Use of ketamine for depression and suicidality in cancer and terminal patients: Review of current data

Aderonke Oyetunji; Christian Huelga; Kailee Bunte; Rachel Tao; Val Bellman; Aderonke Oyetunji; Christian Huelga; Kailee Bunte; Rachel Tao; Val Bellman

doi:10.3934/publichealth.2023043

AIMS Public Health

2023, Volume 10, Issue 3: 610-626. doi: 10.3934/publichealth.2023043

Previous Article Next Article

Review

Use of ketamine for depression and suicidality in cancer and terminal patients: Review of current data

1.
University of Missouri Kansas City, Psychiatry Residency Training Program, Kansas City, MO
2.
Kansas City University, College of Osteopathic Medicine, Kansas City, MO
3.
University of Missouri Kansas City, School of Medicine, Kansas City, MO

Received: 29 May 2023 Revised: 04 July 2023 Accepted: 05 July 2023 Published: 21 July 2023

Depression and suicidality are significant challenges faced by cancer patients, particularly those in advanced stages of the disease or nearing the end of life. Conventional antidepressant therapies often have limited effectiveness or delayed onset of action, making the exploration of alternative treatments crucial. The use of ketamine as a potential treatment for depression and suicidality in cancer and terminal patients has gained considerable attention in recent years. This review article aims to provide a comprehensive analysis of the current data regarding the efficacy and safety of ketamine in this specific population. This review presents an overview of clinical trials and case studies investigating the use of ketamine in this population. It explores the effectiveness of ketamine as a standalone treatment or in combination with other interventions. Furthermore, the article addresses the limitations and future directions of research in this field. It highlights the need for larger, well-controlled studies with long-term follow-up to establish the efficacy, safety and optimal treatment parameters of ketamine for depression and suicidality in palliative care.

Keywords:

Citation: Aderonke Oyetunji, Christian Huelga, Kailee Bunte, Rachel Tao, Val Bellman. Use of ketamine for depression and suicidality in cancer and terminal patients: Review of current data[J]. AIMS Public Health, 2023, 10(3): 610-626. doi: 10.3934/publichealth.2023043

Related Papers:

[1]	Yousaf Khurshid, Muhammad Adil Khan, Yu-Ming Chu . Conformable integral version of Hermite-Hadamard-Fejér inequalities via η-convex functions. AIMS Mathematics, 2020, 5(5): 5106-5120. doi: 10.3934/math.2020328
[2]	M. Emin Özdemir, Saad I. Butt, Bahtiyar Bayraktar, Jamshed Nasir . Several integral inequalities for (α, s,m)-convex functions. AIMS Mathematics, 2020, 5(4): 3906-3921. doi: 10.3934/math.2020253
[3]	Muhammad Uzair Awan, Nousheen Akhtar, Artion Kashuri, Muhammad Aslam Noor, Yu-Ming Chu . 2D approximately reciprocal ρ-convex functions and associated integral inequalities. AIMS Mathematics, 2020, 5(5): 4662-4680. doi: 10.3934/math.2020299
[4]	Yue Wang, Ghulam Farid, Babar Khan Bangash, Weiwei Wang . Generalized inequalities for integral operators via several kinds of convex functions. AIMS Mathematics, 2020, 5(5): 4624-4643. doi: 10.3934/math.2020297
[5]	Hengxiao Qi, Muhammad Yussouf, Sajid Mehmood, Yu-Ming Chu, Ghulam Farid . Fractional integral versions of Hermite-Hadamard type inequality for generalized exponentially convexity. AIMS Mathematics, 2020, 5(6): 6030-6042. doi: 10.3934/math.2020386
[6]	Ghulam Farid, Saira Bano Akbar, Shafiq Ur Rehman, Josip Pečarić . Boundedness of fractional integral operators containing Mittag-Leffler functions via (s,m)-convexity. AIMS Mathematics, 2020, 5(2): 966-978. doi: 10.3934/math.2020067
[7]	Gou Hu, Hui Lei, Tingsong Du . Some parameterized integral inequalities for p-convex mappings via the right Katugampola fractional integrals. AIMS Mathematics, 2020, 5(2): 1425-1445. doi: 10.3934/math.2020098
[8]	Hasan Kara, Hüseyin Budak, Mehmet Eyüp Kiriş . On Fejer type inequalities for co-ordinated hyperbolic ρ-convex functions. AIMS Mathematics, 2020, 5(5): 4681-4701. doi: 10.3934/math.2020300
[9]	Tekin Toplu, Mahir Kadakal, İmdat İşcan . On n-Polynomial convexity and some related inequalities. AIMS Mathematics, 2020, 5(2): 1304-1318. doi: 10.3934/math.2020089
[10]	Naila Mehreen, Matloob Anwar . Some inequalities via Ψ-Riemann-Liouville fractional integrals. AIMS Mathematics, 2019, 4(5): 1403-1415. doi: 10.3934/math.2019.5.1403

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.

Acknowledgments

This study is not funded by any agency and is being conducted by the authors independently.

Conflict of Interest

The authors declare no conflict of interest.

References

[1]	Polsky D, Doshi JA, Marcus S, et al. (2005) Long-term risk for depressive symptoms after a medical diagnosis. Arch Intern Med 165: 1260-1266. https://doi.org/10.1001/archinte.165.11.1260
[2]	Fan CY, Chao HL, Lin CS, et al. (2018) Risk of depressive disorder among patients with head and neck cancer: A nationwide population-based study. Head Neck 40: 312-323. https://doi.org/10.1002/hed.24961
[3]	Rasic DT, Belik SL, Bolton JM, et al. (2008) Cancer, mental disorders, suicidal ideation and attempts in a large community sample. Psychooncology 17: 660-667. https://doi.org/10.1002/pon.1292
[4]	Honda K, Goodwin RD (2004) Cancer and mental disorders in a national community sample: findings from the national comorbidity survey. Psychother Psychosom 73: 235-242. https://doi.org/10.1159/000077742
[5]	(2022) APADiagnostic and statistical manual of mental disorders. Washington: American Psychiatric Association Publishing 187-188.
[6]	Miller K, Massie MJ (2019) Oncology. Washington: American Psychiatric Association Publishing 625.
[7]	Sullivan DR, Forsberg CW, Ganzini L, et al. (2016) Depression symptom trends and health domains among lung cancer patients in the CanCORS study. Lung Cancer 100: 102-109. https://doi.org/10.1016/j.lungcan.2016.08.008
[8]	McIntyre RS (2010) When should you move beyond first-line therapy for depression?. J Clin Psychiatry 1: 16-20. https://doi.org/10.4088/JCP.9104su1c.03
[9]	Sanacora G, Frye MA, McDonald W, et al. (2017) A consensus statement on the use of ketamine in the treatment of mood disorders. JAMA Psychiatry 74: 399-405. https://doi.org/10.1001/jamapsychiatry.2017.0080
[10]	Zhang Y, Ye F, Zhang T, et al. (2021) Structural basis of ketamine action on human NMDA receptors. Nature 596: 301-305. https://doi.org/10.1038/s41586-021-03769-9
[11]	Orhurhu VJ, Vashisht R, Claus LE, et al. Ketamine Toxicity (2023). Available from: https://www.ncbi.nlm.nih.gov/books/NBK541087/
[12]	Rosenblat JD, Carvalho AF, Li M, et al. (2019) Oral ketamine for depression: a systematic review. J Clin Psychiatry 80: 13514. https://doi.org/10.4088/JCP.18r12475
[13]	Dean RL, Hurducas C, Hawton K, et al. (2021) Ketamine and other glutamate receptor modulators for depression in adults with unipolar major depressive disorder. Cochrane Database Syst Rev 9: CD011612. https://doi.org/10.1002/14651858.CD011612
[14]	Andrade Chittaranjan (2017) Ketamine for depression, 3: Does chirality matter?. J Clin Psychiatry 78: e674-e677. https://doi.org/10.4088/JCP.17f11681
[15]	Derakhshanian S, Zhou M, Rath A, et al. (2021) Role of ketamine in the treatment of psychiatric disorders. Health Psychol Res 9: 25091. https://doi.org/10.52965/001c.25091
[16]	Ignácio ZM, Réus GZ, Arent CO, et al. (2016) New perspectives on the involvement of mTOR in depression as well as in the action of antidepressant drugs. Br J Clin Pharmacol 82: 1280-1290. https://doi.org/10.1111/bcp.12845
[17]	Tang L, He Y, Pang Y, et al. (2022) Suicidal ideation in advanced cancer patients without major depressive disorder. Psychooncology 31: 1941-1950. https://doi.org/10.1002/pon.6033
[18]	Sauer C, Grapp M, Bugaj TJ, et al. (2022) Suicidal ideation in patients with cancer: Its prevalence and results of structural equation modelling. Eur J Cancer Care (Engl) 31: e13650. https://doi.org/10.1111/ecc.13650
[19]	Manojna Konda, Rohan Sharma, Arya Mariam Roy, et al. (2019) Risk factors associated with suicide in patients with prostate cancer in the United States. J Clin Onc 37: 84. https://doi.org/10.1200/JCO.2019.37.31_suppl.84
[20]	Hopko DR, Armento ME, Robertson SM, et al. (2011) Brief behavioral activation and problem-solving therapy for depressed breast cancer patients: randomized trial. J Consult Clin Psychol 79: 834-849. https://doi.org/10.1037/a0025450
[21]	Leal-Hernández D A, Sandoval L, Palacios-Espinosa X, et al. (2014) Proposed scales for measuring suicidal ideation in adult cancer patients. Open J Med Psychology 3: 79-86. https://doi.org/10.4236/ojmp.2014.31010
[22]	Tanriverdi D, Cuhadar D, Ciftci S (2014) Does the impairment of functional life increase the probability of suicide in cancer patients?. Asian Pac J Cancer Prev 15: 9549-9553. https://doi.org/10.7314/apjcp.2014.15.21.9549
[23]	Men VY, Emery CR, Lam TC, et al. (2022) Suicidal/self-harm behaviors among cancer patients: a population-based competing risk analysis. Psychol Med 52: 2342-2351. https://doi.org/10.1017/S0033291720004250
[24]	Hagezom HM, Amare T, Hibdye G, et al. (2021) Magnitude and associated factors of suicidal ideation among cancer patients at Ayder Comprehensive Specialized Hospital, Mekelle, Ethiopia, 2019: Cross-sectional study. Cancer Manag Res 13: 4341-4350. https://doi.org/10.2147/CMAR.S268669
[25]	Pitman A, Suleman S, Hyde N, et al. (2018) Depression and anxiety in patients with cancer. BMJ 361: k1415. https://doi.org/10.1136/bmj.k1415
[26]	Ristevska-Dimitrovska G, Stefanovski P, Smichkoska S, et al. (2015) Depression and resilience in breast cancer patients. Open Access Maced J Med Sci 3: 661-665. https://doi.org/10.3889/oamjms.2015.119
[27]	Nakamura Y, Kanemoto E, Kajizono M, et al. (2017) Investigation of Mental Disorders in Lung Cancer Outpatients: A Retrospective Analysis. Yakugaku Zasshi 137: 241-246. https://doi.org/10.1248/yakushi.16-00226
[28]	Tosic Golubovic S, Binic I, Krtinic D, et al. (2022) Risk factors and predictive value of depression and anxiety in cervical cancer patients. Medicina (Kaunas) 58: 507. https://doi.org/10.3390/medicina58040507
[29]	Barrera I, Spiegel D (2014) Review of psychotherapeutic interventions on depression in cancer patients and their impact on disease progression. Int Rev Psychiatry 26: 31-43. https://doi.org/10.3109/09540261.2013.864259
[30]	Huang RW, Chang KP, Marchi F, et al. (2022) The impact of depression on survival of head and neck cancer patients: A population-based cohort study. Front Oncol 12: 871915. https://doi.org/10.3389/fonc.2022.871915
[31]	Widiyono W, Setiyarini S, Effendy C (2019) Self-selected individual music therapy for depression during hospitalization for cancer patients: Randomized controlled clinical trial study. Indones J Cancer 13: 59-68. https://doi.org/10.33371/ijoc.v13i3.632
[32]	Sherrill C, Smith M, Mascoe C, et al. (2017) Effect of treating depressive disorders on mortality of cancer patients. Cureus 9: e1740. https://doi.org/10.7759/cureus.1740
[33]	Lloyd-Williams M, Dennis M, Taylor F (2004) A prospective study to determine the association between physical symptoms and depression in patients with advanced cancer. Palliat Med 18: 558-563. https://doi.org/10.1191/0269216304pm923oa
[34]	Pinquart M, Duberstein PR (2010) Depression and cancer mortality: a meta-analysis. Psychol Med 40: 1797-1810. https://doi.org/10.1017/S0033291709992285
[35]	Rodin G, Lloyd N, Katz M, et al. (2007) The treatment of depression in cancer patients: a systematic review. Supportive Care in Cancer 15: 123-136. https://doi.org/10.1007/s00520-006-0145-3
[36]	Lloyd-Williams M, Payne S, Reeve J, et al. (2013) Antidepressant medication in patients with advanced cancer--an observational study. QJM 106: 995-1001. https://doi.org/10.1093/qjmed/hct133
[37]	Olin J, Masand P (1996) Psychostimulants for depression in hospitalized cancer patients. Psychosomatics 37: 57-62. https://doi.org/10.1016/S0033-3182(96)71599-2
[38]	Stefanczyk-Sapieha L, Oneschuk D, Demas M (2008) Intravenous ketamine “burst” for refractory depression in a patient with advanced cancer. J Palliat Med 11: 1268-1271. https://doi.org/10.1089/jpm.2008.9828
[39]	Irwin SA, Iglewicz A, Nelesen RA, et al. (2013) Daily oral ketamine for the treatment of depression and anxiety in patients receiving hospice care: A 28-day open-label proof-of-concept trial. J Palliat Med 16: 958-965. https://doi.org/10.1089/jpm.2012.0617
[40]	Irwin SA, Iglewicz A (2010) Oral ketamine for the rapid treatment of depression and anxiety in patients receiving hospice care. J Palliat Med 13: 903-908. https://doi.org/10.1089/jpm.2010.9808
[41]	Iglewicz A, Morrison K, Nelesen RA, et al. (2015) Ketamine for the treatment of depression in patients receiving hospice care: A retrospective medical record review of thirty-one cases. Psychosomatics 56: 329-337. https://doi.org/10.1016/j.psym.2014.05.005
[42]	Rosenblat JD, deVries FE, Doyle Z, et al. (2023) A phase II, open-label clinical trial of intranasal ketamine for depression in patients with cancer receiving palliative care (INKeD-PC Study). Cancers (Basel) 15: 400. https://doi.org/10.3390/cancers15020400
[43]	Wilkinson ST, Ballard ED, Bloch MH, et al. (2018) The effect of a single dose of intravenous ketamine on suicidal ideation: A systematic review and individual participant data meta-analysis. Am J Psychiatry 175: 150-158. https://doi.org/10.1176/appi.ajp.2017.17040472
[44]	Fan W, Yang H, Sun Y, et al. (2017) Ketamine rapidly relieves acute suicidal ideation in cancer patients: a randomized controlled clinical trial. Oncotarget 8: 2356-2360. https://doi.org/10.18632/oncotarget.13743
[45]	Grunebaum MF, Galfalvy HC, Choo TH, et al. (2018) Ketamine for rapid reduction of suicidal thoughts in major depression: A midazolam-controlled randomized clinical trial. Am J Psychiatry 175: 327-335. https://doi.org/10.1176/appi.ajp.2017.17060647
[46]	Ballard ED, Ionescu DF, Vande Voort JL, et al. (2014) Improvement in suicidal ideation after ketamine infusion: relationship to reductions in depression and anxiety. J Psychiatr Res 58: 161-166. https://doi.org/10.1016/j.jpsychires.2014.07.027
[47]	Yang C, Qu Y, Fujita Y, et al. (2017) Possible role of the gut microbiota–brain axis in the antidepressant effects of (R)-ketamine in a social defeat stress model. Transl Psychiatry 7: 1294. https://doi.org/10.1038/s41398-017-0031-4
[48]	Chang L, Toki H, Qu Y, et al. (2018) No Sex-specific differences in the acute antidepressant actions of (R)-ketamine in an inflammation model. Int J Neuropsychopharmacol 21: 932-937. https://doi.org/10.1093/ijnp/pyy053
[49]	Qu Y, Yang C, Ren Q, et al. (2017) Comparison of (R)-ketamine and lanicemine on depression-like phenotype and abnormal composition of gut microbiota in a social defeat stress model. Sci Rep 7: 15725. https://doi.org/10.1038/s41598-017-16060-7
[50]	Bonaventura J, Lam S, Carlton M, et al. (2021) Pharmacological and behavioral divergence of ketamine enantiomers: Implications for abuse liability. Mol Psychiatry 26: 6704-6722. https://doi.org/10.1038/s41380-021-01093-2
[51]	Price RB, Mathew SJ (2015) Does ketamine have anti-suicidal properties? Current status and future directions. CNS Drugs 29: 181-188. https://doi.org/10.1007/s40263-015-0232-4
[52]	Abbar M, Demattei C, El-Hage W, et al. (2022) Ketamine for the acute treatment of severe suicidal ideation: double blind, randomised placebo controlled trial. BMJ 376: e067194. https://doi.org/10.1136/bmj-2021-067194
[53]	Vidal S, Gex-Fabry M, Bancila V, et al. (2018) Efficacy and safety of a rapid intravenous injection of ketamine 0.5 mg/kg in treatment-resistant major depression: An open 4-week longitudinal study. J Clin Psychopharmacol 38: 590-597. https://doi.org/10.1097/JCP.0000000000000960
[54]	Weber G, Yao J, Binns S, et al. (2018) Case report of subanesthetic intravenous ketamine infusion for the treatment of neuropathic pain and depression with suicidal features in a pediatric patient. Case Rep Anesthesiol 2018: 9375910. https://doi.org/10.1155/2018/9375910
[55]	Zhu W, Ding Z, Zhang Y, et al. (2016) Risks associated with misuse of ketamine as a rapid-acting antidepressant. Neurosci Bull 32: 557-564. https://doi.org/10.1007/s12264-016-0081-2

This article has been cited by:

1.	Yu-Ming Chu, Muhammad Uzair Awan, Muhammad Zakria Javad, Awais Gul Khan, Bounds for the Remainder in Simpson’s Inequality via n-Polynomial Convex Functions of Higher Order Using Katugampola Fractional Integrals, 2020, 2020, 2314-4629, 1, 10.1155/2020/4189036
2.	Shu-Bo Chen, Saima Rashid, Muhammad Aslam Noor, Rehana Ashraf, Yu-Ming Chu, A new approach on fractional calculus and probability density function, 2020, 5, 2473-6988, 7041, 10.3934/math.2020451
3.	Yousaf Khurshid, Muhammad Adil Khan, Yu-Ming Chu, Conformable integral version of Hermite-Hadamard-Fejér inequalities via η-convex functions, 2020, 5, 2473-6988, 5106, 10.3934/math.2020328
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

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)