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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.

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.

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

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

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].

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)

Conclusion

In this paper, we proved in Theorem 3.1 the Hermite–Hadamard inequality for \(tgs\)-convex functions via Katugampola fractional integrals. From Theorems 3.23.6, 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.

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Acknowledgements

The authors would like thank to the referees for helpful comments and valuable suggestions.

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This research article is supported by National University of Sciences and Technology (NUST), Islamabad, Pakistan.

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The two authors contributed equally to this work. Both authors read and approved the final manuscript.

Correspondence to Naila Mehreen.

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Mehreen, N., Anwar, M. On some Hermite–Hadamard type inequalities for \(tgs\)-convex functions via generalized fractional integrals. Adv Differ Equ 2020, 6 (2020). https://doi.org/10.1186/s13662-019-2457-x

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Keywords

  • Hermite–Hadamard inequality
  • \(tgs\)-convex functions
  • Riemann–Liouville fractional integrals
  • Katugampola fractional integrals
  • ψ-Riemann–Liouville fractional integrals