\alpha}-\kappa_{1}^{\alpha}}\int_{\kappa_{1}}^{\kappa_{2}}h(x)d_{\alpha}x \leq\frac{h(\kappa_{1})+h(\kappa_{2})}{2} \end{equation*}$

if $\alpha\in(0, 1]$ and $h: [\kappa_{1}, \kappa_{2}]\rightarrow\mathbb{R}$ is an $\alpha$ -fractional differentiable function such that $D_{\alpha}(h)$ is increasing. Moreover, if function $h$ is decreasing on $[\kappa_{1}, \kappa_{2}]$ , then

$\begin{equation*} h\left(\frac{\kappa_{1}+\kappa_{2}}{2}\right) \leq\frac{\alpha}{\kappa_{2}^{\alpha}-\kappa_{1}^{\alpha}}\int_{\kappa_{1}}^{\kappa_{2}}h(x)d_{\alpha}x. \end{equation*}$

The main purpose of the article is to establish the conformable fractional integral versions of the Hermite-Hadamard type inequality for $GG$ - and $GA$ -convex functions.

2. Main results

In order to establish our main results, we need a lemma which we present in this section.

Lemma 2.1. Let $\kappa_{1}, \kappa_{2}\in(0, \infty)$ with $\kappa_{1} < \kappa_{2}$ , $\alpha \in (0, 1]$ and $h:[\kappa_{1}, \kappa_{2}]\rightarrow\mathbb{R}$ be an $\alpha$ -fractional differentiable function on $(\kappa_{1}, \kappa_{2})$ such that $D _{\alpha}(h)\in L_{\alpha}([\kappa_{1}, \kappa_{2}])$ . Then the identity

$\begin{equation} \kappa_{2}^{2\alpha}h(\kappa_{2})-\kappa_{1}^{2\alpha}h(\kappa_{1})-2\alpha\int^{\kappa_{2}}_{\kappa_{1}}x^{\alpha}h(x)d_{\alpha}x \end{equation}$

(2.1)

$\begin{equation*} = (\log\kappa_{2}-\log\eta)\int_{0}^{1}(\kappa_{2}^{t}\eta^{1-t})^{3\alpha}D _{\alpha}(h)(\kappa_{2}^{t}\eta^{1-t})t^{1-\alpha}dt \end{equation*}$

$\begin{equation*} +(\log\eta-\log\kappa_{1})\int_{0}^{1}(\eta^{t}\kappa_{1}^{1-t})^{3\alpha}D _{\alpha}(h)(\eta^{t}\kappa_{1}^{1-t})t^{1-\alpha}dt \end{equation*}$

holds for all $\eta\in[\kappa_{1}, \kappa_{2}]$ .

Proof. Integration by parts, we get

$\begin{equation*} I_{1} = \int_{0}^{1}(\kappa_{2}^{t}\eta^{1-t})^{3\alpha}D_{\alpha}(h)(\kappa_{2}^{t}\eta^{1-t})t^{1-\alpha}dt \end{equation*}$

$\begin{equation*} = \int_{0}^{1}(\kappa_{2}^{t}\eta^{1-t})^{2\alpha+1}h'(\kappa_{2}^{t}\eta^{1-t})dt. \end{equation*}$

Let $x = \kappa_{2}^{t}\eta^{1-t}$ . Then $I_{1}$ can be rewritten as

$\begin{equation*} I_{1} = \frac{1}{\log \kappa_{2} -\log \eta}\int_{\eta}^{\kappa_{2}}x^{2\alpha}h^{\prime}(x)dx \end{equation*}$

$\begin{equation*} = \frac{1}{\log\kappa_{2}-\log \eta}\left[\kappa_{2}^{\alpha}h(\kappa_{2}) -\eta^{\alpha}h(\eta)-2\alpha\int^{\kappa_{2}}_{\eta}x^{2\alpha-1}h(x)dx\right] \end{equation*}$

$\begin{equation*} = \frac{1}{\log\kappa_{2}-\log\eta}\left[\kappa_{2}^{\alpha}h(\kappa_{2}) -\eta^{\alpha}h(\eta)-2\alpha\int^{\kappa_{2}}_{\eta}x^{\alpha}h(x)d_{\alpha}x\right]. \end{equation*}$

Similarly, we have

$\begin{equation*} I_{2} = \int_{0}^{1}(\eta^{t}\kappa_{1}^{1-t})^{3\alpha}D _{\alpha}(h)(\eta^{t}\kappa_{1}^{1-t})t^{1-\alpha}dt \end{equation*}$

$\begin{equation*} = \frac{1}{\log\eta-\log\kappa_{1}}\left[\eta^{\alpha}h(\eta)-\kappa_{1}^{\alpha}h(\kappa_{1}) -2\alpha\int_{\kappa_{1}}^{\eta}x^{\alpha}h(x)d_{\alpha}x\right]. \end{equation*}$

Multiplying $I_{1}$ by $(\log\kappa_{2}-\log\eta)$ and $I_{2}$ by $(\log \eta-\log\kappa_{1})$ , then add them we get the desired identity.

Remark 2.1. Let $\alpha = 1$ . Then identity (2.1) reduces to (1.1).

Theorem 2.1. Let $\kappa_{1}, \kappa_{2}\in(0, \infty)$ with $\kappa_{1} < \kappa_{2}$ , $\alpha \in (0, 1]$ , $h:[\kappa_{1}, \kappa_{2}]\rightarrow\mathbb{R}$ be an $\alpha$ -fractional differentiable function on $(\kappa_{1}, \kappa_{2})$ such that $D _{\alpha}(h)\in L_{\alpha}([\kappa_{1}, \kappa_{2}])$ and $|h^{\prime}(x)|$ be a $GG$ -convex function on $[\kappa_{1}, \kappa_{2}]$ . Then the inequality

$\begin{equation} \left|\kappa_{2}^{2\alpha}h(\kappa_{2})-\kappa_{1}^{2\alpha}h(\kappa_{1}) -2\alpha\int^{\kappa_{2}}_{\kappa_{1}}x^{\alpha}h(x)d_{\alpha}x\right| \end{equation}$

(2.2)

$\begin{equation*} \leq(\log\kappa_{2}-\log\eta)L(\kappa_{2}^{2\alpha+1}|h'(\kappa_{2})|, \eta^{2\alpha+1}|h^{\prime}(\eta)|) \end{equation*}$

$\begin{equation*} +(\log\eta-\log\kappa_{1})L(\eta^{2\alpha+1}|h^{\prime}(\eta)|, \kappa_{1}^{2\alpha+1}|h^{\prime}(\kappa_{1})|) \end{equation*}$

holds for all $\eta\in [\kappa_{1}, \kappa_{2}]$ .

Proof. It follows from the $GG$ -convexity of the function $|h^{\prime}(x)|$ on the interval $[\kappa_{1}, \kappa_{2}]$ and Lemma 2.1 that

$\begin{equation*} \left|\kappa_{2}^{2\alpha}h(\kappa_{2})-\kappa_{1}^{2\alpha}h(\kappa_{1}) -2\alpha\int^{\kappa_{2}}_{\kappa_{1}}x^{\alpha}h(x)d_{\alpha}x\right| \end{equation*}$

$\begin{equation*} \leq(\log\kappa_{2}-\log\eta)\int_{0}^{1}(\kappa_{2}^{t}\eta^{1-t})^{2\alpha+1}|h^{\prime}(\kappa_{2}^{t}\eta^{1-t})|dt \end{equation*}$

$\begin{equation*} +(\log\eta-\log\kappa_{1})\int_{0}^{1}(\eta^{t}\kappa_{1}^{1-t})^{2\alpha+1}|h^{\prime}(\eta^{t}\kappa_{1}^{1-t})|dt \end{equation*}$

$\begin{equation*} \leq(\log\kappa_{2}-\log\eta)\int_{0}^{1}(\kappa_{2}^{t}\eta^{1-t})^{2\alpha+1}|h^{\prime}(\kappa_{2})|^{t}|h^{\prime}(\eta)|^{1-t}dt \end{equation*}$

$\begin{equation*} +(\log\eta-\log\kappa_{1})\int_{0}^{1}(\eta^{t}\kappa_{1}^{1-t})^{2\alpha+1}|h^{\prime}(\eta)|^{t}|h'(\kappa_{1})|^{1-t}dt \end{equation*}$

$\begin{equation*} = (\log\kappa_{2}-\log\eta)L(\kappa_{2}^{2\alpha+1}|h^{\prime}(\kappa_{2})|, \eta^{2\alpha+1}|h^{\prime}(\eta)|) \end{equation*}$

$\begin{equation*} +(\log\eta-\log\kappa_{1})L(\eta^{2\alpha+1}|h^{\prime}(\eta)|, \kappa_{1}^{2\alpha+1}|h^{\prime}(\kappa_{1})|). \end{equation*}$

Remark 2.2. Let $\alpha = 1$ . Then inequality (2.2) reduces to (1.2).

Theorem 2.2. Let $\kappa_{1}, \kappa_{2}\in(0, \infty)$ with $\kappa_{1} < \kappa_{2}$ , $\vartheta, \gamma > 1$ with $1/\vartheta+1/\gamma = 1$ , $\alpha \in (0, 1]$ , $h:[\kappa_{1}, \kappa_{2}]\rightarrow\mathbb{R}$ be an $\alpha$ -fractional differentiable function on $(\kappa_{1}, \kappa_{2})$ such that $D _{\alpha}(h)\in L_{\alpha}([\kappa_{1}, \kappa_{2}])$ and $|h^{\prime}(x)|^{\gamma}$ be a $GG$ -convex function on $[\kappa_{1}, \kappa_{2}]$ . Then the inequality

$\begin{equation} \left|\kappa_{2}^{2\alpha}h(\kappa_{2})-\kappa_{1}^{2\alpha}h(\kappa_{1}) -2\alpha\int^{\kappa_{2}}_{\kappa_{1}}x^{\alpha}h(x)d_{\alpha}x\right| \end{equation}$

(2.3)

$\begin{equation*} \leq(\log\kappa_{2}-\log\eta)(L(\kappa_{2}^{(2\alpha+1)\vartheta}, \eta^{(2\alpha+1)\vartheta}))^{\frac{1}{\vartheta}} (L(|h^{\prime}(\kappa_{2})|^{\gamma}, |h^{\prime}(\eta)|^{\gamma}))^{\frac{1}{\gamma}} \end{equation*}$

$\begin{equation*} +(\log\eta-\log \kappa_{1})(L(\eta^{(2\alpha+1)\vartheta}, \kappa_{1}^{(2\alpha+1)\vartheta}))^{\frac{1}{\vartheta}} (L(|h^{\prime}(\eta)|^{\gamma}, |h^{\prime}(\kappa_{1})|^{\gamma}))^{\frac{1}{\gamma}} \end{equation*}$

holds for all $\eta\in [\kappa_{1}, \kappa_{2}]$ .

Proof. From Lemma 2.1, the property of the modulus, $GG$ -convexity of $|h^{\prime}|^\gamma$ and Hölder inequality we clearly see that

$\begin{equation*} \left|\kappa_{2}^{2\alpha}h(\kappa_{2})-\kappa_{1}^{2\alpha}h(\kappa_{1}) -2\alpha\int^{\kappa_{2}}_{\kappa_{1}}x^{\alpha}h(x)d_{\alpha}x\right| \end{equation*}$

$\begin{equation*} \leq(\log\kappa_{2}-\log\eta)\int_{0}^{1}(\kappa_{2}^{t}\eta^{1-t})^{2\alpha+1}|h^{\prime}(\kappa_{2}^{t}\eta^{1-t})|dt \end{equation*}$

$\begin{equation*} +(\log\eta-\log\kappa_{1})\int_{0}^{1}(\eta^{t}\kappa_{1}^{1-t})^{2\alpha+1}|h^{\prime}(\eta^{t}\kappa_{1}^{1-t})|dt \end{equation*}$

$\begin{equation*} \leq(\log\kappa_{2}-\log\eta)\left(\int_{0}^{1}(\kappa_{2}^{t}\eta^{1-t})^{(2\alpha+1) \vartheta}dt\right)^{\frac{1}{\vartheta}}\left(\int_{0}^{1} |h^{\prime}(\kappa_{2}^{t}\eta^{1-t})|^{\gamma}dt\right)^{\frac{1}{\gamma}} \end{equation*}$

$\begin{equation*} +(\log\eta-\log\kappa_{1})\left(\int_{0}^{1}(\eta^{t}\kappa_{1}^{1-t})^{(2\alpha+1)\vartheta}dt\right)^{\frac{1}{\vartheta}} \left(\int_{0}^{1}|h^{\prime}(\eta^{t}\kappa_{1}^{1-t})|^{\gamma}dt\right)^{\frac{1}{\gamma}} \end{equation*}$

$\begin{equation*} \leq(\log\kappa_{2}-\log\eta)\left(\int_{0}^{1}(\kappa_{2}^{t}\eta^{1-t})^{(2\alpha+1)\vartheta}dt\right)^{\frac{1}{\vartheta}} \left(\int_{0}^{1}|h^{\prime}(\kappa_{2})|^{\gamma t}|h^{\prime}(\eta)|^{(1-t)\gamma}dt\right)^{\frac{1}{\gamma}} \end{equation*}$

$\begin{equation*} +(\log\eta-\log\kappa_{1})\left(\int_{0}^{1}(\eta^{t}\kappa_{1}^{1-t})^{(2\alpha+1)\vartheta}dt\right)^{\frac{1}{\vartheta}}\left(\int_{0}^{1} |h^{\prime}(\eta)|^{\gamma t}|h^{\prime}(\kappa_{1})|^{(1-t)\gamma}dt\right)^{\frac{1}{\gamma}} \end{equation*}$

$\begin{equation*} = (\log\kappa_{2}-\log\eta)(L(\kappa_{2}^{(2\alpha+1)\vartheta}, \eta^{(2\alpha+1)\vartheta}))^{\frac{1}{\vartheta}} (L(|h^{\prime}(\kappa_{2})|^{\gamma}, |h^{\prime}(\eta)|^{\gamma}))^{\frac{1}{\gamma}} \end{equation*}$

$\begin{equation*} +(\log\eta-\log\kappa_{1})(L(\eta^{(2\alpha+1)\vartheta}, \kappa_{1}^{(2\alpha+1)\vartheta}))^{\frac{1}{\vartheta}} (L(|h^{\prime}(\eta)|^{\gamma}, |h^{\prime}(\kappa_{1})|^{\gamma}))^{\frac{1}{\gamma}}. \end{equation*}$

Remark 2.3. Let $\alpha = 1$ . Then inequality (2.3) reduces to (1.3).

Theorem 2.3. Let $\kappa_{1}, \kappa_{2}\in(0, \infty)$ with $\kappa_{1} < \kappa_{2}$ , $\gamma > 1$ , $\alpha\in (0, 1]$ , $h:[\kappa_{1}, \kappa_{2}]\rightarrow\mathbb{R}$ be an $\alpha$ -fractional differentiable function on $(\kappa_{1}, \kappa_{2})$ such that $D _{\alpha}(h)\in L_{\alpha}([\kappa_{1}, \kappa_{2}])$ and $|h^{\prime}(x)|^{\gamma}$ be a $GG$ -convex function on $[\kappa_{1}, \kappa_{2}]$ . Then the inequality

$\begin{equation} \left|\kappa_{2}^{2\alpha}h(\kappa_{2})-\kappa_{1}^{2\alpha}h(\kappa_{1}) -2\alpha\int^{\kappa_{2}}_{\kappa_{1}}x^{\alpha}h(x)d_{\alpha}x\right| \end{equation}$

(2.4)

$\begin{equation*} \leq(\log\kappa_{2}-\log\eta)(L(\kappa_{2}^{(2\alpha+1)\gamma}|h^{\prime}(\kappa_{2})|^{\gamma}, \eta^{(2\alpha+1)\gamma} |h^{\prime}(\eta)|^{\gamma}))^{\frac{1}{\gamma}} \end{equation*}$

$\begin{equation*} +(\log\eta-\log\kappa_{1})(L(\eta^{(2\alpha+1)\gamma}|h^{\prime}(\eta)|^{\gamma}, \kappa_{1}^{(2\alpha+1)\gamma} |h^{\prime}(\kappa_{1})|^{\gamma}))^{\frac{1}{\gamma}} \end{equation*}$

holds for all $\eta\in [\kappa_{1}, \kappa_{2}]$ .

Proof. It follows from Lemma 2.1 that

$\begin{equation*} \left|\kappa_{2}^{2\alpha}h(\kappa_{2})-\kappa_{1}^{2\alpha}h(\kappa_{1}) -2\alpha\int^{\kappa_{2}}_{\kappa_{1}}x^{\alpha}h(x)d_{\alpha}x\right| \end{equation*}$

$\begin{equation*} \leq(\log\kappa_{2}-\log\eta)\int_{0}^{1}(\kappa_{2}^{t}\eta^{1-t})^{2\alpha+1}|h^{\prime}(\kappa_{2}^{t}\eta^{1-t})|dt \end{equation*}$

$\begin{equation*} +(\log\eta-\log\kappa_{1})\int_{0}^{1}(\eta^{t}\kappa_{1}^{1-t})^{2\alpha+1}|h^{\prime}(\eta^{t}\kappa_{1}^{1-t})|dt. \end{equation*}$

Let $\vartheta > 1$ such that $\vartheta^{-1}+\gamma^{-1} = 1$ . Then making use of the Hölder integral inequality and the $GG$ -convexity of $|h^{\prime}|^{\gamma}$ , we get

$\begin{equation*} \left|\kappa_{2}^{2\alpha}h(\kappa_{2})-\kappa_{1}^{2\alpha}h(\kappa_{1}) -2\alpha\int^{\kappa_{2}}_{\kappa_{1}}x^{\alpha}h(x)d_{\alpha}x\right| \end{equation*}$

$\begin{equation*} \leq(\log\kappa_{2}-\log\eta)\left(\int_{0}^{1}dt\right)^{\frac{1}{\vartheta}}\left(\int_{0}^{1} (\kappa_{2}^{t}\eta^{1-t})^{(2\alpha+1)\gamma}|h^{\prime}(\kappa_{2}^{t}\eta^{1-t})|^{\gamma}dt\right)^{\frac{1}{\gamma}} \end{equation*}$

$\begin{equation*} +(\log\eta-\log\kappa_{1})\left(\int_{0}^{1}dt\right)^{\frac{1}{\vartheta}} \left(\int_{0}^{1}(\eta^{t}\kappa_{1}^{1-t})^{(2\alpha+1)\gamma} |h^{\prime}(\eta^{t}\kappa_{1}^{1-t})|^{\gamma}dt\right)^{\frac{1}{\gamma}} \end{equation*}$

$\begin{equation*} \leq(\log\kappa_{2}-\log\eta)\left(\int_{0}^{1}dt\right)^{\frac{1}{\vartheta}}\left(\int_{0}^{1} (\kappa_{2}^{t}\eta^{1-t})^{(2\alpha+1)\gamma}|h^{\prime}(\kappa_{2})|^{\gamma t} |h^{\prime}(\eta)|^{(1-t)\gamma}dt\right)^{\frac{1}{\gamma}} \end{equation*}$

$\begin{equation*} +(\log\eta-\log\kappa_{1})\left(\int_{0}^{1}dt\right)^{\frac{1}{\vartheta}} \left(\int_{0}^{1}(\eta^{t}\kappa_{1}^{1-t})^{(2\alpha+1)\gamma}|h^{\prime}(\eta)|^{\gamma t} |h^{\prime}(\kappa_{1})|^{(1-t)\gamma}dt\right)^{\frac{1}{\gamma}} \end{equation*}$

$\begin{equation*} = (\log\kappa_{2}-\log\eta)(L(\kappa_{2}^{(2\alpha+1)\gamma}|h^{\prime}(\kappa_{2})|^{\gamma}, \eta^{(2\alpha+1)\gamma} |h^{\prime}(\eta)|^{\gamma}))^{\frac{1}{\gamma}} \end{equation*}$

Remark 2.4. Let $\alpha = 1$ . Then inequality (2.4) reduces to (1.4).

Theorem 2.4. Let $\kappa_{1}, \kappa_{2}\in(0, \infty)$ with $\kappa_{1} < \kappa_{2}$ , $\gamma > 1$ , $\alpha \in (0, 1]$ , $h:[\kappa_{1}, \kappa_{2}]\rightarrow\mathbb{R}$ be an $\alpha$ -fractional differentiable function on $(\kappa_{1}, \kappa_{2})$ such that $D _{\alpha}(h)\in L_{\alpha}([\kappa_{1}, \kappa_{2}])$ and $|h^{\prime}(x)|^{\gamma}$ be a $GG$ -convex function on $[\kappa_{1}, \kappa_{2}]$ . Then the inequality

$\begin{equation} \left|\kappa_{2}^{2\alpha}h(\kappa_{2})-\kappa_{1}^{2\alpha}h(\kappa_{1}) -2\alpha\int^{\kappa_{2}}_{\kappa_{1}}x^{\alpha}h(x)d_{\alpha}x\right| \end{equation}$

(2.5)

$\begin{equation*} \leq(\log\kappa_{2}-\log \eta)(L(\kappa_{2}^{(2\alpha+1)}, \eta^{(2\alpha+1)}))^{1-\frac{1}{\gamma}} (L(\kappa_{2}^{(2\alpha+1)}|h^{\prime}(\kappa_{2})|^{\gamma}, \eta^{(2\alpha+1)}|h^{\prime}(\eta)|^{\gamma}))^{\frac{1}{\gamma}} \end{equation*}$

$\begin{equation*} +(\log\eta-\log\kappa_{1})(L (\eta^{(2\alpha+1)}, \kappa_{1}^{(2\alpha+1)}))^{1-\frac{1}{\gamma}} (L(\eta^{(2\alpha+1)}|h^{\prime}(\eta)|^{\gamma}, \kappa_{1}^{(2\alpha+1)}|h^{\prime}(\kappa_{1})|^{\gamma}))^{\frac{1}{\gamma}} \end{equation*}$

holds whenever $\eta\in [\kappa_{1}, \kappa_{2}]$ .

Proof. From the $GG$ -convexity of $|h^{\prime}|^{\gamma}$ , power mean inequality, the property of the modulus and Lemma 2.1 we clearly see that

$\begin{equation*} \left|\kappa_{2}^{2\alpha}h(\kappa_{2})-\kappa_{1}^{2\alpha}h(\kappa_{1}) -2\alpha\int^{\kappa_{2}}_{\kappa_{1}}x^{\alpha}h(x)d_{\alpha}x\right| \end{equation*}$

$\begin{equation*} \leq(\log\kappa_{2}-\log\eta)\int_{0}^{1}(\kappa_{2}^{t}\eta^{1-t})^{2\alpha+1}|h^{\prime}(\kappa_{2}^{t}\eta^{1-t})|dt \end{equation*}$

$\begin{equation*} +(\log\eta-\log\kappa_{1})\int_{0}^{1}(\eta^{t}\kappa_{1}^{1-t})^{2\alpha+1}|h^{\prime}(\eta^{t}\kappa_{1}^{1-t})|dt \end{equation*}$

$\begin{equation*} \leq(\log\kappa_{2}-\log\eta)\left(\int_{0}^{1}(\kappa_{2}^{t}\eta^{1-t})^{2\alpha+1}dt\right)^{1-\frac{1}{\gamma}} \left(\int_{0}^{1}(\kappa_{2}^{t}\eta^{1-t})^{2\alpha+1}|h^{\prime}(\kappa_{2}^{t}\eta^{1-t})|^{\gamma}dt\right)^{\frac{1}{\gamma}} \end{equation*}$

$\begin{equation*} +(\log\eta-\log\kappa_{1})\left(\int_{0}^{1}(\eta^{t}\kappa_{1}^{1-t})^{2\alpha+1}dt\right)^{1-\frac{1}{\gamma}} \left(\int_{0}^{1}(\eta^{t}\kappa_{1}^{1-t})^{2\alpha+1}|h^{\prime}(\eta^{t}\kappa_{1}^{1-t})|^{\gamma}dt\right)^{\frac{1}{\gamma}} \end{equation*}$

$\begin{equation*} \leq(\log\kappa_{2}-\log \eta)\left(\int_{0}^{1}(\kappa_{2}^{t}\eta^{1-t})^{2\alpha+1}dt\right)^{1-\frac{1}{\gamma}} \left(\int_{0}^{1}(\kappa_{2}^{t}\eta^{1-t})^{2\alpha+1}|h^{\prime}(\kappa_{2})|^{\gamma t} |h^{\prime}(\eta)|^{(1-t)\gamma}dt\right)^{\frac{1}{\gamma}} \end{equation*}$

$\begin{equation*} +(\log\eta-\log\kappa_{1})\left(\int_{0}^{1}(\eta^{t}\kappa_{1}^{1-t})^{2\alpha+1}dt\right)^{1-\frac{1}{\gamma}} \left(\int_{0}^{1}(\eta^{t}\kappa_{1}^{1-t})^{2\alpha+1}|h^{\prime}(\eta)|^{\gamma t} |h^{\prime}(\kappa_{1})|^{(1-t)\gamma}dt\right)^{\frac{1}{\gamma}} \end{equation*}$

$\begin{equation*} = (\log\kappa_{2}-\log\eta)(L(\kappa_{2}^{(2\alpha+1)}, \eta^{(2\alpha+1)}))^{1-\frac{1}{\gamma}} (L(\kappa_{2}^{(2\alpha+1)}|h^{\prime}(\kappa_{2})|^{\gamma}, \eta^{(2\alpha+1)}|h^{\prime}(\eta)|^{\gamma}))^{\frac{1}{\gamma}} \end{equation*}$

$\begin{equation*} +(\log\eta-\log\kappa_{1})(L(\eta^{(2\alpha+1)}, \kappa_{1}^{(2\alpha+1)}))^{1-\frac{1}{\gamma}} (L(\eta^{(2\alpha+1)}|h^{\prime}(\eta)|^{\gamma}, \kappa_{1}^{(2\alpha+1)}|h^{\prime}(\kappa_{1})|^{\gamma}))^{\frac{1}{\gamma}}. \end{equation*}$

Remark 2.5. Let $\alpha = 1$ . Then inequality (2.5) reduces to (1.5).

Theorem 2.5. Let $\kappa_{1}, \kappa_{2}\in(0, \infty)$ with $\kappa_{1} < \kappa_{2}$ , $\alpha \in (0, 1]$ , $h:[\kappa_{1}, \kappa_{2}]\rightarrow\mathbb{R}$ be an $\alpha$ -fractional differentiable function on $(\kappa_{1}, \kappa_{2})$ such that $D _{\alpha}(h)\in L_{\alpha}([\kappa_{1}, \kappa_{2}])$ and $|h^{\prime}(x)|$ be a $GA$ -convex function on $[\kappa_{1}, \kappa_{2}]$ . Then the inequality

$\begin{equation} \left|\kappa_{2}^{2\alpha}h(\kappa_{2})-\kappa_{1}^{2\alpha}h(\kappa_{1}) -2\alpha\int^{\kappa_{2}}_{\kappa_{1}}x^{\alpha}h(x)d_{\alpha}x\right| \end{equation}$

(2.6)

$\begin{equation*} \leq\frac{|h(\kappa_{2})|}{2\alpha+1}\left[\kappa_{2}^{2\alpha+1}-L (\eta^{2\alpha+1}, \kappa_{2}^{2\alpha+1})\right] +\frac{|h^{\prime}(\eta)|}{2\alpha+1}\left[L(\eta^{2\alpha+1}, \kappa_{2}^{2\alpha+1}) -L(\kappa_{1}^{2\alpha+1}, \eta^{2\alpha+1})\right] \end{equation*}$

$\begin{equation*} +\frac{|h^{\prime}(\kappa_{1})|}{2\alpha+1}\left[L(\kappa_{1}^{2\alpha+1}, \eta^{2\alpha+1})-\eta^{2\alpha+1}\right] \end{equation*}$

holds for each $\eta\in [\kappa_{1}, \kappa_{2}]$ .

Proof. It follows from the $GA$ -convexity of $|h^{\prime}(x)|$ and Lemma 2.1 that

$\begin{equation*} \left|\kappa_{2}^{2\alpha}h(\kappa_{2})-\kappa_{1}^{2\alpha}h(\kappa_{1}) -2\alpha\int^{\kappa_{2}}_{\kappa_{1}}x^{\alpha}h(x)d_{\alpha}x\right| \end{equation*}$

$\begin{equation*} \leq(\log\kappa_{2}-\log\eta)\int_{0}^{1}(\kappa_{2}^{t}\eta^{1-t})^{2\alpha+1}|h^{\prime}(\kappa_{2}^{t}\eta^{1-t})|dt \end{equation*}$

$\begin{equation*} +(\log\eta-\log\kappa_{1})\int_{0}^{1}(\eta^{t}\kappa_{1}^{1-t})^{2\alpha+1}|h^{\prime}(\eta^{t}\kappa_{1}^{1-t})|dt \end{equation*}$

$\begin{equation*} \leq(\log\kappa_{2}-\log\eta)\int_{0}^{1}(\kappa_{2}^{t}\eta^{1-t})^{2\alpha+1}\left[t|h^{\prime}(\kappa_{2})|+(1-t)|h^{\prime}(\eta)|\right]dt \end{equation*}$

$\begin{equation*} +(\log\eta-\log\kappa_{1})\int_{0}^{1}(\eta^{t}\kappa_{1}^{1-t})^{2\alpha+1}\left[t|h^{\prime}(\eta)|+(1-t)|h^{\prime}(\kappa_{1})|\right]dt \end{equation*}$

$\begin{equation*} = \frac{|h^{\prime}(\kappa_{2})|}{2\alpha+1}\left[\kappa_{2}^{2\alpha+1}-L(\eta^{2\alpha+1}, \kappa_{2}^{2\alpha+1})\right] +\frac{|h^{\prime}(\eta)|}{2\alpha+1}\left[L(\eta^{2\alpha+1}, \kappa_{2}^{2\alpha+1})-L(\kappa_{1}^{2\alpha+1}, \eta^{2\alpha+1})\right] \end{equation*}$

$\begin{equation*} +\frac{|h^{\prime}(\kappa_{1})|}{2\alpha+1}\left[L(\kappa_{1}^{2\alpha+1}, \eta^{2\alpha+1})-\eta^{2\alpha+1}\right]. \end{equation*}$

Remark 2.6. Let $\alpha = 1$ . Then inequality (2.6) becomes (1.6).

Theorem 2.6. Let $\kappa_{1}, \kappa_{2}\in(0, \infty)$ with $\kappa_{1} < \kappa_{2}$ , $\alpha \in (0, 1]$ , $\gamma > 1$ , $h:[\kappa_{1}, \kappa_{2}]\rightarrow\mathbb{R}$ be an $\alpha$ -fractional differentiable function on $(\kappa_{1}, \kappa_{2})$ such that $D _{\alpha}(h)\in L_{\alpha}([\kappa_{1}, \kappa_{2}])$ and $|h^{\prime}(x)|^{\gamma}$ be a $GA$ -convex function on $[\kappa_{1}, \kappa_{2}]$ . Then the inequality

$\begin{equation} \left|\kappa_{2}^{2\alpha}h(\kappa_{2})-\kappa_{1}^{2\alpha}h(\kappa_{1}) -2\alpha\int^{\kappa_{2}}_{\kappa_{1}}x^{\alpha}h(x)d_{\alpha}x\right| \end{equation}$

(2.7)

$\begin{equation*} \leq(\log\kappa_{2}-\log\eta)^{1-\frac{1}{\gamma}} (L(\kappa_{2}^{(2\alpha+1)}, \eta^{(2\alpha+1)}))^{1-\frac{1}{\gamma}} \end{equation*}$

$\begin{equation*} \times\left(\frac{|h^{\prime}(\kappa_{2})|^{\gamma}\left[\kappa_{2}^{2\alpha+1} -L(\eta^{2\alpha+1}, \kappa_{2}^{2\alpha+1})\right] +|h^{\prime}(\eta)|^{\gamma}\left[L(\eta^{2\alpha+1}, \kappa_{2}^{2\alpha+1}) -\eta^{2\alpha+1}\right]}{2\alpha+1}\right)^{\frac{1}{\gamma}} \end{equation*}$

$\begin{equation*} +(\log\eta-\log\kappa_{1})^{1-\frac{1}{\gamma}}(L(\eta^{(2\alpha+1)}, \kappa_{1}^{(2\alpha+1)}))^{1-\frac{1}{\gamma}} \end{equation*}$

$\begin{equation*} \times\left(\frac{|h^{\prime}(\eta)|^{\gamma}\left[\eta^{2\alpha+1}-L(\kappa_{1}^{2\alpha+1}), \eta^{2\alpha+1}\right] +|h^{\prime}(\kappa_{1})|^{\gamma}\left[L(\kappa_{1}^{2\alpha+1}, \eta^{2\alpha+1}) -\kappa_{1}^{2\alpha+1}\right]}{2\alpha+1}\right)^{\frac{1}{\gamma}} \end{equation*}$

holds for any $\eta\in [\kappa_{1}, \kappa_{2}]$ .

Proof. From the $GA$ -convexity of $|h^{\prime}|^{\gamma}$ , power mean inequality, the property of the modulus and Lemma 2.1, one has

$\begin{equation*} \left|\kappa_{2}^{2\alpha}h(\kappa_{2})-\kappa_{1}^{2\alpha}h(\kappa_{1}) -2\alpha\int^{\kappa_{2}}_{\kappa_{1}}x^{\alpha}h(x)d_{\alpha}x\right| \end{equation*}$

$\begin{equation*} \leq(\log\kappa_{2}-\log\eta)\int_{0}^{1}(\kappa_{2}^{t}\eta^{1-t})^{2\alpha+1}|h^{\prime}(\kappa_{2}^{t}\eta^{1-t})|dt \end{equation*}$

$\begin{equation*} +(\log\eta-\log\kappa_{1})\int_{0}^{1}(\eta^{t}\kappa_{1}^{1-t})^{2\alpha+1}|h'(\eta^{t}\kappa_{1}^{1-t})|dt \end{equation*}$

$\begin{equation*} \leq(\log\kappa_{2}-\log\eta)\left(\int_{0}^{1}(\kappa_{2}^{t}\eta^{1-t})^{2\alpha+1}dt\right)^{1-\frac{1}{\gamma}}\left(\int_{0}^{1} (\kappa_{2}^{t}\eta^{1-t})^{2\alpha+1}|h^{\prime}(\kappa_{2}^{t}\eta^{1-t})|^{\gamma}dt\right)^{\frac{1}{\gamma}} \end{equation*}$

$\begin{equation*} \leq(\log\kappa_{2}-\log\eta)\left(\int_{0}^{1}(\kappa_{2}^{t}\eta^{1-t})^{2\alpha+1}dt\right)^{1-\frac{1}{\gamma}} \left(\int_{0}^{1}(\kappa_{2}^{t}\eta^{1-t})^{2\alpha+1}\left[t|h^{\prime}(\kappa_{2})|^{\gamma}+(1-t) |h^{\prime}(\eta)|^{\gamma}\right]dt\right)^{\frac{1}{\gamma}} \end{equation*}$

$\begin{equation*} +(\log\eta-\log\kappa_{1})\left(\int_{0}^{1}(\eta^{t}\kappa_{1}^{1-t})^{2\alpha+1}dt\right)^{1-\frac{1}{\gamma}} \left(\int_{0}^{1}(\eta^{t}\kappa_{1}^{1-t})^{2\alpha+1} \left[t|h^{\prime}(\eta)|^{\gamma }+(1-t)|h^{\prime}(\kappa_{1})|^{\gamma}\right]dt\right)^{\frac{1}{\gamma}} \end{equation*}$

$\begin{equation*} = (\log\kappa_{2}-\log\eta)^{1-\frac{1}{\gamma}}(L(\kappa_{2}^{(2\alpha+1)}, \eta^{(2\alpha+1)}))^{1-\frac{1}{\gamma}} \end{equation*}$

$\begin{equation*} \times\left(\frac{|h^{\prime}(\kappa_{2})|^{\gamma}\left[\kappa_{2}^{2\alpha+1}-L (\eta^{2\alpha+1}, \kappa_{2}^{2\alpha+1})\right] +|h^{\prime}(\eta)|^{\gamma}\left[L(\eta^{2\alpha+1}, \kappa_{2}^{2\alpha+1})-\eta^{2\alpha+1}\right]}{2\alpha+1}\right)^{\frac{1}{\gamma}} \end{equation*}$

$\begin{equation*} +(\log\eta-\log\kappa_{1})^{1-\frac{1}{\gamma}}(L(\eta^{(2\alpha+1)}, \kappa_{1}^{(2\alpha+1)}))^{1-\frac{1}{\gamma}} \end{equation*}$

$\begin{equation*} \times\left(\frac{|h^{\prime}(\eta)|^{\gamma}\left[\eta^{2\alpha+1}-L(\kappa_{1}^{2\alpha+1}), \eta^{2\alpha+1}\right] +|h^{\prime}(\kappa_{1})|^{\gamma}\left[L(\kappa_{1}^{2\alpha+1}, \eta^{2\alpha+1})-\kappa_{1}^{2\alpha+1}\right]} {2\alpha+1}\right)^{\frac{1}{\gamma}}. \end{equation*}$

Remark 2.7. Let $\alpha = 1$ . Then inequality (2.7) reduces to (1.7).

Theorem 2.7. Let $\kappa_{1}, \kappa_{2}\in(0, \infty)$ with $\kappa_{1} < \kappa_{2}$ , $\vartheta, \gamma > 1$ with $1/\vartheta+1/\gamma = 1$ , $\alpha \in (0, 1]$ , $h: [\kappa_{1}, \kappa_{2}]\rightarrow\mathbb{R}$ be an $\alpha$ -fractional differentiable function on $(\kappa_{1}, \kappa_{2})$ such that $D_{\alpha}(h)\in L_{\alpha}([\kappa_{1}, \kappa_{2}])$ and $|h^{\prime}(x)|^{\gamma}$ be a $GA$ -convex function on $[\kappa_{1}, \kappa_{2}]$ . Then the inequality

$\begin{equation} \left|\kappa_{2}^{2\alpha}h(\kappa_{2})-\kappa_{1}^{2\alpha}h(\kappa_{1}) -2\alpha\int^{\kappa_{2}}_{\kappa_{1}}x^{\alpha}h(x)d_{\alpha}x\right| \end{equation}$

(2.8)

$\begin{equation*} \leq\frac{(\log\kappa_{2}-\log\eta)^{1-\frac{1}{\gamma}}}{\vartheta^{\frac{1}{\gamma}}} \left(L\left(\kappa_{2}^{\frac{(\gamma-\vartheta)(2\alpha+1)}{\gamma-1}}, \eta^{\frac{(\gamma-\vartheta) (2\alpha+1)}{\gamma-1}}\right)\right)^{\frac{\gamma-1}{\gamma}}(A_{\gamma}(\kappa_{2}, \eta))^{\frac{1}{\gamma}} \end{equation*}$

$\begin{equation*} +\frac{(\log\eta-\log\kappa_{1})^{1-\frac{1}{\gamma}}}{\vartheta^{\frac{1}{\gamma}}} \left(L\left(\eta^{\frac{(\gamma-\vartheta)(2\alpha+1)}{\gamma-1}}, \kappa_{1}^{\frac{(\gamma-\vartheta) (2\alpha+1)}{\gamma-1}}\right)\right)^{\frac{\gamma-1} {\gamma}}(A_{\gamma}(\eta, \kappa_{1}))^{\frac{1}{\gamma}} \end{equation*}$

holds for any $\eta\in [\kappa_{1}, \kappa_{2}]$ , where

$\begin{equation*} A_{\gamma}(\kappa_{2}, \eta) = \frac{|h^{\prime}(\kappa_{2})|^\gamma\left[\kappa_{2}^{\gamma(2\alpha+1)} -L\left(\eta^{\gamma(2\alpha+1)}, \kappa_{2}^{\gamma(2\alpha+1)}\right)\right]+|h^{\prime}(\eta)|^\gamma \left[L\left(\eta^{\gamma(2\alpha+1)}, \kappa_{2}^{\gamma(2\alpha+1)}\right)-\eta^{\gamma(2\alpha+1)}\right]}{2\alpha+1}, \end{equation*}$

$\begin{equation*} A_{\gamma}(\eta, \kappa_{1}) = \frac{|h^{\prime}(\eta)|^\gamma\left[\eta^{\gamma(2\alpha+1)} -L\left(\kappa_{1}^{\gamma(2\alpha+1)}, \eta^{\gamma(2\alpha+1)}\right)\right]+|h^{\prime}(\kappa_{1})|^\gamma \left[L(\kappa_{1}^{\gamma(2\alpha+1)}, \eta^{\gamma(2\alpha+1)})-\kappa_{1}^{\gamma(2\alpha+1)}\right]}{2\alpha+1}. \end{equation*}$

Proof. It follows from Lemma 2.1, the $GA$ -convexity of $|h^{\prime}|^\gamma$ , power mean inequality, Hölder integral inequality and the property of the modulus that

$\begin{equation*} \left|\kappa_{2}^{2\alpha}h(\kappa_{2})-\kappa_{1}^{2\alpha}h(\kappa_{1}) -2\alpha\int^{\kappa_{2}}_{\kappa_{1}}x^{\alpha}h(x)d_{\alpha}x\right| \end{equation*}$

$\begin{equation*} \leq(\log\kappa_{2}-\log\eta)\int_{0}^{1}(\kappa_{2}^{t}\eta^{1-t})^{2\alpha+1}|h^{\prime}(\kappa_{2}^{t}\eta^{1-t})|dt \end{equation*}$

$\begin{equation*} +(\log\eta-\log\kappa_{1})\int_{0}^{1}(\eta^{t}\kappa_{1}^{1-t})^{2\alpha+1}|h^{\prime}(\eta^{t}\kappa_{1}^{1-t})|dt \end{equation*}$

$\begin{equation*} \leq(\log \kappa_{2}-\log\eta) \left(\int_{0}^{1}(\kappa_{2}^{(2\alpha+1)t} \eta^{(2\alpha+1)(1-t)})^{\frac{\gamma-\vartheta}{\gamma-1}}dt\right)^{\frac{\gamma-1}{\gamma}} \end{equation*}$

$\begin{equation*} \times\left(\int_{0}^{1}\left(\kappa_{2}^{(2\alpha+1)t}\eta^{(2\alpha+1)(1-t)}\right)^{\vartheta} |h^{\prime}(\kappa_{2}^{t}\eta^{1-t})|^{\gamma}dt\right)^{\frac{1}{\gamma}} \end{equation*}$

$\begin{equation*} +(\log\eta-\log\kappa_{1})\left(\int_{0}^{1}(\eta^{(2\alpha+1)t} \kappa_{1}^{(2\alpha+1)(1-t)})^{\frac{\gamma-\vartheta} {\gamma-1}}dt\right)^{\frac{\gamma-1}{\gamma}} \end{equation*}$

$\begin{equation*} \times\left(\int_{0}^{1}\left(\eta^{(2\alpha+1)t}\kappa_{1}^{(2\alpha+1)(1-t)}\right)^{\vartheta} |h^{\prime}(\eta^{t}\kappa_{1}^{1-t})|^{\gamma}dt\right)^{\frac{1}{\gamma}} \end{equation*}$

$\begin{equation*} \leq(\log\kappa_{2}-\log\eta)\left(\int_{0}^{1}(\kappa_{2}^{(2\alpha+1)t} \eta^{(2\alpha+1)(1-t)})^{\frac{\gamma-\vartheta}{\gamma-1}}dt\right)^{\frac{\gamma-1}{\gamma}} \end{equation*}$

$\begin{equation*} \times\left(\int_{0}^{1}\left(\kappa_{2}^{(2\alpha+1)t} \eta^{(2\alpha+1)(1-t)}\right)^{\vartheta}\left[t|h^{\prime}(\kappa_{2})|^{\gamma} +(1-t)|h^{\prime}(\eta)|^{\gamma}\right]dt\right)^{\frac{1}{\gamma}} \end{equation*}$

$\begin{equation*} +(\log\eta-\log\kappa_{1})\left(\int_{0}^{1}(\eta^{(2\alpha+1)t} \kappa_{1}^{(2\alpha+1)(1-t)})^{\frac{\gamma-\vartheta}{\gamma-1}}dt\right)^{\frac{\gamma-1}{\gamma}} \end{equation*}$

$\begin{equation*} \times\left(\int_{0}^{1}\left(\eta^{(2\alpha+1)t} \kappa_{1}^{(2\alpha+1)(1-t)}\right)^{\vartheta}\left[t|h^{\prime}(\eta)|^{\gamma} +(1-t)|h^{\prime}(\kappa_{1})|^{\gamma}\right]dt\right)^{\frac{1}{\gamma}} \end{equation*}$

$\begin{equation*} = \frac{(\log\kappa_{2}-\log\eta)^{1-\frac{1}{\gamma}}}{\vartheta^{\frac{1}{\gamma}}} \left(L\left(\kappa_{2}^{\frac{(\gamma-\vartheta)(2\alpha+1)}{\gamma-1}}, \eta^{\frac{(\gamma-\vartheta) (2\alpha+1)}{\gamma-1}}\right)\right)^{\frac{\gamma-1}{\gamma}}(A_{\gamma}(\kappa_{2}, \eta))^{\frac{1}{\gamma}} \end{equation*}$

$\begin{equation*} +\frac{(\log\eta-\log\kappa_{1})^{1-\frac{1}{\gamma}}}{\vartheta^{\frac{1}{\gamma}}} \left(L\left(\eta^{\frac{(\gamma-\vartheta)(2\alpha+1)}{\gamma-1}}, \kappa_{1}^{\frac{(\gamma-\vartheta) (2\alpha+1)}{\gamma-1}}\right)\right)^{\frac{\gamma-1}{\gamma}}(A_{\gamma}(\eta, \kappa_{1}))^{\frac{1}{\gamma}}. \end{equation*}$

Remark 2.8. Let $\alpha = 1$ . Then inequality (2.8) becomes (1.8).

Theorem 2.8. Let $\kappa_{1}, \kappa_{2}\in(0, \infty)$ with $\kappa_{1} < \kappa_{2}$ , $\gamma > 1$ , $\alpha \in (0, 1]$ , $h: [\kappa_{1}, \kappa_{2}]\rightarrow\mathbb{R}$ be an $\alpha$ -fractional differentiable function on $(\kappa_{1}, \kappa_{2})$ such that $D _{\alpha}(h)\in L_{\alpha}([\kappa_{1}, \kappa_{2}])$ and $|h^{\prime}(x)|^{\gamma}$ be a $GA$ -convex function on $[\kappa_{1}, \kappa_{2}]$ . Then the inequality

$\begin{equation} \left|\kappa_{2}^{2\alpha}h(\kappa_{2})-\kappa_{1}^{2\alpha}h(\kappa_{1}) -2\alpha\int^{\kappa_{2}}_{\kappa_{1}}x^{\alpha}h(x)d_{\alpha}x\right| \end{equation}$

(2.9)

$\begin{equation*} \leq(\log\kappa_{2}-\log\eta)^{1-\frac{1}{\gamma}}\left(L\left(\kappa_{2}^{\frac{\gamma(2\alpha+1)}{\gamma-1}}, \eta^{\frac{\gamma(2\alpha+1)} {\gamma-1}}\right)\right)^{1-\frac{1}{\gamma}}\left(A(|h^{\prime}(\kappa_{2})|^{\gamma}, |h^{\prime}(\eta)|^{\gamma})\right)^{\frac{1}{\gamma}} \end{equation*}$

$\begin{equation*} +(\log\eta-\log\kappa_{1})^{1-\frac{1}{\gamma}}\left(L\left(\eta^{\frac{\gamma(2\alpha+1)} {\gamma-1}}, \kappa_{1}^{\frac{\gamma(2\alpha+1)}{\gamma-1}}\right)\right)^{1-\frac{1}{\gamma}} \left(A\left(|h^{\prime}(\eta)|^{\gamma}, |h^{\prime}(\kappa_{1})|^{\gamma}\right)\right)^{\frac{1}{\gamma}} \end{equation*}$

holds for any $\eta\in [\kappa_{1}, \kappa_{2}]$ .

Proof. From Lemma 2.1, the $GG$ -convexity of $|h^{\prime}|^{\gamma}$ , Hölder inequality and the property of the modulus, we have

$\begin{equation*} \left|\kappa_{2}^{2\alpha}h(\kappa_{2})-\kappa_{1}^{2\alpha}h(\kappa_{1}) -2\alpha\int^{\kappa_{2}}_{\kappa_{1}}x^{\alpha}h(x)d_{\alpha}x\right| \end{equation*}$

$\begin{equation*} \leq(\log\kappa_{2}-\log\eta)\int_{0}^{1}(\kappa_{2}^{t}\eta^{1-t})^{2\alpha+1}|h^{\prime}(\kappa_{2}^{t}\eta^{1-t})|dt \end{equation*}$

$\begin{equation*} +(\log\eta-\log\kappa_{1})\int_{0}^{1}(\eta^{t}\kappa_{1}^{1-t})^{2\alpha+1}|h^{\prime}(\eta^{t}\kappa_{1}^{1-t})|dt \end{equation*}$

$\begin{equation*} \leq(\log\kappa_{2}-\log\eta)\left(\int_{0}^{1}(\kappa_{2}^{t}\eta^{1-t})^{2\alpha+1}dt\right)^{1-\frac{1} {\gamma}}\left(\int_{0}^{1}|h^{\prime}(\kappa_{2}^{t}\eta^{1-t})|^{\gamma}dt\right)^{\frac{1}{\gamma}} \end{equation*}$

$\begin{equation*} +(\log\eta-\log\kappa_{1})\left(\int_{0}^{1}(\eta^{t}\kappa_{1}^{1-t})^{2\alpha+1}dt\right)^{1-\frac{1} {\gamma}}\left(\int_{0}^{1}|h^{\prime}(\eta^{t}\kappa_{1}^{1-t})|^{\gamma}dt\right)^{\frac{1}{\gamma}} \end{equation*}$

$\begin{equation*} \leq(\log\kappa_{2}-\log\eta)\left(\int_{0}^{1}(\kappa_{2}^{t}\eta^{1-t})^{2\alpha+1}dt\right)^{1-\frac{1}{\gamma}} \left(\int_{0}^{1}\left[t|h^{\prime}(\kappa_{2})|^{\gamma }+(1-t)|h^{\prime}(\eta)|^{\gamma}\right]dt\right)^{\frac{1}{\gamma}} \end{equation*}$

$\begin{equation*} +(\log\eta-\log\kappa_{1})\left(\int_{0}^{1}(\eta^{t}\kappa_{1}^{1-t})^{2\alpha+1}dt\right)^{1-\frac{1}{\gamma}} \left(\int_{0}^{1}\left[t|h^{\prime}(\eta)|^{\gamma }+(1-t)|h^{\prime}(\kappa_{1})|^{\gamma}\right]dt\right)^{\frac{1}{\gamma}} \end{equation*}$

$\begin{equation*} = (\log\kappa_{2}-\log\eta)^{1-\frac{1}{\gamma}}\left(L\left(\kappa_{2}^{\frac{\gamma(2\alpha+1)}{\gamma-1}}, \eta^{\frac{\gamma(2\alpha+1)} {\gamma-1}}\right)\right)^{1-\frac{1}{\gamma}}\left(A(|h^{\prime}(\kappa_{2})|^{\gamma}, |h^{\prime}(\eta)|^{\gamma})\right)^{\frac{1}{\gamma}} \end{equation*}$

Remark 2.9. Let $\alpha = 1$ . Then inequality (2.9) leads to (1.9).

Theorem 2.9. Let $\kappa_{1}, \kappa_{2}\in(0, \infty)$ with $\kappa_{1} < \kappa_{2}$ , $\gamma > 1$ , $\alpha\in (0, 1]$ , $h:[\kappa_{1}, \kappa_{2}]\rightarrow\mathbb{R}$ be an $\alpha$ -fractional differentiable function on $(\kappa_{1}, \kappa_{2})$ such that $D _{\alpha}(h)\in L_{\alpha}([\kappa_{1}, \kappa_{2}])$ and $|h^{\prime}(x)|^{\gamma}$ be a $GA$ -convex function on $[\kappa_{1}, \kappa_{2}]$ . Then the inequality

$\begin{equation} \left|\kappa_{2}^{2\alpha}h(\kappa_{2})-\kappa_{1}^{2\alpha}h(\kappa_{1}) -2\alpha\int^{\kappa_{2}}_{\kappa_{1}}x^{\alpha}h(x)d_{\alpha}x\right| \end{equation}$

(2.10)

$\begin{equation*} \leq\frac{(\log\kappa_{2}-\log\eta)^{1-\frac{1}{\gamma}}}{\gamma^{\frac{1}{\gamma}}}B_{\gamma}(\kappa_{2}, \eta) +\frac{(\log\eta-\log\kappa_{1})^{1-\frac{1}{\gamma}}}{\gamma^{\frac{1}{\gamma}}}B_{\gamma}(\eta, \kappa_{1}) \end{equation*}$

holds for any $\eta\in [\kappa_{1}, \kappa_{2}]$ , where

$\begin{equation*} B_{\gamma}(\kappa_{2}, \eta) = \left(\frac{|h^{\prime}(\kappa_{2})|^\gamma\left[\kappa_{2}^{\gamma(2\alpha+1)} -L\left(\eta^{\gamma(2\alpha+1)}, \kappa_{2}^{\gamma(2\alpha+1)}\right)\right]+|h^{\prime}(\eta)|^\gamma \left[L(\eta^{\gamma(2\alpha+1)}, \kappa_{2}^{\gamma(2\alpha+1)})-\eta^{\gamma(\alpha+1)}\right]}{2\alpha+1}\right)^{\frac{1}{\gamma}}, \end{equation*}$

$\begin{equation*} B_{\gamma}(\eta, \kappa_{1}) = \left(\frac{|h^{\prime}(\eta)|^\gamma\left[\eta^{\gamma(2\alpha+1)} -L\left(\kappa_{1}^{\gamma(2\alpha+1)}, \eta^{\gamma(2\alpha+1)}\right)\right]+|h^{\prime}(\kappa_{1})|^\gamma \left[L(\kappa_{1}^{\gamma(2\alpha+1)}, \eta^{\gamma(2\alpha+1)})-\kappa_{1}^{\gamma(\alpha+1)}\right]} {2\alpha+1}\right)^{\frac{1}{\gamma}}. \end{equation*}$

Proof. It follows from Lemma 2.1, the $GA$ -convexity of $|h^{\prime}|^\gamma$ , power mean inequality and property of the modulus that

$\begin{equation*} \left|\kappa_{2}^{2\alpha}h(\kappa_{2})-\kappa_{1}^{2\alpha}h(\kappa_{1}) -2\alpha\int^{\kappa_{2}}_{\kappa_{1}}x^{\alpha}h(x)d_{\alpha}x\right| \end{equation*}$

$\begin{equation*} \leq(\log\kappa_{2}-\log\eta)\int_{0}^{1}(\kappa_{2}^{t}\eta^{1-t})^{2\alpha+1}|h^{\prime}(\kappa_{2}^{t}\eta^{1-t})|dt \end{equation*}$

$\begin{equation*} +(\log\eta-\log\kappa_{1})\int_{0}^{1}(\eta^{t}\kappa_{1}^{1-t})^{2\alpha+1}|h^{\prime}(\eta^{t}\kappa_{1}^{1-t})|dt \end{equation*}$

$\begin{equation*} \leq(\log\kappa_{2}-\log\eta)\left(\int_{0}^{1}dt\right)^{1-\frac{1}{\gamma}}\left(\int_{0}^{1}(\kappa_{2}^{t}\eta^{1-t})^{2\alpha+1} |h^{\prime}(\kappa_{2}^{t}\eta^{1-t})|^{\gamma}dt\right)^{\frac{1}{\gamma}} \end{equation*}$

$\begin{equation*} +(\log\eta-\log\kappa_{1})\left(\int_{0}^{1}dt\right)^{1-\frac{1}{\gamma}}\left(\int_{0}^{1} (\eta^{t}\kappa_{1}^{1-t})^{2\alpha+1}|h^{\prime}(\eta^{t}\kappa_{1}^{1-t})|^{\gamma}dt\right)^{\frac{1}{\gamma}} \end{equation*}$

$\begin{equation*} \leq(\log\kappa_{2}-\log\eta)\left(\int_{0}^{1}dt\right)^{1-\frac{1}{\gamma}} \left(\int_{0}^{1}(\kappa_{2}^{t}\eta^{1-t})^{2\alpha+1}\left[t|h^{\prime}(\kappa_{2})|^{\gamma} +(1-t)|h^{\prime}(\eta)|^{\gamma}\right]dt\right)^{\frac{1}{\gamma}} \end{equation*}$

$\begin{equation*} +(\log\eta-\log\kappa_{1})\left(\int_{0}^{1}dt\right)^{1-\frac{1}{\gamma}}\left(\int_{0}^{1}(\eta^{t}\kappa_{1}^{1-t})^{2\alpha+1} \left[t|h^{\prime}(\eta)|^{\gamma }+(1-t)|h^{\prime}(\kappa_{1})|^{\gamma}\right]dt\right)^{\frac{1}{\gamma}} \end{equation*}$

$\begin{equation*} = \frac{(\log\kappa_{2}-\log\eta)^{1-\frac{1}{\gamma}}}{\gamma^{\frac{1}{\gamma}}}B_{\gamma}(\kappa_{2}, \eta) +\frac{(\log\eta-\log\kappa_{1})^{1-\frac{1}{\gamma}}}{\gamma^{\frac{1}{\gamma}}}B_{\gamma}(\eta, \kappa_{1}). \end{equation*}$

Remark 2.10. Let $\alpha = 1$ . Then inequality (2.10) reduces to (1.10).

3. Conclusion

We have generalized the Hermite-Hadamard type inequalities for $GG$ - and $GA$ -convex functions established by Ardıç, Akdemir and Yıdız in ^[61] to the conformable fractional integrals. Our ideas and approach may lead to a lot of follow-up research.

Acknowledgements

The authors would like to thank the anonymous referees for their valuable comments and suggestions, which led to considerable improvement of the article.

The work was supported by the Natural Science Foundation of China (Grant Nos. 61673169, 11971142, 11701176, 11626101, 11601485).

Conflict of interest

The authors declare no conflict of interest.

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AIMS Mathematics

Conformable fractional integral inequalities for GG- and GA-convex functions

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Acknowledgements

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Conformable fractional integral inequalities for GG- and GA-convex functions

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