Evaluation of the bioenergy potential of agricultural and agroindustrial waste generated in southeastern Mexico

Nathaly A. Díaz Molina; José A. Sosa Olivier; José R. Laines Canepa; Rudy Solis Silvan; Donato A. Figueiras Jaramillo; Nathaly A. Díaz Molina; José A. Sosa Olivier; José R. Laines Canepa; Rudy Solis Silvan; Donato A. Figueiras Jaramillo

doi:10.3934/energy.2024046

AIMS Energy

2024, Volume 12, Issue 5: 984-1009. doi: 10.3934/energy.2024046

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Research article Special Issues

Evaluation of the bioenergy potential of agricultural and agroindustrial waste generated in southeastern Mexico

1.
Pilot Plant 3 solid waste and atmospheric treatment, Academic Division of Biological Sciences, Juarez Autonomous University of Tabasco, Mexico
2.
Comalcalco Higher Technological Institute, Comalcalco, Tabasco, Mexico
3.
Coprocessing Laboratory, Geocycle® SA. of CV. Macuspana, Mexico

Received: 22 May 2024 Revised: 31 July 2024 Accepted: 12 August 2024 Published: 29 August 2024

The generation of large volumes of agricultural and agroindustrial waste in the state of Tabasco represents a significant waste management challenge. We aimed to determine the bioenergy potential of five types of biomasses: Banana rachis, coconut shell, cocoa pod husk, sugarcane bagasse, and palm kernel shell, generated in agricultural and agroindustrial processes. This research involved characterizing and evaluating the energy quality of these biomasses by determining their calorific values and assessing their viability as fuel alternative sources. Additionally, we explored these biomasses' calorific value potential to reduce the inadequate disposal of wastes, reduce environmental impact, and provide alternative uses for these materials, which are typically discarded or have limited added value in the southeast region. The yield of waste generation per amount of production was estimated, with cocoa pod husk biomass and sugarcane bagasse, banana rachis, coconut shell, and palm kernel shell generating 0.685, 0.283, 0.16, 0.135, and 0.0595 kg of biomass per kg of crop, respectively. The bioenergy potential was evaluated through direct measurements using a calorimeter bomb, and indirect measurements using stoichiometric calculations. Four stoichiometric methods based on predictive equations were employed to determine the energy content of the biomasses from their elemental composition (Dulong, Friedl, Channiwala, Boie). The biomasses with the highest calorific values were coconut shell and cocoa pod husk, with values of 16.47 ± 0.24 and 16.02 ± 1.54 MJ/kg, respectively. Moreover, banana rachis had the lowest calorific value at 13.68 ± 3.22 MJ/kg. The calorific values of the sugarcane bagasse and palm kernel shell were 13.91 ± 0.98 and 15.29 ± 1.02, respectively. The factorial experimental design and statistical analysis revealed trends and magnitudes in the evaluation of energy determination methods and types of waste. The predictive equation of Dulong showed the highest similarity to the experimental values, especially for coconut shell (16.02 ± 0.08 MJ/kg). The metal content in biomasses such as palm kernel shell and coconut shell were below the limits established in ISO 17225:2014. Finally, our results indicated that coconut shell has superior characteristics for potential use as an alternative fuel, whereas banana rachis requires exploring alternative utilization options.

Keywords:

Citation: Nathaly A. Díaz Molina, José A. Sosa Olivier, José R. Laines Canepa, Rudy Solis Silvan, Donato A. Figueiras Jaramillo. Evaluation of the bioenergy potential of agricultural and agroindustrial waste generated in southeastern Mexico[J]. AIMS Energy, 2024, 12(5): 984-1009. doi: 10.3934/energy.2024046

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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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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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