On some Hermite–Hadamard type inequalities for $tgs$-convex functions via generalized fractional integrals

Abstract

In this research article, we establish some Hermite–Hadamard type inequalities for $tgs$-convex functions via Katugampola fractional integrals and ψ-Riemann–Liouville fractional integrals. Through these results we give some new Hermite–Hadamard type inequalities for $tgs$-convex functions via Riemann–Liouville fractional integrals and classical integrals.

1 Introduction

The convex function and its generalization play an important role in optimization theory and in other field of sciences. These functions have many integral inequalities (see [1, 10, 16]). The Hermite–Hadamard inequality [4, 5] for convex functions $\chi :\mathcal{H} \rightarrow \mathbb{R}$ on an interval $\mathcal{H}$ of the real line is defined by

$$ \chi \biggl(\frac{h_{1}+h_{2}}{2} \biggr)\leq \frac{1}{h _{2}-h_{1}} \int ^{h_{2}}_{h_{1}}\chi (g)\,dg\leq \frac{\chi (h_{1})+ \chi (h_{2})}{2}, $$

(1)

for all $h_{1},h_{2}\in \mathcal{H}$ with $h_{1}< h_{2}$. Several applications are found by using the Hermite–Hadamard inequality (see [2, 3, 6, 12, 14]).

Fractional calculus [8] has played a key role in different scientific fields due to its long term memory methods. In [15], Sarikaya et al. proved some Hermite–Hadamard type integral inequalities for fractional integrals and also gave some applications. In [10, 11, 13], the authors have established several Hermite–Hadamard type inequalities for new fractional conformable integral operators, Katugampola fractional integrals and ψ-Riemann–Liouville fractional integrals, respectively.

Motivated by Liu et al. [9] and by [11, 13], we prove Hermite–Hadamard type inequalities using ψ-Riemann–Liouville fractional integrals and Katugampola fractional integrals.

2 Preliminaries

In this section, we give some definitions and relevant results essential for this research article.

Definition 2.1

([18])

Let $\chi :\mathcal{H}\subseteq \mathbb{R}\rightarrow \mathbb{R}$ be a nonnegative function. Then χ is called $tgs$-convex, if it satisfies the following inequality:

$$ \chi \bigl(r h_{1}+(1-r)h_{2}\bigr)\leq r(1-r)\bigl[ \chi (h_{1})+\chi (h_{2})\bigr], $$

(2)

for all $h_{1},h_{2}\in \mathcal{H}$ and $r\in [0,1]$.

Definition 2.2

([8])

Let $\chi \in L[h_{1},h_{2}]$. The right-hand side and left-hand side Riemann–Liouville fractional integrals $J^{\alpha }_{h_{1}+}\chi$ and $J^{\alpha }_{h_{2}-}\chi$ of order $\alpha > 0$ with $h_{2} > h_{1} \geq 0$ are defined by

$$ J^{\alpha }_{h_{1}+}\chi (g)=\frac{1}{\varGamma (\alpha )} \int _{h_{1}} ^{g}(g-t)^{\alpha -1}\chi (t)\,dt,\quad \ g>h_{1} $$

and

$$ J^{\alpha }_{h_{2}-}\chi (g)=\frac{1}{\varGamma (\alpha )} \int _{g}^{h _{2}}(t-g)^{\alpha -1}\chi (t)\,dt,\quad g< h_{2}, $$

respectively, where $\varGamma (\cdot )$ is the Gamma function defined by $\varGamma (\alpha )=\int _{0}^{\infty }e^{-t}t^{\alpha -1}\,dt$.

Definition 2.3

([7])

Let $[h_{1},h_{2}]\subset \mathbb{R}$ be a finite interval. Then, the left- and right-side Katugampola fractional integrals of order $\alpha (>0)$ of $\chi \in X^{p}_{c}(h_{1},h_{2})$ are defined by

$$ {}^{\rho }I^{\alpha }_{h_{1}+}\chi (g)= \frac{\rho ^{1-\alpha }}{\varGamma ( \alpha )} \int _{h_{1}}^{g}\bigl(g^{\rho }-t^{\rho } \bigr)^{\alpha -1}t^{\rho -1} \chi (t)\,dt $$

and

$$ {}^{\rho }I^{\alpha }_{h_{2}-}\chi (g)= \frac{\rho ^{1-\alpha }}{\varGamma ( \alpha )} \int _{g}^{h_{2}}\bigl(t^{\rho }-g^{\rho } \bigr)^{\alpha -1}t^{\rho -1} \chi (t)\,dt, $$

with $h_{1}< g< h_{2}$ and $\rho >0$. Here $X^{p}_{c}(h_{1},h_{2})$ ($c\in \mathbb{R}, 1\leq p\leq \infty$) is the space of those complex valued Lebesgue measurable functions χ on $[h_{1},h_{2}]$ for which $\|\chi \|_{X^{p}_{c}}<\infty$, where the norm is defined by

$$ \Vert \chi \Vert _{X^{p}_{c}}= \biggl( \int _{h_{1}}^{h_{2}} \bigl\vert t^{c} \chi (t) \bigr\vert ^{p} \frac{dt}{t} \biggr)^{1/p}< \infty , $$

for $1\leq p<\infty$, $c\in \mathbb{R}$ and, for the case $p=\infty$,

$$ \Vert \chi \Vert _{X^{\infty }_{c}}= \mathop{\operatorname{ess}\, \operatorname{sup}} _{h_{1}\leq t\leq h_{2}}\bigl[t^{c} \bigl\vert \chi (t) \bigr\vert \bigr]. $$

Here $\operatorname{ess}\, \operatorname{sup}$ stands for essential supremum.

Definition 2.4

([8, 17])

Let $(h_{1},h_{2})\ (-\infty \leq h_{1}< h_{2}\leq \infty )$ be a finite or infinite real interval and $\gamma >0$. Let $\psi (x)$ be an increasing and positive monotone function on $(h_{1},h_{2}]$ with continuous derivative on $(h_{1},h_{2})$. Then the left- and right-sided ψ-Riemann–Liouville fractional integrals of a function χ with respect to ψ on $[h_{1},h_{2}]$ are defined by

$$\begin{aligned} &\mathcal{I}^{\gamma :\psi }_{h_{1}+}\chi (g)=\frac{1}{\varGamma (\gamma )} \int _{h_{1}}^{g}\psi '(z) \bigl(\psi (g)-\psi (z)\bigr)^{\gamma -1}\chi (z)\,dz, \\ &\mathcal{I}^{\gamma :\psi }_{h_{2}-}\chi (g)=\frac{1}{\varGamma (\gamma )} \int _{g}^{h_{2}}\psi '(z) \bigl(\psi (z)-\psi (g)\bigr)^{\gamma -1}\chi (z)\,dz, \end{aligned}$$

respectively.

Liu et al. [9] established Hermite–Hadamard type inequalities via ψ-Riemann–Liouville fractional integrals for convex functions.

Lemma 2.1

([9])

Let $\chi : [h_{1},h_{2}]\rightarrow \mathbb{R}$be a differentiable mapping, for $0\leq h_{1}< h_{2}$, and $\chi \in L_{1}[h _{1},h_{2}]$. Let $\psi (g)$be an increasing and positive monotone function on $(h_{1},h_{2}]$, with continuous derivative $\psi '(g)$on $(h_{1},h_{2})$and $\gamma \in (0,1)$. Then the following equality for fractional integral holds:

$$\begin{aligned} & \frac{\chi (h_{1})+\chi (h_{2})}{2}-\frac{\varGamma (\gamma +1)}{2(h _{2}-h_{1})^{\gamma }}\bigl[ \mathcal{I}^{\gamma :\psi }_{\psi ^{-1}(h_{1})+}( \chi \circ \psi ) \bigl(\psi ^{-1}(h_{2})\bigr) \\ &\qquad{} +\mathcal{I}^{\gamma :\psi }_{\psi ^{-1}(h_{2})-}(\chi \circ \psi ) \bigl(\psi ^{-1}(h_{1})\bigr)\bigr] \\ &\quad =\frac{1}{2(h_{2}-h_{1})^{\gamma }} \int _{\psi ^{-1}(h_{1})}^{\psi ^{-1}(h_{2})}\bigl[\bigl(\psi (g)-h_{1}\bigr)^{\gamma }-\bigl(h_{2}-\psi (g) \bigr)^{\gamma }\bigr]\bigl( \chi '\circ \psi \bigr) (g)\psi '(g)\,dg. \end{aligned}$$

(3)

Lemma 2.2

([9])

Let $\chi : [h_{1},h_{2}]\rightarrow \mathbb{R}$be a differentiable mapping, for $0\leq h_{1}< h_{2}$, and $\chi \in L_{1}[h _{1},h_{2}]$. Let $\psi (g)$be an increasing and positive monotone function on $(h_{1},h_{2}]$, with continuous derivative $\psi '(g)$on $(h_{1},h_{2})$and $\gamma \in (0,1)$. Then the following equality for fractional integral holds:

$$\begin{aligned} & \frac{\varGamma (\gamma +1)}{2(h_{2}-h_{1})^{\gamma }}\bigl[\mathcal{I}^{ \gamma :\psi }_{\psi ^{-1}(h_{1})+}( \chi \circ \psi ) \bigl(\psi ^{-1}(h_{2})\bigr)+ \mathcal{I}^{\gamma :\psi }_{\psi ^{-1}(h_{2})-}(\chi \circ \psi ) \bigl( \psi ^{-1}(h_{1})\bigr)\bigr] \\ &\qquad{} -\chi \biggl( \frac{h_{1}+h_{2}}{2} \biggr) \\ &\quad = \int _{\psi ^{-1}(h_{1})}^{\psi ^{-1}(h_{2})}k\bigl(\chi '\circ \psi \bigr) (g) \psi '(g)\,dg \\ &\qquad{} + \frac{1}{2(h_{2}-h_{1})^{\gamma }} \int _{\psi ^{-1}(h_{1})}^{\psi ^{-1}(h_{2})}\bigl[\bigl(\psi (g)-h_{1}\bigr)^{\gamma }-\bigl(h _{2}-\psi (g) \bigr)^{\gamma }\bigr]\bigl(\chi '\circ \psi \bigr) (g)\psi '(g)\,dg, \end{aligned}$$

(4)

where

$$\begin{aligned} k= \textstyle\begin{cases} \frac{1}{2} , &\psi ^{-1} (\frac{h_{1}+h_{2}}{2} ) \leq z\leq \psi ^{-1}(h_{2}), \\ -\frac{1}{2}, & \psi ^{-1}(h_{1})< z< \psi ^{-1} (\frac{h _{1}+h_{2}}{2} ). \end{cases}\displaystyle \end{aligned}$$

3 Inequalities via Katugampola fractional integrals

In this section, we find a Hermite–Hadamard inequality for a $tgs$-convex function via Katugampola fractional integrals.

Theorem 3.1

Let $\alpha >0$and $\rho >0$. Let $\chi :[h_{1}^{\rho },h _{2}^{\rho }]\subset \mathbb{R}\rightarrow \mathbb{R}$be a nonnegative function with $0\leq h_{1}< h_{2}$and $\chi \in X^{p}_{c}(h_{1}^{ \rho },h_{2}^{\rho })$. Ifχis also a $tgs$-convex function on $[h_{1}^{\rho },h_{2}^{\rho }]$, then the following inequalities hold:

$$\begin{aligned} &2\chi \biggl(\frac{h_{1}^{\rho }+h_{2}^{\rho }}{2} \biggr) \\ &\quad\leq \frac{\rho ^{\alpha }\varGamma (\alpha +1)}{2(h_{2}^{\rho }-h_{1} ^{\rho })^{\alpha }} \bigl[{}^{\rho }I^{\alpha }_{h_{1}+} \chi \bigl(h_{2} ^{\rho }\bigr)+{}^{\rho }I^{\alpha }_{h_{2}-} \chi \bigl(h_{1}^{\rho }\bigr) \bigr] \\ &\quad\leq \frac{\alpha (\chi (h_{1}^{\rho })+\chi (h_{2}^{\rho }))}{ \rho (\alpha +1)(\alpha +2)}. \end{aligned}$$

(5)

Proof

Let $r\in [0,1]$. Consider $x,y\in [h_{1},h_{2}]$, $h_{1}\geq 0$, defined by $x^{\rho }=r^{\rho }h_{1}^{\rho }+(1-r^{\rho })h_{2}^{ \rho }$, $y^{\rho }=r^{\rho }h_{2}^{\rho }+(1-r^{\rho })h_{1}^{\rho }$. Since χ is a $tgs$-convex function on $[h_{1}^{\rho },h_{2}^{ \rho }]$, we have

$$ \chi \biggl(\frac{x^{\rho }+y^{\rho }}{2} \biggr)\leq \frac{\chi (x ^{\rho })+\chi (y^{\rho })}{4}. $$

Then we have

$$ 4\chi \biggl(\frac{h_{1}^{\rho }+h_{2}^{\rho }}{2} \biggr) \leq \chi \bigl(r^{\rho }h_{1}^{\rho }+\bigl(1-r^{\rho } \bigr)h_{2}^{\rho }\bigr)+\chi \bigl(r ^{\rho }h_{2}^{\rho }+ \bigl(1-r^{\rho }\bigr)h_{1}^{\rho }\bigr). $$

(6)

Multiplying both sides of (6) by $r^{\alpha \rho -1}$, $\alpha >0$ and then integrating the resulting inequality with respect to r over $[0,1]$, we obtain

$$\begin{aligned} \frac{4}{\alpha \rho }\chi \biggl( \frac{h_{1}^{\rho }+h_{2}^{\rho }}{2} \biggr)\leq{}& \int ^{1}_{0}r^{\alpha \rho -1}\chi \bigl(r^{\rho }h_{1}^{\rho }+\bigl(1-r ^{\rho }\bigr)h_{2}^{\rho }\bigr)\,dr \\ &{} + \int ^{1}_{0}r^{\alpha \rho -1}\chi \bigl(r^{\rho }h_{2}^{ \rho }+\bigl(1-r^{\rho } \bigr)h_{1}^{\rho }\bigr)\,dr \\ ={}& \int _{h_{2}}^{h_{1}} \biggl(\frac{h_{2}^{\rho }-g^{\rho }}{h_{2} ^{\rho }-h_{1}^{\rho }} \biggr)^{\alpha -1}\chi \bigl(g^{\rho }\bigr)\frac{g ^{\rho -1}}{h_{1}^{\rho }-h_{2}^{\rho }} \,dg \\ &{} + \int _{h_{1}}^{h_{2}} \biggl(\frac{k^{\rho }-h_{1}^{ \rho }}{h_{2}^{\rho }-h_{1}^{\rho }} \biggr)^{\alpha -1}\chi \bigl(k^{ \rho }\bigr)\frac{k^{\rho -1}}{h_{2}^{\rho }-h_{1}^{\rho }} \,dk \\ ={}&\frac{\rho ^{\alpha -1}\varGamma (\alpha )}{(h_{2}^{\rho }-h_{1}^{ \rho })^{\alpha }} \bigl[{}^{\rho }I^{\alpha }_{h_{1}+} \chi \bigl(h_{2}^{ \rho }\bigr)+{}^{\rho }I^{\alpha }_{h_{2}-} \chi \bigl(h_{1}^{\rho }\bigr) \bigr]. \end{aligned}$$

(7)

This establishes the first inequality. For the proof of the second inequality in (5), we first observe that, for a $tgs$-convex function χ, we have

$$ \chi \bigl(r^{\rho }h_{1}^{\rho }+ \bigl(1-r^{\rho }\bigr)h_{2}^{\rho }\bigr)\leq r^{ \rho }\bigl(1-r^{\rho }\bigr) \bigl(\chi \bigl(h_{1}^{\rho }\bigr)+\chi \bigl(h_{2}^{\rho } \bigr)\bigr) $$

and

$$ \chi \bigl(r^{\rho }h_{2}^{\rho }+ \bigl(1-r^{\rho }\bigr)h_{1}^{\rho }\bigr)\leq r^{ \rho }\bigl(1-r^{\rho }\bigr) \bigl(\chi \bigl(h_{1}^{\rho }\bigr)+\chi \bigl(h_{2}^{\rho } \bigr)\bigr). $$

By adding these inequalities, we get

$$ \chi \bigl(r^{\rho }h_{1}^{\rho }+ \bigl(1-r^{\rho }\bigr)h_{2}^{\rho }\bigr)+ \chi \bigl(r^{\rho }h_{2}^{\rho }+\bigl(1-r^{\rho } \bigr)h_{1}^{\rho }\bigr)\leq 2r^{ \rho } \bigl(1-r^{\rho }\bigr) \bigl(\chi \bigl(h_{1}^{\rho } \bigr)+\chi \bigl(h_{2}^{\rho }\bigr)\bigr). $$

(8)

Multiplying both sides of (8) by $r^{\alpha \rho -1}$, $\alpha >0$ and then integrating the resulting inequality with respect to r over $[0,1]$, we obtain

$$ \frac{\rho ^{\alpha -1}\varGamma (\alpha )}{(h_{2}^{\rho }-h _{1}^{\rho })^{\alpha }} \bigl[{}^{\rho }I^{\alpha }_{h_{1}+} \chi \bigl(h _{2}^{\rho }\bigr)+{}^{\rho }I^{\alpha }_{h_{2}-} \chi \bigl(h_{1}^{\rho }\bigr) \bigr] \leq 2 \int _{0}^{1}r^{\alpha \rho +\rho -1 } \bigl(1-r^{\rho }\bigr) \bigl(\chi \bigl(h_{1} ^{\rho }\bigr)+\chi \bigl(h_{2}^{\rho }\bigr) \bigr)\,dr. $$

(9)

Since

$$ \int ^{1}_{0}\bigl(r^{\alpha \rho +\rho -1}-r^{\alpha \rho +2\rho -1} \bigr)\,dt=\frac{1}{ \rho (\alpha +1)(\alpha +2)}, $$

(9) becomes

$$ \frac{\rho ^{\alpha -1}\varGamma (\alpha )}{(h_{2}^{\rho }-h _{1}^{\rho })^{\alpha }} \bigl[{}^{\rho }I^{\alpha }_{h_{1}+} \chi \bigl(h _{2}^{\rho }\bigr)+{}^{\rho }I^{\alpha }_{h_{2}-} \chi \bigl(h_{1}^{\rho }\bigr) \bigr] \leq \frac{2(\chi (h_{1}^{\rho })+\chi (h_{2}^{\rho }))}{\rho (\alpha +1)(\alpha +2)} . $$

(10)

Thus (7) and (10) give (5). □

Remark 3.1

(1) By letting $\rho \rightarrow 1$ in (5) of Theorem 3.1 we get inequality 3.1 of Theorem 3.1 in [18].

(2) By letting $\rho \rightarrow 1$ and $\alpha =1$ in (5) of Theorem 3.1 we get inequality 2.2 of Theorem 2.1 in [18].

Theorem 3.2

Let $\alpha >0$and $\rho >0$. Let $\chi :[h_{1}^{\rho },h _{2}^{\rho }]\subset \mathbb{R}\rightarrow \mathbb{R}$be a differentiable and nonnegative mapping on $(h_{1}^{\rho },h_{2}^{ \rho })$with $0\leq h_{1}< h_{2}$. If $|\chi '|$is $tgs$-convex on $[h_{1}^{\rho },h_{2}^{\rho }]$, then the following inequality holds:

$$\begin{aligned} & \biggl\vert \frac{\chi (h_{1}^{\rho })+\chi (h_{2}^{\rho })}{2}- \frac{ \rho ^{\alpha }\varGamma (\alpha +1)}{2(h_{2}^{\rho }-h_{1}^{\rho })^{ \alpha }} \bigl[{}^{\rho }I^{\alpha }_{h_{1}+} \chi \bigl(h_{2}^{\rho }\bigr)+^{ \rho }I^{\alpha }_{h_{2}-} \chi \bigl(h_{1}^{\rho }\bigr) \bigr] \biggr\vert \\ &\quad \leq \frac{h_{2}^{\rho }-h_{1}^{\rho }}{(\alpha +2)(\alpha +3)}\bigl[ \bigl\vert \chi ' \bigl(h_{1}^{\rho }\bigr) \bigr\vert + \bigl\vert \chi '\bigl(h_{2}^{\rho }\bigr) \bigr\vert \bigr] . \end{aligned}$$

(11)

Proof

From (7) one can have

$$\begin{aligned} &\frac{\rho ^{\alpha -1}\varGamma (\alpha )}{(h_{2}^{\rho }-h_{1}^{\rho })^{ \alpha }} \bigl[{}^{\rho }I^{\alpha }_{h_{1}+} \chi \bigl(h_{2}^{\rho }\bigr)+^{ \rho }I^{\alpha }_{h_{2}-} \chi \bigl(h_{1}^{\rho }\bigr) \bigr] \\ &\quad = \int ^{1}_{0}r^{\alpha \rho -1}\chi \bigl(r^{\rho }h_{1}^{\rho }+\bigl(1-r ^{\rho }\bigr)h_{2}^{\rho }\bigr)\,dr + \int ^{1}_{0}r^{\alpha \rho -1}\chi \bigl(r^{ \rho }h_{2}^{\rho }+\bigl(1-r^{\rho } \bigr)h_{1}^{\rho }\bigr)\,dr. \end{aligned}$$

(12)

By integrating by parts, we then get

$$\begin{aligned} &\frac{\chi (h_{1}^{\rho })+\chi (h_{2}^{\rho })}{\alpha \rho }-\frac{ \rho ^{\alpha -1}\varGamma (\alpha )}{(h_{2}^{\rho }-h_{1}^{\rho })^{ \alpha }} \bigl[{}^{\rho }I^{\alpha }_{h_{1}+} \chi \bigl(h_{2}^{\rho }\bigr)+^{ \rho }I^{\alpha }_{h_{2}-} \chi \bigl(h_{1}^{\rho }\bigr) \bigr] \\ &\quad =\frac{h_{2}^{\rho }-h_{1}^{\rho }}{\alpha } \int _{0}^{1}r^{\rho ( \alpha +1)-1} \bigl[\chi '\bigl(r^{\rho }h_{2}^{\rho }+ \bigl(1-r^{\rho }\bigr)h_{1} ^{\rho }\bigr)-\chi '\bigl(r^{\rho }h_{1}^{\rho }+ \bigl(1-r^{\rho }\bigr)h_{2}^{\rho }\bigr) \bigr] \,dr. \end{aligned}$$

(13)

By using the triangle inequality and the $tgs$-convexity of $|\chi '|$, we obtain

$$\begin{aligned} & \biggl\vert \frac{\chi (h_{1}^{\rho })+\chi (h_{2}^{\rho })}{\alpha \rho }-\frac{\rho ^{\alpha -1}\varGamma (\alpha )}{(h_{2}^{\rho }-h_{1} ^{\rho })^{\alpha }} \bigl[{}^{\rho }I^{\alpha }_{h_{1}+}\chi \bigl(h_{2} ^{\rho }\bigr)+{}^{\rho }I^{\alpha }_{h_{2}-} \chi \bigl(h_{1}^{\rho }\bigr) \bigr] \biggr\vert \\ &\quad\leq \frac{h_{2}^{\rho }-h_{1}^{\rho }}{\alpha } \int _{0}^{1}r^{ \rho (\alpha +1)-1} \bigl\vert \chi '\bigl(r^{\rho }h_{2}^{\rho }+ \bigl(1-r^{\rho }\bigr)h _{1}^{\rho }\bigr)-\chi '\bigl(r^{\rho }h_{1}^{\rho }+ \bigl(1-r^{\rho }\bigr)h_{2}^{ \rho }\bigr) \bigr\vert \,dr \\ &\quad\leq \frac{h_{2}^{\rho }-h_{1}^{\rho }}{\alpha } \int _{0}^{1}r^{ \rho (\alpha +1)-1} \bigl[\chi '\bigl(r^{\rho }h_{2}^{\rho }+ \bigl(1-r^{\rho }\bigr)h _{1}^{\rho }\bigr)+\chi '\bigl(r^{\rho }h_{1}^{\rho }+ \bigl(1-r^{\rho }\bigr)h_{2}^{ \rho }\bigr) \bigr] \,dr \\ &\quad=\frac{2(h_{2}^{\rho }-h_{1}^{\rho })}{\alpha } \int _{0}^{1}r^{ \rho (\alpha +1)-1} r^{\rho }\bigl(1-r^{\rho }\bigr) \bigl[ \bigl\vert \chi '\bigl(h_{1}^{ \rho }\bigr) \bigr\vert + \bigl\vert \chi '\bigl(h_{2}^{\rho }\bigr) \bigr\vert \bigr]\,dr \\ &\quad=\frac{2(h_{2}^{\rho }-h_{1}^{\rho })}{\alpha } \frac{ \vert \chi '(h_{1} ^{\rho }) \vert + \vert \chi '(h_{2}^{\rho }) \vert }{\rho (\alpha +2)(\alpha +3)} . \end{aligned}$$

(14)

Multiplying both sides of the above inequality by $\frac{\alpha \rho }{2}$, we get the required inequality (11). □

Corollary 3.3

Consider the similar assumptions of Theorem 3.2.

1. If $\rho =1$, then

$$\begin{aligned} & \biggl\vert \frac{\chi (h_{1})+\chi (h_{2})}{2}- \frac{\varGamma (\alpha +1)}{2(h _{2}-h_{1})^{\alpha }} \bigl[J^{\alpha }_{h_{1}+}\chi (h_{2})+J^{ \alpha }_{h_{2}-}\chi (h_{1}) \bigr] \biggr\vert \\ &\quad \leq \frac{h_{2}-h_{1}}{(\alpha +2)(\alpha +3)}\bigl[ \bigl\vert \chi '(h_{1}) \bigr\vert + \bigl\vert \chi '(h_{2}) \bigr\vert \bigr] . \end{aligned}$$

(15)

2. If $\rho =\alpha =1$, then

$$\begin{aligned} \biggl\vert \frac{\chi (h_{1})+\chi (h_{2})}{2}- \frac{1}{h_{2}-h_{1}} \int _{h_{1}}^{h_{2}}\chi (g)\,dg \biggr\vert \leq \frac{h_{2}-h_{1}}{12}\bigl[ \bigl\vert \chi '(h_{1}) \bigr\vert + \bigl\vert \chi '(h_{2}) \bigr\vert \bigr] . \end{aligned}$$

(16)

For more results we need the following lemma, also proved in [11].

Lemma 3.1

([11])

Let $\alpha >0$and $\rho >0$. Let $\chi :[h_{1}^{\rho },h _{2}^{\rho }]\subset \mathbb{R}_{+}=[0,\infty )\rightarrow \mathbb{R}$be a differentiable mapping on $(h_{1}^{\rho },h_{2}^{\rho })$with $0\leq h_{1}< h_{2}$. Then the following equality holds if the fractional integrals exist:

$$\begin{aligned} &\frac{\chi (h_{1}^{\rho })+\chi (h_{2}^{\rho })}{2}-\frac{ \rho ^{\alpha }\varGamma (\alpha +1)}{2(h_{2}^{\rho }-h_{1}^{\rho })^{ \alpha }} \bigl[{}^{\rho }I^{\alpha }_{h_{1}+}\chi \bigl(h_{2}^{\rho }\bigr)+^{ \rho }I^{\alpha }_{h_{2}-} \chi \bigl(h_{1}^{\rho }\bigr) \bigr] \\ &\quad =\frac{\rho (h_{2}^{\rho }-h_{1}^{\rho })}{2} \int _{0}^{1} \bigl[\bigl(1-r ^{\rho } \bigr)^{\alpha }-\bigl(r^{\rho }\bigr)^{\alpha } \bigr] r^{\rho -1}\chi '\bigl(r ^{\rho }h_{1}^{\rho }+ \bigl(1-r^{\rho }\bigr)h_{2}^{\rho }\bigr)\,dr. \end{aligned}$$

(17)

Proof

By using the similar arguments as in the proof of Lemma 2 in [15]. First consider

$$\begin{aligned} & \int _{0}^{1}\bigl(1-r^{\rho } \bigr)^{\alpha }r^{\rho -1}\chi '\bigl(r^{\rho }h_{1} ^{\rho }+\bigl(1-r^{\rho }\bigr)h_{2}^{\rho } \bigr)\,dr \\ &\quad=\frac{(1-r^{\rho })^{\alpha }\chi (r^{\rho }h_{1}^{\rho }+(1-r^{ \rho })h_{2}^{\rho })}{\rho (h_{1}^{\rho }-h_{2}^{\rho })}\bigg|_{0} ^{1} \\ &\qquad{} +\frac{\alpha }{h_{1}^{\rho }-h_{2}^{\rho }} \int _{0} ^{1}\bigl(1-r^{\rho } \bigr)^{\alpha -1}r^{\rho -1}\chi \bigl(r^{\rho }h_{1}^{\rho }+ \bigl(1-r ^{\rho }\bigr)h_{2}^{\rho }\bigr)\,dr \\ &\quad=\frac{\chi (h_{2}^{\rho })}{\rho (h_{2}^{\rho }-h_{1}^{\rho })}-\frac{ \alpha }{h_{2}^{\rho }-h_{1}^{\rho }} \int _{h_{2}}^{h_{1}} \biggl(\frac{g ^{\rho }-h_{1}^{\rho }}{h_{2}^{\rho }-h_{1}^{\rho }} \biggr)^{\alpha -1}\cdot \frac{g^{\rho -1}}{h_{1}^{\rho }-h_{2}^{\rho }}\,dg \\ &\quad=\frac{\chi (h_{2}^{\rho })}{\rho (h_{2}^{\rho }-h_{1}^{\rho })}-\frac{ \rho ^{\alpha -1}\varGamma (\alpha +1)}{(h_{2}^{\rho }-h_{1}^{\rho })^{ \alpha +1}}\cdot {}^{\rho }I_{h_{2}-}^{\alpha } \chi \bigl(g^{\rho }\bigr)\bigg|_{g=h _{1}}. \end{aligned}$$

(18)

Similarly, we can show that

$$\begin{aligned} &\int _{0}^{1}r^{\rho \alpha }\cdot r^{\rho -1}\chi '\bigl(r ^{\rho }h_{1}^{\rho }+ \bigl(1-r^{\rho }\bigr)h_{2}^{\rho }\bigr)\,dr \\ &\quad = - \frac{\chi (h _{1}^{\rho })}{\rho (h_{2}^{\rho }-h_{1}^{\rho })}+\frac{\rho ^{\alpha -1}\varGamma (\alpha +1)}{(h_{2}^{\rho }-h_{1}^{\rho })^{\alpha +1}} \cdot {}^{\rho }I_{h_{1}+}^{\alpha } \chi \bigl(g^{\rho }\bigr)\bigg|_{g=h_{2}}. \end{aligned}$$

(19)

Thus from (18) and (19) we get (17). □

Theorem 3.4

Let $\alpha >0$and $\rho >0$. Let $\chi :[h_{1}^{\rho },h _{2}^{\rho }]\subset \mathbb{R}_{+}\rightarrow \mathbb{R}$be a differentiable and nonnegative mapping on $(h_{1}^{\rho },h_{2}^{ \rho })$such that $\chi '\in L_{1}[h_{1},h_{2}]$with $0\leq h_{1}< h _{2}$. If $|\chi '|^{q}$is $tgs$-convex on $[h_{1}^{\rho },h_{2}^{ \rho }]$for some fixed $q\geq 1$, then the following inequality holds:

$$\begin{aligned} & \biggl\vert \frac{\chi (h_{1}^{\rho })+\chi (h_{2}^{\rho })}{2}- \frac{ \rho ^{\alpha }\varGamma (\alpha +1)}{2(h_{2}^{\rho }-h_{1}^{\rho })^{ \alpha }} \bigl[{}^{\rho }I^{\alpha }_{h_{1}+} \chi \bigl(h_{2}^{\rho }\bigr)+^{ \rho }I^{\alpha }_{h_{2}-} \chi \bigl(h_{1}^{\rho }\bigr) \bigr] \biggr\vert \\ &\quad \leq \frac{ (h_{2}^{\rho }-h_{1}^{\rho })}{2} \biggl(\frac{2}{ \alpha +1} \biggr)^{1-1/q} \\ & \qquad{}\times \biggl( \biggl[\beta (2,\alpha +2)+\frac{1}{( \alpha +2)(\alpha +3)} \biggr] \bigl[ \bigl\vert \chi '\bigl(h_{1}^{\rho }\bigr) \bigr\vert ^{q}+ \bigl\vert \chi '\bigl(h _{2}^{\rho }\bigr) \bigr\vert ^{q}\bigr] \biggr)^{1/q}. \end{aligned}$$

(20)

Proof

Using Lemma 3.1 and the power mean inequality and the $tgs$-convexity of $|\chi '|^{q}$, we obtain

$$\begin{aligned} & \bigl\vert I_{\chi }(\alpha ,\rho ,h_{1},h_{2}) \bigr\vert \\ &\quad= \biggl\vert \frac{\rho (h_{2}^{\rho }-h_{1}^{\rho })}{2} \int _{0}^{1} \bigl\{ \bigl(1-r^{\rho } \bigr)^{\alpha }-\bigl(r^{\rho }\bigr)^{\alpha } \bigr\} r ^{\rho -1}\chi '\bigl(r^{\rho }h_{1}^{\rho }+ \bigl(1-r^{\rho }\bigr)h_{2}^{\rho }\bigr)\,dr \biggr\vert \\ &\quad\leq \frac{\rho (h_{2}^{\rho }-h_{1}^{\rho })}{2} \biggl( \int _{0} ^{1} \bigl\vert \bigl(1-r^{\rho }\bigr)^{\alpha }-\bigl(r^{\rho } \bigr)^{\alpha } \bigr\vert r ^{\rho -1}\,dr \biggr)^{1-1/q} \\ & \qquad{}\times \biggl( \int _{0}^{1} \bigl\vert \bigl(1-r^{\rho }\bigr)^{\alpha }-\bigl(r^{\rho } \bigr)^{\alpha } \bigr\vert r^{\rho -1} \bigl\vert \chi '\bigl(r^{\rho }h_{1}^{ \rho }+ \bigl(1-r^{\rho }\bigr)h_{2}^{\rho }\bigr) \bigr\vert ^{q}\,dr \biggr)^{1/q} \\ &\quad\leq \frac{\rho (h_{2}^{\rho }-h_{1}^{\rho })}{2} \biggl( \int _{0} ^{1} \bigl\{ \bigl(1-r^{\rho } \bigr)^{\alpha }+\bigl(r^{\rho }\bigr)^{\alpha } \bigr\} r^{\rho -1}\,dr \biggr)^{1-1/q} \\ &\qquad{} \times \biggl( \int _{0}^{1} \bigl\{ \bigl(1-r^{\rho } \bigr)^{ \alpha }+\bigl(r^{\rho }\bigr)^{\alpha } \bigr\} r^{\rho -1}r^{\rho }\bigl(1-r^{ \rho }\bigr)\bigl[ \bigl\vert \chi '\bigl(h_{1}^{\rho }\bigr) \bigr\vert ^{q}+ \bigl\vert \chi '\bigl(h_{2}^{\rho } \bigr) \bigr\vert ^{q}\bigr]\,dr \biggr) ^{1/q}. \end{aligned}$$

(21)

By using the change of variable $t=r^{\rho }$, we get

$$\begin{aligned} & \int _{0}^{1} \bigl\{ \bigl(1-r^{\rho } \bigr)^{\alpha }+\bigl(r^{\rho }\bigr)^{\alpha } \bigr\} r^{\rho -1}\,dr \\ &\quad= \int _{0}^{1}\bigl(1-r^{\rho } \bigr)^{\alpha }r^{\rho -1}\,dr+ \int _{0}^{1}\bigl(r ^{\rho } \bigr)^{\alpha }r^{\rho -1}\,dr \\ &\quad=\frac{2}{\rho (\alpha +1)}, \end{aligned}$$

(22)

$$\begin{aligned} & \int _{0}^{1} \bigl\{ \bigl(1-r^{\rho } \bigr)^{\alpha }+\bigl(r^{\rho }\bigr)^{\alpha } \bigr\} r^{\rho -1}r ^{\rho }\bigl(1-r^{\rho }\bigr)\,dr \\ &\quad= \int _{0}^{1}\bigl(1-r^{\rho } \bigr)^{\alpha }r^{\rho -1}r^{\rho }\bigl(1-r^{ \rho } \bigr)\,dr+ \int _{0}^{1}\bigl(r^{\rho } \bigr)^{\alpha }r^{\rho -1}r^{\rho }\bigl(1-r ^{\rho }\bigr)\,dr \\ &\quad=\frac{1}{\rho }\beta (2,\alpha +2)+\frac{1}{\rho (\alpha +2)( \alpha +3)}. \end{aligned}$$

(23)

Hence using (23) and (22) in (21) we get (20). □

Corollary 3.5

Consider the similar assumptions of Theorem 3.4.

1. If $\rho =1$, then

$$\begin{aligned} & \biggl\vert \frac{\chi (h_{1})+\chi (h_{2})}{2}- \frac{\varGamma (\alpha +1)}{2(h _{2}-h_{1})^{\alpha }} \bigl[J^{\alpha }_{h_{1}+}\chi (h_{2})+J^{ \alpha }_{h_{2}-}\chi (h_{1}) \bigr] \biggr\vert \\ &\quad \leq \frac{ (h_{2}-h_{1})}{2} \biggl(\frac{2}{\alpha +1} \biggr) ^{1-1/q} \\ &\qquad{} \times \biggl( \biggl[\beta (2,\alpha +2)+\frac{1}{( \alpha +2)(\alpha +3)} \biggr] \bigl[ \bigl\vert \chi '(h_{1}) \bigr\vert ^{q}+ \bigl\vert \chi '(h_{2}) \bigr\vert ^{q}\bigr] \biggr) ^{1/q}. \end{aligned}$$

(24)

2. If $\rho =\alpha =1$, then

$$\begin{aligned} & \biggl\vert \frac{\chi (h_{1})+\chi (h_{2})}{2}- \frac{1}{h_{2}-h_{1}} \int _{h_{1}}^{h_{2}}\chi (g)\,dg \biggr\vert \\ & \quad\leq \frac{ (h_{2}-h_{1})}{2} \biggl(\frac{2( \vert \chi '(h _{1}) \vert ^{q}+ \vert \chi '(h_{2}) \vert ^{q})}{3} \biggr)^{1/q}. \end{aligned}$$

(25)

Theorem 3.6

Let $\alpha >0$and $\rho >0$. Let $\chi :[h_{1}^{\rho },h _{2}^{\rho }]\subset \mathbb{R}_{+}\rightarrow \mathbb{R}$be a differentiable and nonnegative mapping on $(h_{1}^{\rho },h_{2}^{ \rho })$such that $\chi '\in L_{1}[h_{1},h_{2}]$with $0\leq h_{1}< h _{2}$. If $|\chi '|^{q}$is $tgs$-convex on $[h_{1}^{\rho },h_{2}^{ \rho }]$for some fixed $q\geq 1$, then the following inequality holds:

$$\begin{aligned} & \biggl\vert \frac{\chi (h_{1}^{\rho })+\chi (h_{2}^{\rho })}{2}-\frac{ \rho ^{\alpha }\varGamma (\alpha +1)}{2(h_{2}^{\rho }-h_{1}^{\rho })^{ \alpha }} \bigl[{}^{\rho }I^{\alpha }_{h_{1}+}\chi \bigl(h_{2}^{\rho }\bigr)+^{ \rho }I^{\alpha }_{h_{2}-} \chi \bigl(h_{1}^{\rho }\bigr) \bigr] \biggr\vert \\ &\quad \leq \frac{ (h_{2}^{\rho }-h_{1}^{\rho })}{2} \biggl( \biggl[\beta (2,\alpha +2)+ \frac{1}{(\alpha +2)(\alpha +3)} \biggr] \bigl[ \bigl\vert \chi ' \bigl(h_{1} ^{\rho }\bigr) \bigr\vert ^{q}+ \bigl\vert \chi '\bigl(h_{2}^{\rho }\bigr) \bigr\vert ^{q}\bigr] \biggr)^{1/q}. \end{aligned}$$

(26)

Proof

Using Lemma 3.1 and the power mean inequality and the $tgs$-convexity of $|\chi '|^{q}$, we obtain

$$\begin{aligned} & \biggl\vert \frac{\chi (h_{1}^{\rho })+\chi (h_{2}^{\rho })}{2}- \frac{ \rho ^{\alpha }\varGamma (\alpha +1)}{2(h_{2}^{\rho }-h_{1}^{\rho })^{ \alpha }} \bigl[{}^{\rho }I^{\alpha }_{h_{1}+} \chi \bigl(h_{2}^{\rho }\bigr)+^{ \rho }I^{\alpha }_{h_{2}-} \chi \bigl(h_{1}^{\rho }\bigr) \bigr] \biggr\vert \\ &\quad= \biggl\vert \frac{\rho (h_{2}^{\rho }-h_{1}^{\rho })}{2} \int _{0}^{1} \bigl\{ \bigl(1-r^{\rho } \bigr)^{\alpha }-\bigl(r^{\rho }\bigr)^{\alpha } \bigr\} r ^{\rho -1}\chi '\bigl(r^{\rho }h_{1}^{\rho }+ \bigl(1-r^{\rho }\bigr)h_{2}^{\rho }\bigr)\,dr \biggr\vert \\ &\quad\leq \frac{\rho (h_{2}^{\rho }-h_{1}^{\rho })}{2} \biggl( \int _{0} ^{1} r^{\rho -1}\,dr \biggr)^{1-1/q} \\ &\qquad{} \times \biggl( \int _{0}^{1} \bigl\vert \bigl(1-r^{\rho }\bigr)^{\alpha }-\bigl(r^{\rho } \bigr)^{\alpha } \bigr\vert r^{\rho -1} \bigl\vert \chi '\bigl(r^{\rho }h_{1}^{ \rho }+ \bigl(1-r^{\rho }\bigr)h_{2}^{\rho }\bigr) \bigr\vert ^{q}\,dr \biggr)^{1/q} \\ &\quad\leq \frac{\rho (h_{2}^{\rho }-h_{1}^{\rho })}{2} \biggl(\frac{1}{ \rho } \biggr)^{1-1/q} \\ &\qquad{} \times \biggl( \int _{0}^{1} \bigl\{ \bigl(1-r^{\rho } \bigr)^{ \alpha }+\bigl(r^{\rho }\bigr)^{\alpha } \bigr\} r^{\rho -1}r^{\rho }\bigl(1-r^{ \rho }\bigr)\bigl[ \bigl\vert \chi '\bigl(h_{1}^{\rho }\bigr) \bigr\vert ^{q}+ \bigl\vert \chi '\bigl(h_{2}^{\rho } \bigr) \bigr\vert ^{q}\bigr]\,dr \biggr) ^{1/q}. \end{aligned}$$

(27)

Since by using the change of variable $t=r^{\rho }$, we get

$$\begin{aligned} & \int _{0}^{1} \bigl\{ \bigl(1-r^{\rho } \bigr)^{\alpha }+\bigl(r^{\rho }\bigr)^{\alpha } \bigr\} r^{\rho -1}r ^{\rho }\bigl(1-r^{\rho }\bigr)\,dr \\ &\quad= \int _{0}^{1}\bigl(1-r^{\rho } \bigr)^{\alpha }r^{\rho -1}r^{\rho }\bigl(1-r^{ \rho } \bigr)\,dr+ \int _{0}^{1}\bigl(r^{\rho } \bigr)^{\alpha }r^{\rho -1}r^{\rho }\bigl(1-r ^{\rho }\bigr)\,dr \\ &\quad=\frac{1}{\rho }\beta (2,\alpha +2)+\frac{1}{\rho (\alpha +2)( \alpha +3)}. \end{aligned}$$

(28)

Hence using (28) in (27) we get(26). □

Corollary 3.7

Consider the similar assumptions of Theorem 3.6. If $\rho =1$, then

$$\begin{aligned} & \biggl\vert \frac{\chi (h_{1})+\chi (h_{2})}{2}-\frac{\varGamma (\alpha +1)}{2(h _{2}-h_{1})^{\alpha }} \bigl[J^{\alpha }_{h_{1}+}\chi (h_{2})+J^{ \alpha }_{h_{2}-} \chi (h_{1}) \bigr] \biggr\vert \\ & \quad \leq \frac{ (h_{2}-h_{1})}{2} \biggl( \biggl[\beta (2,\alpha +2)+ \frac{1}{( \alpha +2)(\alpha +3)} \biggr] \bigl[ \bigl\vert \chi '(h_{1}) \bigr\vert ^{q}+ \bigl\vert \chi '(h_{2}) \bigr\vert ^{q}\bigr] \biggr) ^{1/q}. \end{aligned}$$

(29)

Theorem 3.8

Let $\chi _{1}$, $\chi _{2}$be real valued, symmetric about $\frac{h_{1}^{\rho }+h_{2}^{\rho }}{2}$, nonnegative and $tgs$-convex functions on $[h_{1}^{\rho },h_{2}^{\rho }]$, where $\rho >0$. Then, for all $h_{1},h_{2}>0$and $\alpha >0$, we have

$$ \frac{\rho ^{\alpha }\ ^{\rho }I^{\alpha }_{h_{1}+}(\chi _{1}(h_{2}^{\rho })\chi _{2}(h_{2}^{\rho }))}{(h_{2}^{\rho }-h_{1}^{ \rho })^{\alpha }}\leq \frac{2\alpha (\alpha +1)[M(h_{1}^{\rho },h _{2}^{\rho })+N(h_{1}^{\rho },h_{2}^{\rho })]}{\varGamma (\alpha +5)} $$

(30)

and

$$\begin{aligned} &8\chi _{1} \biggl( \frac{h_{1}^{\rho }+h_{2}^{\rho }}{2} \biggr)\chi _{2} \biggl(\frac{h_{1}^{\rho }+h_{2}^{\rho }}{2} \biggr) \\ &\quad \leq \frac{\rho ^{\alpha }\ ^{\rho }I^{\alpha }_{h_{1}+}(\chi _{1}(h _{2}^{\rho })\chi _{2}(h_{2}^{\rho }))}{(h_{2}^{\rho }-h_{1}^{\rho })^{ \alpha }}+\frac{2\alpha (\alpha +1)[M(h_{1}^{\rho },h_{2}^{\rho })+N(h _{1}^{\rho },h_{2}^{\rho })]}{\varGamma (\alpha +5)}, \end{aligned}$$

(31)

where $M(h_{1}^{\rho },h_{2}^{\rho })=\chi _{1}(h_{1})\chi _{2}(h_{1})+ \chi _{1}(h_{2})\chi _{2}(h_{2})$and $N(h_{1}^{\rho },h_{2}^{\rho })= \chi _{1}(h_{1})\chi _{2}(h_{2})+\chi _{1}(h_{2})\chi _{2}(h_{1})$.

Proof

Since $\chi _{1}$ and $\chi _{2}$ are $tgs$-convex functions on $[h_{1},h_{2}]$, we can have

$$ \chi _{1}\bigl(r^{\rho }h_{1}^{\rho }+ \bigl(1-r^{\rho }\bigr)h_{2}^{\rho }\bigr)\leq r^{ \rho }\bigl(1-r^{\rho }\bigr) \bigl(\chi _{1} \bigl(h_{1}^{\rho }\bigr)+\chi _{1} \bigl(h_{2}^{\rho }\bigr)\bigr) $$

and

$$ \chi _{2}\bigl(r^{\rho }h_{1}^{\rho }+ \bigl(1-r^{\rho }\bigr)h_{2}^{\rho }\bigr)\leq r^{ \rho }\bigl(1-r^{\rho }\bigr) \bigl(\chi _{2} \bigl(h_{1}^{\rho }\bigr)+\chi _{2} \bigl(h_{2}^{\rho }\bigr)\bigr), $$

From the above, we obtain

$$\begin{aligned} &\chi _{1}\bigl(r^{\rho }h_{1}^{\rho }+ \bigl(1-r^{\rho }\bigr)h_{2}^{ \rho }\bigr)\chi _{2}\bigl(r^{\rho }h_{1}^{\rho }+ \bigl(1-r^{\rho }\bigr)h_{2}^{\rho }\bigr) \\ &\quad \leq r^{2\rho }\bigl(1-r^{\rho }\bigr)^{2}\bigl(\chi _{1}\bigl(h_{1}^{\rho }\bigr)+\chi _{1}\bigl(h _{2}^{\rho }\bigr)\bigr) \bigl( \chi _{2}\bigl(h_{1}^{\rho }\bigr)+\chi _{2}\bigl(h_{2}^{\rho }\bigr)\bigr). \end{aligned}$$

(32)

Multiplying both sides of (32) by $\frac{r^{\alpha \rho -1}}{ \varGamma (\alpha )}$, $\alpha >0$ and then integrating the resulting inequality with respect to r over $[0,1]$, we obtain

$$\begin{aligned} &\frac{1}{\varGamma (\alpha )} \int _{0}^{1}r^{\alpha \rho -1}\chi _{1}\bigl(r ^{\rho }h_{1}^{\rho }+ \bigl(1-r^{\rho }\bigr)h_{2}^{\rho }\bigr)\chi _{2}\bigl(r^{\rho }h _{1}^{\rho }+ \bigl(1-r^{\rho }\bigr)h_{2}^{\rho }\bigr)\,dr \\ &\quad \leq \frac{(\chi _{1}(h_{1}^{\rho })+\chi _{1}(h_{2}^{\rho }))(\chi _{2}(h_{1}^{\rho })+\chi _{2}(h_{2}^{\rho }))}{\varGamma (\alpha )} \int _{0}^{1}r^{2\rho } \bigl(1-r^{\rho }\bigr)^{2}\,dr. \end{aligned}$$

(33)

By the change of variable $t=r^{\rho }$, we get

$$ \int _{0}^{1}r^{2\rho } \bigl(1-r^{\rho }\bigr)^{2}\,dr=\frac{2\alpha (\alpha +1)}{ \rho \varGamma (\alpha +5)}. $$

(34)

Also by letting $x^{\rho }=r^{\rho }h_{1}^{\rho }+(1-r^{\rho })h_{2} ^{\rho }$, we obtain

$$\begin{aligned} &\frac{1}{\varGamma (\alpha )} \int _{0}^{1}r^{\alpha \rho -1}\chi _{1}\bigl(r ^{\rho }h_{1}^{\rho }+ \bigl(1-r^{\rho }\bigr)h_{2}^{\rho }\bigr)\chi _{2}\bigl(r^{\rho }h _{1}^{\rho }+ \bigl(1-r^{\rho }\bigr)h_{2}^{\rho }\bigr)\,dr \\ &\quad =\frac{\rho ^{\alpha -1}\ ^{\rho }I^{\alpha }_{h_{1}+}(\chi _{1}(h _{2}^{\rho })\chi _{2}(h_{2}^{\rho }))}{(h_{2}^{\rho }-h_{1}^{\rho })^{ \alpha }}. \end{aligned}$$

(35)

Hence from (33)–(35), we get (30).

Again using the $tgs$-convexity of $\chi _{1}$ and $\chi _{2}$ on $[h_{1}^{\rho },h_{2}^{\rho }]$, we find

$$\begin{aligned} &\chi _{1} \biggl(\frac{h_{1}^{\rho }+h_{2}^{\rho }}{2} \biggr)\chi _{2} \biggl(\frac{h_{1}^{\rho }+h_{2}^{\rho }}{2} \biggr) \\ &\quad\leq \chi _{1} \biggl(\frac{r^{\rho }h_{1}^{\rho }+(1-r^{\rho })h_{2} ^{\rho }}{2}+ \frac{r^{\rho }h_{2}^{\rho }+(1-r^{\rho })h_{1}^{\rho }}{2} \biggr) \\ & \qquad{}\times \chi _{2} \biggl(\frac{r^{\rho }h_{1}^{\rho }+(1-r ^{\rho })h_{2}^{\rho }}{2}+ \frac{r^{\rho }h_{2}^{\rho }+(1-r^{\rho })h _{1}^{\rho }}{2} \biggr) \\ &\quad\leq \frac{1}{4}\bigl[\chi _{1}\bigl(r^{\rho }h_{1}^{\rho }+ \bigl(1-r^{\rho }\bigr)h_{2} ^{\rho }\bigr)+\chi _{1}\bigl(r^{\rho }h_{2}^{\rho }+ \bigl(1-r^{\rho }\bigr)h_{1}^{\rho }\bigr)\bigr] \\ & \qquad{}\times \frac{1}{4}\bigl[\chi _{2}\bigl(r^{\rho }h_{1}^{\rho }+ \bigl(1-r ^{\rho }\bigr)h_{2}^{\rho }\bigr)+\chi _{1}\bigl(r^{\rho }h_{2}^{\rho }+ \bigl(1-r^{\rho }\bigr)h _{1}^{\rho }\bigr)\bigr] \\ &\quad=\frac{1}{16}\bigl[\chi _{1}\bigl(r^{\rho }h_{1}^{\rho }+ \bigl(1-r^{\rho }\bigr)h_{2} ^{\rho }\bigr)\chi _{2}(r^{\rho }h_{1}^{\rho }+ \bigl(1-r^{\rho }\bigr)h_{2}^{\rho } \\ &\qquad{} +\chi _{1}\bigl(r^{\rho }h_{2}^{\rho }+ \bigl(1-r^{\rho }\bigr)h_{1}^{ \rho }\bigr)\chi _{2}(r^{\rho }h_{2}^{\rho }+ \bigl(1-r^{\rho }\bigr)h_{1}^{\rho } \\ &\qquad{} +\chi _{1}\bigl(r^{\rho }h_{1}^{\rho }+ \bigl(1-r^{\rho }\bigr)h_{2}^{ \rho }\bigr)\chi _{2}(r^{\rho }h_{2}^{\rho }+ \bigl(1-r^{\rho }\bigr)h_{1}^{\rho } \\ &\qquad{} +\chi _{1}\bigl(r^{\rho }h_{2}^{\rho }+ \bigl(1-r^{\rho }\bigr)h_{1}^{ \rho }\bigr)\chi _{2}\bigl(r^{\rho }h_{1}^{\rho }+ \bigl(1-r^{\rho }\bigr)h_{2}^{\rho }\bigr)\bigr]. \end{aligned}$$

(36)

Multiplying both sides of (36) by $\frac{r^{\alpha \rho -1}}{ \varGamma (\alpha )}$, $\alpha >0$ and then integrating the resulting inequality with respect to r over $[0,1]$, we obtain

$$\begin{aligned} &\frac{1}{\rho \varGamma (\alpha +1)}\chi _{1} \biggl( \frac{h_{1}^{\rho }+h _{2}^{\rho }}{2} \biggr)\chi _{2} \biggl(\frac{h_{1}^{\rho }+h_{2}^{ \rho }}{2} \biggr) \\ &\quad\leq \frac{1}{16\varGamma (\alpha )} \biggl[ \int _{0}^{1}r^{\alpha \rho -1} \chi _{1}\bigl(r^{\rho }h_{1}^{\rho }+ \bigl(1-r^{\rho }\bigr)h_{2}^{\rho }\bigr)\chi _{2}(r ^{\rho }h_{1}^{\rho }+ \bigl(1-r^{\rho }\bigr)h_{2}^{\rho }\,dr \\ & \qquad{}+ \int _{0}^{1}r^{\alpha \rho -1}\chi _{1}\bigl(r^{\rho }h_{2} ^{\rho }+ \bigl(1-r^{\rho }\bigr)h_{1}^{\rho }\bigr)\chi _{2}(r^{\rho }h_{2}^{\rho }+\bigl(1-r ^{\rho }\bigr)h_{1}^{\rho }\,dr \\ &\qquad{} + \int _{0}^{1}r^{\alpha \rho -1}\chi _{1}\bigl(r^{\rho }h_{1} ^{\rho }+ \bigl(1-r^{\rho }\bigr)h_{2}^{\rho }\bigr)\chi _{2}(r^{\rho }h_{2}^{\rho }+\bigl(1-r ^{\rho }\bigr)h_{1}^{\rho }\,dr \\ & \qquad{}+ \int _{0}^{1}r^{\alpha \rho -1}\chi _{1}\bigl(r^{\rho }h_{2} ^{\rho }+ \bigl(1-r^{\rho }\bigr)h_{1}^{\rho }\bigr)\chi _{2}\bigl(r^{\rho }h_{1}^{\rho }+ \bigl(1-r ^{\rho }\bigr)h_{2}^{\rho }\bigr)\,dr \biggr]. \end{aligned}$$

That is,

$$\begin{aligned} &8\chi _{1} \biggl(\frac{h_{1}^{\rho }+h_{2}^{\rho }}{2} \biggr)\chi _{2} \biggl(\frac{h_{1}^{\rho }+h_{2}^{\rho }}{2} \biggr) \\ &\quad\leq \frac{\rho \varGamma (\alpha +1)}{2(h_{2}^{\rho }-h_{1}^{\rho })^{ \alpha }}\ ^{\rho }I^{\alpha }_{h_{1}+} \bigl[\chi _{1}\bigl(h_{2}^{\rho }\bigr)\chi _{2}\bigl(h_{2}^{\rho }\bigr)+\chi _{1}\bigl(h_{2}^{\rho }\bigr)\chi _{2}\bigl(h_{1}^{\rho }\bigr)\bigr] \\ & \qquad{}+ ^{\rho }I^{\alpha }_{h_{1}+}\bigl[\chi _{1}\bigl(h_{1}^{\rho }\bigr) \chi _{2}\bigl(h_{1}^{\rho }\bigr)+\chi _{1}\bigl(h_{1}^{\rho }\bigr)\chi _{2}\bigl(h_{2}^{\rho }\bigr)\bigr]. \end{aligned}$$

After some calculations we get the required inequality (31). □

Remark 3.2

1. By letting $\rho =1$ in Theorem 3.8 the inequalities (30) and (31) give the inequalities $(3.11)$ and $(3.12)$, respectively, in Theorem 3.2 of [18].

2. By letting $\rho =\alpha = 1$ in Theorem 3.8 the inequality (30) becomes the inequality in Theorem $(2.2)$ of [18].

4 Inequalities via ψ-Riemann–Liouville fractional integrals

First we establish the Hermite–Hadamard inequality via ψ-Riemann–Liouville fractional integrals.

Theorem 4.1

Let $\chi : [h_{1},h_{2}]\rightarrow \mathbb{R}$be a positive function, for $0\leq h_{1}< h_{2}$, and $\chi \in L_{1}[h_{1},h _{2}]$. Let $\psi (z)$be an increasing and positive monotone function on $(h_{1},h_{2}]$, with continuous derivative $\psi '(z)$on $(h_{1},h_{2})$. Letχbe a $tgs$-convex function, then the following inequalities for a fractional integral hold:

$$\begin{aligned} &2\chi \biggl(\frac{h_{1}+h_{2}}{2} \biggr) \\ & \quad\leq \frac{\varGamma (\gamma +1)}{2(h_{2}-h_{1})^{\gamma }}\bigl[ \mathcal{I}^{\gamma :\psi }_{\psi ^{-1}(h_{1})+}( \chi \circ \psi ) \bigl( \psi ^{-1}(h_{2})\bigr)+ \mathcal{I}^{\gamma :\psi }_{\psi ^{-1}(h_{2})-}( \chi \circ \psi ) \bigl(\psi ^{-1}(h_{1})\bigr)\bigr] \\ &\quad \leq \frac{\gamma [\chi (h_{1})+\chi (h_{2})]}{(\gamma +1)(\gamma +2)}. \end{aligned}$$

(37)

Proof

Since χ is $tgs$-convex, we have

$$ \chi \biggl( \frac{u+v}{2} \biggr) \leq \frac{\chi (u)+\chi (v)}{2^{2}}. $$

Let $u=rh_{1}+(1-r)h_{2}$ and $v=rh_{2}+(1-r)h_{1}$, we get

$$ 4\chi \biggl( \frac{h_{1}+h_{2}}{2} \biggr) \leq \chi \bigl(rh _{1}+(1-r)h_{2}\bigr)+\chi \bigl(rh_{2}+(1-r)h_{1} \bigr). $$

(38)

Multiplying by $r^{\gamma -1}$ on both sides of inequality (38) and then integrating with respect to r over $[0,1]$ imply

$$ \frac{4}{\gamma }\chi \biggl( \frac{h_{1}+h_{2}}{2} \biggr) \leq \int _{0}^{1}r^{\gamma -1}\chi \bigl(rh _{1}+(1-r)h_{2}\bigr)\,dr+ \int _{0}^{1}r^{\gamma -1}\chi \bigl(rh_{2}+(1-r)h_{1}\bigr)\,dr. $$

(39)

Now consider

$$\begin{aligned} &\frac{\varGamma (\gamma +1)}{2(h_{2}-h_{1})^{\gamma }}\bigl[\mathcal{I}^{ \gamma :\psi }_{\psi ^{-1}(h_{1})+}( \chi \circ \psi ) \bigl(\psi ^{-1}(h_{2})\bigr)+ \mathcal{I}^{\gamma :\psi }_{\psi ^{-1}(h_{2})-}(\chi \circ \psi ) \bigl( \psi ^{-1}(h_{1})\bigr)\bigr] \\ &\quad= \frac{\varGamma (\gamma +1)}{2(h_{2}-h_{1})^{\gamma }\varGamma (\gamma )} \biggl[ \int _{\psi ^{-1}(h_{1})}^{\psi ^{-1}(h_{2})}\psi '(g) \bigl(h_{2}- \psi (g)\bigr)^{\gamma -1}(\chi \circ \psi ) (g) \,dg \\ &\qquad{} + \int _{\psi ^{-1}(h_{1})}^{\psi ^{-1}(h_{2})}\psi '(g) \bigl( \psi (g)-h_{1}\bigr)^{\gamma -1}(\chi \circ \psi ) (g)\,dg \biggr] \\ &\quad=\frac{\gamma }{2} \biggl[ \int _{\psi ^{-1}(h_{1})}^{\psi ^{-1}(h_{2})} \biggl(\frac{h_{2}-\psi (g)}{h_{2}-h_{1}} \biggr)^{\gamma -1}\chi \bigl( \psi (g)\bigr)\frac{\psi '(g)}{h_{2}-h_{1}}\,dg \\ &\qquad{} + \int _{\psi ^{-1}(h_{1})}^{\psi ^{-1}(h_{2})} \biggl(\frac{ \psi (g)-h_{1}}{h_{2}-h_{1}} \biggr)^{\gamma -1}\chi \bigl(\psi (g)\bigr)\frac{ \psi '(g)}{h_{2}-h_{1}}\,dg \biggr] \\ &\quad=\frac{\gamma }{2} \biggl[ \int _{0}^{1}r^{\gamma -1}\chi \bigl(rh_{1}+(1-r)h _{2}\bigr)\,dr+ \int _{0}^{1}r^{\gamma -1}\chi \bigl(rh_{2}+(1-r)h_{1}\bigr)\,dr \biggr] \\ &\quad\geq 2\chi \biggl( \frac{h_{1}+h_{2}}{2} \biggr), \end{aligned}$$

(40)

by using (39). Thus first inequality of (37) is proved.

For the next inequality we consider

$$ \chi \bigl(rh_{1}+(1-r)h_{2}\bigr)\leq r(1-r)\bigl[ \chi (h_{1})+\chi (h_{2})\bigr] $$

and

$$ \chi \bigl(rh_{2}+(1-r)h_{1}\bigr)\leq r(1-r)\bigl[ \chi (h_{2})+\chi (h_{1})\bigr]. $$

We add

$$ \chi \bigl(rh_{1}+(1-r)h_{2}\bigr)+ \chi \bigl(rh_{2}+(1-r)h_{1}\bigr)\leq 2r(1-r)\bigl[ \chi (h_{1})+\chi (h_{2})\bigr]. $$

(41)

Multiplying by $r^{\gamma -1}$ on both sides of inequality (41) and then integrating with respect to r over $[0,1]$ imply

$$\begin{aligned} & \int _{0}^{1}r^{\gamma -1}\chi \bigl(rh_{1}+(1-r)h_{2}\bigr)\,dr+ \int _{0}^{1}r ^{\gamma -1}\chi \bigl(rh_{2}+(1-r)h_{1}\bigr)\,dr \\ &\quad \leq \frac{2[\chi (h_{1})+\chi (h_{2})]}{(\gamma +1)(\gamma +2)}. \end{aligned}$$

That is,

$$\begin{aligned} &\frac{\varGamma (\gamma +1)}{(h_{2}-h_{1})^{\gamma }}\bigl[\mathcal{I}^{ \gamma :\psi }_{\psi ^{-1}(h_{1})+}( \chi \circ \psi ) \bigl(\psi ^{-1}(h_{2})\bigr)+ \mathcal{I}^{\gamma :\psi }_{\psi ^{-1}(h_{2})-}(\chi \circ \psi ) \bigl( \psi ^{-1}(h_{1})\bigr)\bigr] \\ &\quad \leq \frac{\gamma [\chi (h_{1})+\chi (h_{2})]}{(\gamma +1)(\gamma +2)}. \end{aligned}$$

Hence the proof is completed. □

Remark 4.1

(1) By letting $\psi (g)=g$ in (37) of Theorem 4.1 we get inequality 3.1 of Theorem 3.1 in [18].

(2) By letting $\psi (g)=g$ and $\gamma =1$ in (37) of Theorem 4.1 we get inequality 2.2 of Theorem 2.1 in [18].

For the next two results we use Lemma 2.1 and Lemma 2.2, respectively.

Theorem 4.2

Let $\chi : [h_{1},h_{2}]\rightarrow \mathbb{R}$be a nonnegative differentiable mapping, for $0\leq h_{1}< h_{2}$. Let $\psi (g)$be an increasing and positive monotone function on $(h_{1},h_{2}]$, with continuous derivative $\psi '(g)$on $(h_{1},h _{2})$and $\gamma \in (0,1)$. If $|\chi '|^{q}$is $tgs$-convex and $q\geq 1$, then the following inequality for fractional integral holds:

$$\begin{aligned} & \biggl\vert \frac{\chi (h_{1})+\chi (h_{2})}{2}- \frac{\varGamma (\gamma +1)}{2(h _{2}-h_{1})^{\gamma }}\bigl[\mathcal{I}^{\gamma :\psi }_{\psi ^{-1}(h_{1})+}( \chi \circ \psi ) \bigl(\psi ^{-1}(h_{2})\bigr) \\ &\qquad{} +\mathcal{I}^{\gamma :\psi }_{\psi ^{-1}(h_{2})-}(\chi \circ \psi ) \bigl(\psi ^{-1}(h_{1})\bigr)\bigr] \biggr\vert \\ &\quad \leq \frac{h_{2}-h_{1}}{2} \biggl[\frac{2}{\gamma +1} \biggl(1- \frac{1}{2^{ \gamma }} \biggr) \biggr]^{\frac{q-1}{q}} \biggl( \frac{2( \vert \chi '(h _{1}) \vert ^{q}+ \vert \chi (h_{2}) \vert ^{q})}{(\gamma +2)(\gamma +3)} \biggr)^{ \frac{1}{q}}. \end{aligned}$$

(42)

Proof

First note that, for every $g\in (\psi ^{-1}(h_{1}),\psi ^{-1}(h_{2}))$, we have $h_{1}<\psi (g)<h_{2}$. Let $r=\frac{h_{2}-\psi (g)}{h_{2}-h _{1}}$, then we have $\psi (g)=rh_{1}+(1-r)h_{2}$. Applying Lemma 2.1 and the $tgs$-convexity of $|\chi '|$, we obtain

$$\begin{aligned} & \biggl\vert \frac{\chi (h_{1})+\chi (h_{2})}{2}- \frac{\varGamma (\gamma +1)}{2(h _{2}-h_{1})^{\gamma }}\bigl[\mathcal{I}^{\gamma :\psi }_{\psi ^{-1}(h_{1})+}( \chi \circ \psi ) \bigl(\psi ^{-1}(h_{2})\bigr) \\ &\qquad{} +\mathcal{I}^{\gamma :\psi }_{\psi ^{-1}(h_{2})-}(\chi \circ \psi ) \bigl(\psi ^{-1}(h_{1})\bigr)\bigr] \biggr\vert \\ &\quad\leq \frac{1}{2(h_{2}-h_{1})^{\gamma }} \int _{\psi ^{-1}(h_{1})}^{ \psi ^{-1}(h_{2})} \bigl\vert \bigl(\psi (g)-h_{1}\bigr)^{\gamma }-\bigl(h_{2}-\psi (g) \bigr)^{\gamma } \bigr\vert \bigl\vert \bigl(\chi '\circ \psi \bigr) (g) \bigr\vert \,d\psi (g) \\ &\quad=\frac{h_{2}-h_{1}}{2} \int _{0}^{1} \bigl\vert (1-r)^{\gamma }-r^{\gamma } \bigr\vert \bigl\vert \chi '\bigl(rh_{1}+(1-r)h_{2} \bigr) \bigr\vert \,dr \\ &\quad\leq \frac{h_{2}-h_{1}}{2} \int _{0}^{1} \bigl\vert (1-r)^{\gamma }-r^{\gamma } \bigr\vert r(1-r)\bigl[ \bigl\vert \chi '(h_{2})+| \chi '(h_{2}) \bigr\vert \bigr]\,dr \\ &\quad\leq \frac{h_{2}-h_{1}}{2} \int _{0}^{1}\bigl[(1-r)^{\gamma }+r^{\gamma } \bigr]r(1-r)\bigl[ \bigl\vert \chi '(h_{2})+|\chi '(h_{2}) \bigr\vert \bigr]\,dr \\ &\quad=\frac{h_{2}-h_{1}}{(\gamma +2)(\gamma +3)} \bigl[ \bigl\vert \chi '(h_{2})+| \chi '(h _{2}) \bigr\vert \bigr]. \end{aligned}$$

(43)

Since

$$ \int _{0}^{1}\bigl[(1-r)^{\gamma }+r^{\gamma } \bigr]r(1-r)\,dr=\frac{2}{(\gamma +2)( \gamma +3)}, $$

we get the required inequality (42) for $q=1$.

Now consider the case when $q>1$. Again using Lemma 2.1, the power mean inequality and the s-convexity of $|\chi '|^{q}$ on $[a_{1},a _{2}]$, we get

$$\begin{aligned} & \biggl\vert \frac{\chi (h_{1})+\chi (h_{2})}{2}- \frac{\varGamma (\gamma +1)}{2(h _{2}-h_{1})^{\gamma }}\bigl[\mathcal{I}^{\gamma :\psi }_{\psi ^{-1}(h_{1})+}( \chi \circ \psi ) \bigl(\psi ^{-1}(h_{2})\bigr) \\ &\qquad{} +\mathcal{I}^{\gamma :\psi }_{\psi ^{-1}(h_{2})-}(\chi \circ \psi ) \bigl(\psi ^{-1}(h_{1})\bigr)\bigr] \biggr\vert \\ &\quad\leq \frac{1}{2(h_{2}-h_{1})^{\gamma }} \int _{\psi ^{-1}(h_{1})}^{ \psi ^{-1}(h_{2})} \bigl\vert \bigl(\psi (g)-h_{1}\bigr)^{\gamma }-\bigl(h_{2}-\psi (g) \bigr)^{\gamma } \bigr\vert \bigl\vert \bigl(\chi '\circ \psi \bigr) (g) \bigr\vert \,d\psi (g) \\ &\quad=\frac{h_{2}-h_{1}}{2} \int _{0}^{1} \bigl\vert (1-r)^{\gamma }-r^{\gamma } \bigr\vert \bigl\vert \chi '\bigl(rh_{1}+(1-r)h_{2} \bigr) \bigr\vert \,dr \\ &\quad=\frac{h_{2}-h_{1}}{2} \biggl( \int _{0}^{1} \bigl\vert (1-r)^{\gamma }-r^{ \gamma } \bigr\vert \,dr \biggr)^{1-\frac{1}{q}} \\ &\qquad{} \times \biggl( \int _{0}^{1} \bigl\vert (1-r)^{\gamma }-r^{\gamma } \bigr\vert \bigl\vert \chi '\bigl(rh_{1}+(1-r)h_{2} \bigr) \bigr\vert ^{q}\,dr \biggr)^{\frac{1}{q}} \\ &\quad=\frac{h_{2}-h_{1}}{2} \biggl( \int _{0}^{1} \bigl\vert (1-r)^{\gamma }-r^{ \gamma } \bigr\vert \,dr \biggr)^{\frac{q-1}{q}} \\ &\qquad{} \times \biggl( \int _{0}^{1}\bigl[(1-r)^{\gamma }+r^{\gamma } \bigr]r(1-r)\bigl[ \bigl\vert \chi '(h_{2}) \bigr\vert ^{q}+\chi '(h_{2}) \vert ^{q}\bigr]\,dr \biggr)^{\frac{1}{q}} \\ &\quad=\frac{h_{2}-h_{1}}{2} \biggl[\frac{2}{\gamma +1} \biggl(1- \frac{1}{2^{ \gamma }} \biggr) \biggr]^{\frac{q-1}{q}} \biggl( \frac{2( \vert \chi '(h _{1}) \vert ^{q}+ \vert \chi (h_{2}) \vert ^{q})}{(\gamma +2)(\gamma +3)} \biggr)^{ \frac{1}{q}}. \end{aligned}$$

(44)

We have

$$\begin{aligned} \int _{0}^{1} \bigl\vert (1-r)^{\gamma }-r^{\gamma } \bigr\vert \,dr &= \int _{0}^{1/2}\bigl[(1-r)^{ \gamma }-r^{\gamma } \bigr]\,dr+ \int _{1/2}^{1}\bigl[r^{\gamma }-(1-r)^{\gamma } \bigr]\,dr \\ &=\frac{2}{\gamma +1} \biggl(1-\frac{1}{2^{\gamma }} \biggr). \end{aligned}$$

This completes the proof. □

Corollary 4.3

Under the similar conditions of Theorem 4.2.

1. If $\psi (g)=g$, then we get

$$\begin{aligned} & \biggl\vert \frac{\chi (h_{1})+\chi (h_{2})}{2}- \frac{\varGamma (\gamma +1)}{2(h _{2}-h_{1})^{\gamma }}\bigl[J^{\gamma }_{h_{1}+}\chi (h_{2})+J^{\gamma } _{h_{2}-}\chi (h_{1})\bigr] \biggr\vert \\ &\quad\leq \frac{h_{2}-h_{1}}{2} \biggl[\frac{2}{\gamma +1} \biggl(1- \frac{1}{2^{ \gamma }} \biggr) \biggr]^{\frac{q-1}{q}} \biggl( \frac{2( \vert \chi '(h _{1}) \vert ^{q}+ \vert \chi (h_{2}) \vert ^{q})}{(\gamma +2)(\gamma +3)} \biggr)^{ \frac{1}{q}}. \end{aligned}$$

(45)

2. If $\psi (g)=g$and $\gamma =1$, then we get

$$\begin{aligned} & \biggl\vert \frac{\chi (h_{1})+\chi (h_{2})}{2}- \frac{2}{(h_{2}-h_{1})} \int _{h_{1}}^{h_{2}}\chi (g)\,dg \biggr\vert \\ &\quad\leq \frac{h_{2}-h_{1}}{2} \biggl[\frac{1}{2} \biggr]^{ \frac{q-1}{q}} \biggl( \frac{ \vert \chi '(h_{1}) \vert ^{q}+ \vert \chi (h_{2}) \vert ^{q}}{3} \biggr)^{\frac{1}{q}}. \end{aligned}$$

(46)

Theorem 4.4

Let $\chi : [h_{1},h_{2}]\rightarrow \mathbb{R}$be a nonnegative differentiable mapping, for $0\leq h_{1}< h_{2}$. Let $\psi (g)$be an increasing and positive monotone function on $(h_{1},h_{2}]$, with continuous derivative $\psi '(g)$on $(h_{1},h _{2})$and $\gamma \in (0,1)$. If $|\chi '|$is $tgs$-convex, then the following inequality for fractional integral holds:

$$\begin{aligned} & \biggl\vert \frac{\varGamma (\gamma +1)}{2(h_{2}-h_{1})^{\gamma }}\bigl[ \mathcal{I}^{\gamma :\psi }_{\psi ^{-1}(h_{1})+}(\chi \circ \psi ) \bigl( \psi ^{-1}(h_{2})\bigr)+\mathcal{I}^{\gamma :\psi }_{\psi ^{-1}(h_{2})-}( \chi \circ \psi ) \bigl(\psi ^{-1}(h_{1})\bigr)\bigr] \\ & \qquad{}-\chi \biggl( \frac{h_{1}+h_{2}}{2} \biggr) \biggr\vert \\ &\quad \leq \frac{\chi (h_{2})-\chi (h_{1})}{2}+\frac{h_{2}-h_{1}}{(\gamma +2)(\gamma +3)} \bigl( \bigl\vert \chi '(h_{1}) \bigr\vert + \bigl\vert \chi (h_{2}) \bigr\vert \bigr) . \end{aligned}$$

(47)

Proof

From Lemma 2.2 and the $tgs$-convexity of $|\chi '|$, we have

$$\begin{aligned} & \biggl\vert \frac{\varGamma (\gamma +1)}{2(h_{2}-h_{1})^{\gamma }}\bigl[ \mathcal{I}^{\gamma :\psi }_{\psi ^{-1}(h_{1})+}(\chi \circ \psi ) \bigl( \psi ^{-1}(h_{2})\bigr)+\mathcal{I}^{\gamma :\psi }_{\psi ^{-1}(h_{2})-}( \chi \circ \psi ) \bigl(\psi ^{-1}(h_{1})\bigr)\bigr] \\ &\qquad{} -\chi \biggl( \frac{h_{1}+h_{2}}{2} \biggr) \biggr\vert \\ &\quad= \biggl\vert \int _{\psi ^{-1}(h_{1})}^{\psi ^{-1}(h_{2})}k\bigl(\chi '\circ \psi \bigr) (g)\psi '(g)\,dg \\ &\qquad{} + \frac{1}{2(h_{2}-h_{1})^{\gamma }} \int _{\psi ^{-1}(h_{1})}^{\psi ^{-1}(h_{2})}\bigl[\bigl(\psi (g)-h_{1}\bigr)^{\gamma }-\bigl(h _{2}-\psi (g) \bigr)^{\gamma }\bigr]\bigl(\chi '\circ \psi \bigr) (g)\psi '(g)\,dg \biggr\vert \\ &\quad\leq \biggl\vert \int _{\psi ^{-1}(h_{1})}^{\psi ^{-1}(h_{2})}k\bigl(\chi '\circ \psi \bigr) (g)\psi '(g)\,dg \biggr\vert \\ &\qquad{} + \biggl\vert \frac{1}{2(h_{2}-h_{1})^{\gamma }} \int _{\psi ^{-1}(h_{1})}^{\psi ^{-1}(h_{2})}\bigl[\bigl(\psi (z)-h_{1}\bigr)^{\gamma }-\bigl(h _{2}-\psi (g) \bigr)^{\gamma }\bigr]\bigl(\chi '\circ \psi \bigr) (g)\psi '(g)\,dg \biggr\vert \\ &\quad:=S_{1}+S_{2}, \end{aligned}$$

(48)

where

$$\begin{aligned} &S_{1}:= \biggl\vert \int _{\psi ^{-1}(h_{1})}^{\psi ^{-1}(h_{2})}k\bigl(\chi '\circ \psi \bigr) (g)\psi '(g)\,dg \biggr\vert , \\ &S_{2}:= \biggl\vert \frac{1}{2(h_{2}-h_{1})^{\gamma }} \int _{\psi ^{-1}(h_{1})} ^{\psi ^{-1}(h_{2})}\bigl[\bigl(\psi (z)-h_{1}\bigr)^{\gamma }-\bigl(h_{2}-\psi (g) \bigr)^{ \gamma }\bigr]\bigl(\chi '\circ \psi \bigr) (g)\psi '(g)\,dg \biggr\vert , \end{aligned}$$

and k is defined as in Lemma 2.2. Note that

$$ S_{1}=\frac{\chi (h_{2})-\chi (h_{1})}{2}, $$

(49)

and from Theorem 4.2 for the case $q=1$, we have

$$ S_{2}\leq \frac{h_{2}-h_{1}}{(\gamma +2)(\gamma +3)} \bigl( \bigl( \bigl\vert \chi '(h_{1}) \bigr\vert + \bigl\vert \chi (h_{2}) \bigr\vert \bigr) \bigr). $$

(50)

Hence by using (49) and (50) in (48), we get (47). □

Corollary 4.5

Assume the similar conditions of Theorem 4.4.

1. If $\psi (g)=g$, then we get

$$\begin{aligned} & \biggl\vert \frac{\varGamma (\gamma +1)}{2(h_{2}-h_{1})^{\gamma }} \bigl[J^{ \gamma }_{h_{1}+}\chi (h_{2})+J^{\gamma }_{h_{2}-} \chi (h_{1})\bigr]- \chi \biggl( \frac{h_{1}+h_{2}}{2} \biggr) \biggr\vert \\ &\quad\leq \frac{\chi (h_{2})-\chi (h_{1})}{2}+\frac{h_{2}-h_{1}}{(\gamma +2)(\gamma +3)} \bigl( \bigl( \bigl\vert \chi '(h_{1}) \bigr\vert + \bigl\vert \chi (h_{2}) \bigr\vert \bigr) \bigr). \end{aligned}$$

(51)

2. If $\psi (g)=g$and $\gamma =1$, then we get

$$\begin{aligned} & \biggl\vert \frac{2}{(h_{2}-h_{1})} \int _{h_{1}}^{h_{2}}\chi (g)\,dg- \chi \biggl( \frac{h_{1}+h_{2}}{2} \biggr) \biggr\vert \\ &\quad\leq \frac{\chi (h_{2})-\chi (h_{1})}{2}+\frac{h_{2}-h_{1}}{6} \bigl( \bigl\vert \chi '(h_{1}) \bigr\vert + \bigl\vert \chi (h_{2}) \bigr\vert \bigr). \end{aligned}$$

(52)

5 Conclusion

In this paper, we proved in Theorem 3.1 the Hermite–Hadamard inequality for $tgs$-convex functions via Katugampola fractional integrals. From Theorems 3.2–3.6, we established a Hermite–Hadamard type inequality for $tgs$-convex functions via Katugampola fractional integrals. From Corollaries 3.3 and 3.5 we obtained a new Hermite–Hadamard type inequality for $tgs$-convex functions via Riemann–Liouville fractional and classical integrals. Also from Corollary 3.7 we obtained a new Hermite–Hadamard type inequality for $tgs$-convex functions via Riemann–Liouville fractional integrals.

On the other hand, from Theorem 4.1 we obtained the Hermite–Hadamard inequality for $tgs$-convex functions via ψ-Riemann–Liouville fractional integrals. From Theorems 4.2 and 4.4, we established a Hermite–Hadamard type inequality for $tgs$-convex functions via ψ-Riemann–Liouville fractional integrals. From Corollaries 4.3 and 4.5 we obtained a new Hermite–Hadamard type inequality for $tgs$-convex functions via Riemann–Liouville fractional and classical integrals.