Exact solutions and conservation laws for the time-fractional nonlinear dirac system: A study of classical and nonclassical lie symmetries

Farzaneh Alizadeh; Samad Kheybari; Kamyar Hosseini; Farzaneh Alizadeh; Samad Kheybari; Kamyar Hosseini

doi:10.3934/math.2025532

AIMS Mathematics

2025, Volume 10, Issue 5: 11757-11782. doi: 10.3934/math.2025532

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Research article Special Issues

Exact solutions and conservation laws for the time-fractional nonlinear dirac system: A study of classical and nonclassical lie symmetries

1.
Department of Mathematics, Near East University TRNC, Mersin10, Nicosia 99138, Turkey
2.
Mathematics Research Center, Near East University TRNC, Mersin 10, Nicosia 99138, Turkey
3.
Research Center of Applied Mathematics, Khazar University, Baku, Azerbaijan
4.
Faculty of Art and Science, University of Kyrenia, TRNC, Mersin 10, Turkey
5.
Faculty of Engineering and Natural Sciences, Istanbul Okan University, Istanbul, Turkey

Received: 31 March 2025 Revised: 08 May 2025 Accepted: 13 May 2025 Published: 21 May 2025
MSC : 35R11, 70H33, 76M60

Time-fractional Dirac-type systems arise in quantum field theory, plasma physics, and condensed matter systems where fractional calculus captures nonlocal interactions. In this study, we employ classical and nonclassical Lie symmetry methods to analyze the underlying symmetry structure of the system. By deriving infinitesimal generators and performing similarity reductions, we transform the governing fractional partial differential equations (FPDEs) into fractional ordinary differential equations (FODEs). Exact solutions are constructed using the power series method. Furthermore, we establish conservation laws in the fractional setting, ensuring the physical consistency of the system. Our findings offer new insights into the interplay among symmetry, conservation principles, and exact solutions in fractional quantum field models, expanding the analytical toolkit for studying nonlinear relativistic wave equations.

Keywords:

Citation: Farzaneh Alizadeh, Samad Kheybari, Kamyar Hosseini. Exact solutions and conservation laws for the time-fractional nonlinear dirac system: A study of classical and nonclassical lie symmetries[J]. AIMS Mathematics, 2025, 10(5): 11757-11782. doi: 10.3934/math.2025532

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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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4.	Humaira Kalsoom, Muhammad Idrees, Artion Kashuri, Muhammad Uzair Awan, Yu-Ming Chu, Some New $(p_1p_2,q_1q_2)$ -Estimates of Ostrowski-type integral inequalities via n-polynomials s-type convexity, 2020, 5, 2473-6988, 7122, 10.3934/math.2020456
5.	Ming-Bao Sun, Yu-Ming Chu, Inequalities for the generalized weighted mean values of g-convex functions with applications, 2020, 114, 1578-7303, 10.1007/s13398-020-00908-1
6.	Shuang-Shuang Zhou, Saima Rashid, Muhammad Aslam Noor, Khalida Inayat Noor, Farhat Safdar, Yu-Ming Chu, New Hermite-Hadamard type inequalities for exponentially convex functions and applications, 2020, 5, 2473-6988, 6874, 10.3934/math.2020441
7.	Ming-Bao Sun, Xin-Ping Li, Sheng-Fang Tang, Zai-Yun Zhang, Schur Convexity and Inequalities for a Multivariate Symmetric Function, 2020, 2020, 2314-8896, 1, 10.1155/2020/9676231
8.	Muhammad Uzair Awan, Sadia Talib, Artion Kashuri, Muhammad Aslam Noor, Yu-Ming Chu, Estimates of quantum bounds pertaining to new q-integral identity with applications, 2020, 2020, 1687-1847, 10.1186/s13662-020-02878-5
9.	Xi-Fan Huang, Miao-Kun Wang, Hao Shao, Yi-Fan Zhao, Yu-Ming Chu, Monotonicity properties and bounds for the complete p-elliptic integrals, 2020, 5, 2473-6988, 7071, 10.3934/math.2020453
10.	Saima Rashid, Aasma Khalid, Gauhar Rahman, Kottakkaran Sooppy Nisar, Yu-Ming Chu, On New Modifications Governed by Quantum Hahn’s Integral Operator Pertaining to Fractional Calculus, 2020, 2020, 2314-8896, 1, 10.1155/2020/8262860
11.	Muhammad Uzair Awan, Sadia Talib, Artion Kashuri, Muhammad Aslam Noor, Khalida Inayat Noor, Yu-Ming Chu, A new q-integral identity and estimation of its bounds involving generalized exponentially μ-preinvex functions, 2020, 2020, 1687-1847, 10.1186/s13662-020-03036-7
12.	Sabir Hussain, Javairiya Khalid, Yu Ming Chu, Some generalized fractional integral Simpson’s type inequalities with applications, 2020, 5, 2473-6988, 5859, 10.3934/math.2020375
13.	Eze R. Nwaeze, Muhammad Adil Khan, Yu-Ming Chu, Fractional inclusions of the Hermite–Hadamard type for m-polynomial convex interval-valued functions, 2020, 2020, 1687-1847, 10.1186/s13662-020-02977-3
14.	Muhammad Uzair Awan, Sadia Talib, Muhammad Aslam Noor, Yu-Ming Chu, Khalida Inayat Noor, Some Trapezium-Like Inequalities Involving Functions Having Strongly n-Polynomial Preinvexity Property of Higher Order, 2020, 2020, 2314-8896, 1, 10.1155/2020/9154139
15.	Ling Zhu, New Cusa-Huygens type inequalities, 2020, 5, 2473-6988, 5320, 10.3934/math.2020341
16.	Jian-Mei Shen, Saima Rashid, Muhammad Aslam Noor, Rehana Ashraf, Yu-Ming Chu, Certain novel estimates within fractional calculus theory on time scales, 2020, 5, 2473-6988, 6073, 10.3934/math.2020390
17.	Hu Ge-JiLe, Saima Rashid, Muhammad Aslam Noor, Arshiya Suhail, Yu-Ming Chu, Some unified bounds for exponentially $tgs$ -convex functions governed by conformable fractional operators, 2020, 5, 2473-6988, 6108, 10.3934/math.2020392
18.	Tie-Hong Zhao, Zai-Yin He, Yu-Ming Chu, On some refinements for inequalities involving zero-balanced hypergeometric function, 2020, 5, 2473-6988, 6479, 10.3934/math.2020418
19.	Sadia Khalid, Josip Pečarić, Refinements of some Hardy–Littlewood–Pólya type inequalities via Green’s functions and Fink’s identity and related results, 2020, 2020, 1029-242X, 10.1186/s13660-020-02498-3
20.	Li Xu, Yu-Ming Chu, Saima Rashid, A. A. El-Deeb, Kottakkaran Sooppy Nisar, On New Unified Bounds for a Family of Functions via Fractionalq-Calculus Theory, 2020, 2020, 2314-8896, 1, 10.1155/2020/4984612
21.	Arshad Iqbal, Muhammad Adil Khan, Noor Mohammad, Eze R. Nwaeze, Yu-Ming Chu, Revisiting the Hermite-Hadamard fractional integral inequality via a Green function, 2020, 5, 2473-6988, 6087, 10.3934/math.2020391
22.	Thabet Abdeljawad, Saima Rashid, Zakia Hammouch, İmdat İşcan, Yu-Ming Chu, Some new Simpson-type inequalities for generalized p-convex function on fractal sets with applications, 2020, 2020, 1687-1847, 10.1186/s13662-020-02955-9
23.	Shu-Bo Chen, Saima Rashid, Muhammad Aslam Noor, Zakia Hammouch, Yu-Ming Chu, New fractional approaches for n-polynomial P-convexity with applications in special function theory, 2020, 2020, 1687-1847, 10.1186/s13662-020-03000-5
24.	Humaira Kalsoom, Muhammad Idrees, Dumitru Baleanu, Yu-Ming Chu, New Estimates of q1q2-Ostrowski-Type Inequalities within a Class of n-Polynomial Prevexity of Functions, 2020, 2020, 2314-8896, 1, 10.1155/2020/3720798
25.	Saad Ihsan Butt, Muhammad Umar, Saima Rashid, Ahmet Ocak Akdemir, Yu-Ming Chu, New Hermite–Jensen–Mercer-type inequalities via k-fractional integrals, 2020, 2020, 1687-1847, 10.1186/s13662-020-03093-y
26.	Yu-Ming Chu, Muhammad Uzair Awan, Sadia Talib, Muhammad Aslam Noor, Khalida Inayat Noor, New post quantum analogues of Ostrowski-type inequalities using new definitions of left–right $(p,q)$ -derivatives and definite integrals, 2020, 2020, 1687-1847, 10.1186/s13662-020-03094-x
27.	Imran Abbas Baloch, Aqeel Ahmad Mughal, Yu-Ming Chu, Absar Ul Haq, Manuel De La Sen, A variant of Jensen-type inequality and related results for harmonic convex functions, 2020, 5, 2473-6988, 6404, 10.3934/math.2020412
28.	Thabet Abdeljawad, Saima Rashid, A. A. El-Deeb, Zakia Hammouch, Yu-Ming Chu, Certain new weighted estimates proposing generalized proportional fractional operator in another sense, 2020, 2020, 1687-1847, 10.1186/s13662-020-02935-z
29.	Hu Ge-JiLe, Saima Rashid, Fozia Bashir Farooq, Sobia Sultana, Shanhe Wu, Some Inequalities for a New Class of Convex Functions with Applications via Local Fractional Integral, 2021, 2021, 2314-8888, 1, 10.1155/2021/6663971
30.	Yu‐ming Chu, Saima Rashid, Jagdev Singh, A novel comprehensive analysis on generalized harmonically ψ ‐convex with respect to Raina's function on fractal set with applications , 2021, 0170-4214, 10.1002/mma.7346
31.	Xue Wang, Absar ul Haq, Muhammad Shoaib Saleem, Sami Ullah Zakir, Mohsan Raza, The Strong Convex Functions and Related Inequalities, 2022, 2022, 2314-8888, 1, 10.1155/2022/4056201
32.	SAIMA RASHID, AASMA KHALID, YELIZ KARACA, YU-MING CHU, REVISITING FEJÉR–HERMITE–HADAMARD TYPE INEQUALITIES IN FRACTAL DOMAIN AND APPLICATIONS, 2022, 30, 0218-348X, 10.1142/S0218348X22401338
33.	Waewta Luangboon, Kamsing Nonlaopon, Jessada Tariboon, Sotiris K. Ntouyas, Simpson- and Newton-Type Inequalities for Convex Functions via (p,q)-Calculus, 2021, 9, 2227-7390, 1338, 10.3390/math9121338
34.	Waewta Luangboon, Kamsing Nonlaopon, Jessada Tariboon, Sotiris K. Ntouyas, Hüseyin Budak, On generalizations of some integral inequalities for preinvex functions via $(p,q)$ -calculus, 2022, 2022, 1029-242X, 10.1186/s13660-022-02896-9
35.	Khuram Ali Khan, Saeeda Fatima, Ammara Nosheen, Rostin Matendo Mabela, Kenan Yildirim, New Developments of Hermite–Hadamard Type Inequalities via s-Convexity and Fractional Integrals, 2024, 2024, 2314-4785, 1, 10.1155/2024/1997549
36.	Ghada AlNemer, Samir H. Saker, Gehad M. Ashry, Mohammed Zakarya, Haytham M. Rezk, Mohammed R. Kenawy, Some Hardy's inequalities on conformable fractional calculus, 2024, 57, 2391-4661, 10.1515/dema-2024-0027
37.	Tahir Ullah Khan, Muhammad Adil Khan, Hermite-Hadamard inequality for new generalized conformable fractional operators, 2021, 6, 2473-6988, 23, 10.3934/math.2021002

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