Skip to main content

Hermite–Hadamard-type inequalities involving ψ-Riemann–Liouville fractional integrals via s-convex functions

Abstract

In this paper, we establish some new Hermite–Hadamard-type inequalities involving ψ-Riemann–Liouville fractional integrals via s-convex functions in the second sense. Meanwhile, we present many useful estimates on these types of new Hermite–Hadamard-type inequalities. Finally, we give some applications to special means of real numbers.

Introduction

The classical Hermite–Hadamard inequality is as follows:

$$\begin{aligned} g \biggl(\frac{a+b}{2} \biggr)\leq \frac{1}{b-a} \int _{a}^{b}g (s )\,ds\leq \frac{g (a )+g (b )}{2} \end{aligned}$$
(1)

for convex functions \(g: [a, b]\subset R\rightarrow R\) (see [1]).

In the past decade, fractional calculus has been regarded as one of the best tools to describe long-memory processes. Many researchers are interested in such a model. The most important of these models are described by differential equations with fractional derivatives. Their evolution is much more complex than the classical integer-order case, and the corresponding theory is also more difficult in the integer-order case. The theory of fractional integral inequalities plays an important role in mathematics.

The Hermite–Hadamard integral inequality for convex functions is one of the most famous inequalities. Ten recently published papers [211] are focused on the generalizations and variants for the convexity and Hermite–Hadamard inequality. Many mathematicians devoted to the promotion and expansion of (1). For more information, refer to [1, 1218] and closely related references.

With a wide application of fractional integration and Hermite–Hadamard inequality, many researchers extended their research to the Hermite–Hadamard inequality, including fractional integration rather than ordinary integration; see [1927]. Sarikaya et al. [19] derived an interesting Hermite–Hadamard-type inequality, which contains the fractional integral instead of the ordinary one. The study attracted many researchers to consider the problem. So far, some new integral inequalities have been obtained by using fractional calculus. Sousa et al. [28] introduced fractional integral operators with ψ-Riemann–Liouville kernel and proved similar inequalities.

In addition to the classical convex functions, Hudzik and Maligranda [29] introduced the definition of s-convex functions in the second sense.

Definition 1.1

(see [30, Definition 1.4])

A function \(g: I \subseteq R_{+}\rightarrow R_{+}\) is said to be s-convex in the second sense on I if inequality \(g (\lambda x+ (1-\lambda )y )\leq \lambda ^{s}g (x )+ (1-\lambda )^{s}g (y )\) for all \(x, y\in I\) and \(\lambda \in [0, 1]\) and for some fixed \(s\in (0, 1]\).

Definition 1.2

(see [28, Definition 4])

Let \((a, b)\ (-\infty \leq a < b \leq \infty )\) be a finite or infinite interval of the real line R, and let \(\alpha > 0\). Also, let \(\psi (x)\) be an increasing positive function on \((a, b]\) with continuous derivative \(\psi '(x)\) on \((a, b)\). Then the left- and right-sided ψ-Riemann–Liouville fractional integrals of a function f with respect to the function ψ on \([a, b]\) are defined by

$$\begin{aligned} &I_{a^{+}}^{\alpha : \psi }g(x)=\frac{1}{\varGamma (\alpha )} \int _{a}^{x} \psi '(t) \bigl(\psi (x)- \psi (t)\bigr)^{\alpha -1}g(t)\,dt, \\ &I_{b^{-}}^{\alpha : \psi }g(x)=\frac{1}{\varGamma (\alpha )} \int _{x}^{b} \psi '(t) \bigl(\psi (t)- \psi (x)\bigr)^{\alpha -1}g(t)\,dt, \end{aligned}$$

respectively, where Γ is the gamma function.

Lemma 1.3

Let\(h: [b,c] \rightarrow R\)be a differentiable mapping on\((b,c)\)with\(b < c\). Also, let\(h\in L[b,c]\). Then we have the following equality for fractional integrals:

$$\begin{aligned} &\frac{h(b)+h(c)}{2}-\frac{\varGamma (\alpha +1)}{2(c-b)^{\alpha }} \bigl[I_{ \psi ^{-1}(b)^{+}}^{\alpha : \psi }\bigl( {h^{\circ }\psi }\bigr) \bigl( \psi ^{-1}(c)\bigr)+I_{ \psi ^{-1}(c)^{-}}^{\alpha : \psi }\bigl( {h^{\circ }\psi }\bigr) \bigl(\psi ^{-1}(b)\bigr) \bigr] \\ &\quad= \frac{c-b}{2} \int _{0}^{1}\bigl((1-t)^{\alpha }-t^{\alpha } \bigr)h'\bigl(tb+(1-t)c\bigr)\,dt. \end{aligned}$$
(2)

Proof

From [31] we have

$$\begin{aligned} &\frac{h(b)+h(c)}{2}-\frac{\varGamma (\alpha +1)}{2(c-b)^{\alpha }} \bigl[I_{ \psi ^{-1}(b)^{+}}^{\alpha : \psi }\bigl( {h^{\circ }\psi }\bigr) \bigl(\psi ^{-1}(c)\bigr)+I_{ \psi ^{-1}(c)^{-}}^{\alpha : \psi } \bigl( {h^{\circ }\psi }\bigr) \bigl(\psi ^{-1}(b)\bigr) \bigr] \\ &\quad= \frac{1}{2(c-b)^{\alpha }} \int _{\psi ^{-1}(b)}^{\psi ^{-1}(c)}\bigl[\bigl( \psi (\nu )-b \bigr)^{\alpha }-\bigl(c-\psi (\nu )\bigr)^{\alpha }\bigr]\bigl( {h^{\prime \,\circ }\psi }\bigr) (\nu )\psi '(\nu )\,d \nu \\ &\quad= \frac{1}{2} \int _{\psi ^{-1}(b)}^{\psi ^{-1}(c)} \biggl[ \biggl( \frac{\psi (\nu )-b}{c-b} \biggr)^{\alpha }- \biggl( \frac{c-\psi (\nu )}{c-b} \biggr)^{\alpha } \biggr] \bigl( {h^{\prime \,\circ }\psi }\bigr) (\nu )\psi '(\nu )\,d \nu \\ &\qquad \biggl(\text{let } t=\frac{c-\psi (\nu )}{c-b}\biggr) \\ &\quad = \frac{c-b}{2} \int _{0}^{1}\bigl((1-t)^{\alpha }-t^{\alpha } \bigr)h'\bigl(tb+(1-t)c\bigr)\,dt. \end{aligned}$$

The proof is completed. □

Lemma 1.4

Let\(h: [b,c] \rightarrow R\)be a differentiable mapping on\((b,c)\)with\(b < c\). If\(h\in L[b,c]\), then we have the following equality for fractional integrals:

$$\begin{aligned} &\frac{\varGamma (\alpha +1)}{2(c-b)^{\alpha }} \bigl[I_{\psi ^{-1}(b)^{+}}^{ \alpha : \psi } \bigl({h^{\circ }\psi }\bigr) \bigl( \psi ^{-1}(c) \bigr)+I_{\psi ^{-1}(c)^{-}}^{\alpha : \psi }\bigl( {h^{\circ }\psi }\bigr) \bigl( \psi ^{-1}(b)\bigr) \bigr]- {h \biggl(\frac{b+c}{2} \biggr)} \\ &\quad= \frac{c-b}{2} \int _{0}^{1}\bigl(k+t^{\alpha }-(1-t)^{\alpha } \bigr)h'\bigl(tb+(1-t)c\bigr)\,dt {,} \end{aligned}$$
(3)

where

$$ k= \textstyle\begin{cases} 1,& 0\leq t< \frac{1}{2}, \\ -1,& \frac{1}{2}\leq t< 1. \end{cases} $$

Proof

Note that

$$\begin{aligned} &\frac{c-b}{2} \int _{0}^{1}kh'\bigl(tb+(1-t)c\bigr) \,dt \\ &\quad= \frac{c-b}{2} \int _{0}^{\frac{1}{2}}h'\bigl(tb+(1-t)c\bigr) \,dt-\frac{c-b}{2} \int _{\frac{1}{2}}^{1}h'\bigl(tb+(1-t)c\bigr) \,dt \\ &\quad= \frac{h(c)-{h (\frac{b+c}{2} )}}{2}+ \frac{h(b)-{h (\frac{b+c}{2} )}}{2} \\ &\quad=\frac{h(b)+h(c)}{2}- {h \biggl(\frac{b+c}{2} \biggr)}. \end{aligned}$$

By Lemma 1.3 we have

$$\begin{aligned} &\frac{\varGamma (\alpha +1)}{2(c-b)^{\alpha }} \bigl[I_{\psi ^{-1}(b)^{+}}^{ \alpha : \psi }\bigl({h^{\circ } \psi }\bigr) \bigl( \psi ^{-1}(c)\bigr)+I_{\psi ^{-1}(c)^{-}}^{\alpha : \psi } \bigl( {h^{\circ }\psi }\bigr) \bigl(\psi ^{-1}(b)\bigr) \bigr]- {h \biggl(\frac{b+c}{2} \biggr)} \\ &\quad= \biggl[\frac{h(b)+h(c)}{2}- {h \biggl(\frac{b+c}{2} \biggr)} \biggr] \\ &\qquad{}- \biggl\{ \frac{h(b)+h(c)}{2}- \frac{\varGamma (\alpha +1)}{2(c-b)^{\alpha }} \bigl[I_{\psi ^{-1}(b)^{+}}^{ \alpha : \psi } \bigl({h^{\circ }\psi }\bigr) \bigl( \psi ^{-1}(c) \bigr)+I_{\psi ^{-1}(c)^{-}}^{\alpha : \psi }\bigl( {h^{\circ }\psi }\bigr) \bigl( \psi ^{-1}(b)\bigr) \bigr] \biggr\} \\ &\quad= \frac{c-b}{2} \int _{0}^{1}kh'\bigl(tb+(1-t)c\bigr) \,dt-\frac{c-b}{2} \int _{0}^{1}\bigl((1-t)^{\alpha }-t^{\alpha } \bigr)h'\bigl(tb+(1-t)c\bigr)\,dt \\ &\quad= \frac{c-b}{2} \int _{0}^{1}\bigl(k+t^{\alpha }-(1-t)^{\alpha } \bigr)h'\bigl(tb+(1-t)c\bigr)\,dt. \end{aligned}$$

The proof is completed. □

Lemma 1.5

(see [32, Definition 1.1])

Let\((\varOmega ,\varLambda ,\mu )\)be a measure space with\(0 <\mu (\varOmega )< 1\), and let\(\phi : I\rightarrow R\)be a convex function defined on an open intervalIinR. If\(f: \varOmega \rightarrow I\)is such thatf, \(\phi ^{\circ }f\in L(\varOmega ,\varLambda ,\mu )\), then

$$\begin{aligned} \phi \biggl(\frac{1}{\mu (\varOmega )} \int _{\varOmega }f\,d\mu \biggr) \leq \frac{1}{\mu (\varOmega )} \int _{\varOmega }\phi (f)\,d\mu . \end{aligned}$$
(4)

In the case whereΩis strictly convex onI, we have equality in (4) if and only iffis constant almost everywhere onΩ.

Remark 1.6

Inequality (4) is reversed if ϕ is, that is,

$$\begin{aligned} \phi \biggl(\frac{1}{\mu (\varOmega )} \int _{\varOmega }f\,d\mu \biggr)\geq \frac{1}{\mu (\varOmega )} \int _{\varOmega }\phi (f)\,d\mu . \end{aligned}$$
(5)

The main purpose of this paper is to introduce some new Hermite–Hadamard-type inequalities involving ψ-Riemann–Liouville fractional integrals via s-convex functions in the second sense. For these functions, we establish some results related to the left end of new inequalities similar to inequality (1). We give some applications to special mean of a positive real number.

Main results

We now in a position to establish some inequalities of Hermite–Hadamard type involving ψ-Riemann–Liouville fractional integrals (with \(\alpha \in (0,1)\)) via s-convex functions.

Theorem 2.1

Let\(\alpha \in (0,1)\), let\(h: [b, c]\rightarrow R\)be a positive function with\(0 \leq b < c\)and\(h\in L[b, c]\), and letψbe an increasing positive function on\([b, c]\)having a continuous derivative\(\psi '\)on\((b, c)\). Ifhis ans-convex function on\([b, c]\), then we have the following inequality for fractional integrals:

$$\begin{aligned} 2^{s-1}{h \biggl(\frac{b+c}{2} \biggr)} & \leq \frac{\varGamma (\alpha +1)}{2(c-b)^{\alpha }} \bigl[I_{\psi ^{-1}(b)^{+}}^{ \alpha : \psi } \bigl({h^{\circ }\psi }\bigr) \bigl( \psi ^{-1}(c) \bigr)+I_{\psi ^{-1}(c)^{-}}^{\alpha : \psi }\bigl( {h^{\circ }\psi }\bigr) \bigl( \psi ^{-1}(b)\bigr) \bigr] \\ &\leq \biggl[\frac{3\alpha }{\alpha +s}- \frac{\alpha }{(\alpha +s)2^{\alpha +s}} \biggr]\frac{h(b)+h(c)}{2}. \end{aligned}$$
(6)

Proof

Since h is an s-convex function on \([b, c]\), for every \(x,y\in [b,c]\) with \(\lambda =\frac{1}{2}\), we have

$$ {h \biggl(\frac{x+y}{2} \biggr)}\leq \frac{1}{2^{s}}h(x)+ \frac{1}{2^{s}}h(y), $$

that is, with \(x=tb+(1-t)c, y=(1-t)b+tc\),

$$\begin{aligned} 2^{s}{h \biggl(\frac{b+c}{2} \biggr)} \leq h\bigl(tb+(1-t)c\bigr)+h\bigl((1-t)b+tc\bigr). \end{aligned}$$
(7)

Multiplying both sides of (7) by \(t^{\alpha -1}\) and then integrating the resulting inequality with respect to t over \([0, 1]\), we obtain

$$\begin{aligned} & \int _{0}^{1}t^{\alpha -1}h\bigl(tb+(1-t)c\bigr) \,dt+ \int _{0}^{1}t^{\alpha -1}h\bigl((1-t)b+tc\bigr) \,dt \\ &\quad\geq \int _{0}^{1}t^{\alpha +s-1} {h \biggl( \frac{b+c}{2} \biggr)}\,dt \\ &\quad\geq \frac{2^{s}}{\alpha } {h \biggl(\frac{b+c}{2} \biggr)}. \end{aligned}$$

Next,

$$\begin{aligned} &\frac{\varGamma (\alpha +1)}{2(c-b)^{\alpha }} \bigl[I_{\psi ^{-1}(b)^{+}}^{ \alpha : \psi }\bigl({h^{\circ } \psi }\bigr) \bigl( \psi ^{-1}(c)\bigr)+I_{\psi ^{-1}(c)^{-}}^{\alpha : \psi } \bigl( {h^{\circ }\psi }\bigr) \bigl(\psi ^{-1}(b)\bigr) \bigr] \\ &\quad= \frac{\varGamma (\alpha +1)}{2(c-b)^{\alpha }} \biggl[ \frac{1}{\varGamma (\alpha )} \int _{\psi ^{-1}(b)}^{\psi ^{-1}(c)}\psi '(t) \bigl( \psi \bigl(\psi ^{-1}(c)\bigr)-\psi (t)\bigr)^{\alpha -1}\bigl( {h^{\circ }\psi }\bigr) (t)\,dt \\ &\qquad{}+\frac{1}{\varGamma (\alpha )} \int _{\psi ^{-1}(b)}^{\psi ^{-1}(c)} \psi '(t) \bigl(\psi (t)-\psi \bigl(\psi ^{-1}(b)\bigr)\bigr)^{\alpha -1}\bigl( {h^{\circ }\psi }\bigr) (t)\,dt \biggr] \\ &\quad= \frac{\varGamma (\alpha +1)}{2(c-b)^{\alpha }}\times \frac{1}{\varGamma (\alpha )} \biggl[ \int _{\psi ^{-1}(b)}^{\psi ^{-1}(c)} \psi '(t) \bigl(c-\psi (t)\bigr)^{\alpha -1}h\bigl(\psi (t)\bigr)\,dt \\ &\qquad{}+ \int _{\psi ^{-1}(b)}^{\psi ^{-1}(c)}\psi '(t) \bigl(\psi (t)-b\bigr)^{ \alpha -1}h\bigl(\psi (t)\bigr)\,dt \biggr] \\ &\qquad \bigl(\text{let } m=\psi (t)\bigr) \\ &\quad= \frac{\alpha }{2(c-b)} \biggl[ \int _{b}^{c} { \biggl(\frac{c-m}{c-b} \biggr)}^{ \alpha -1}h(m)\,dm + \int _{b}^{c} { \biggl(\frac{m-b}{c-b} \biggr)}^{ \alpha -1}h(m)\,dm \biggr] \\ &\qquad \biggl(\text{let } u=\frac{c-m}{c-b}, v=\frac{m-b}{c-b}, \text{ then let } t=u \text{ and } t=v\biggr) \\ &\quad= \frac{\alpha }{2}\biggl[ \int _{0}^{1}t^{\alpha -1}h\bigl(tb+(1-t)c\bigr) \,dt+ \int _{0}^{1}t^{ \alpha -1}h\bigl((1-t)b+tc\bigr) \,dt\biggr] \\ &\quad\geq \frac{\alpha }{2}\times \frac{2^{s}}{\alpha } {h \biggl(\frac{b+c}{2} \biggr)} \\ &\quad= 2^{s-1} {h \biggl(\frac{b+c}{2} \biggr)}, \end{aligned}$$

so the left-hand side inequality in (6) is proved.

To prove the right-hand side inequality in (6), since h is an s-convex function, for \(t \in [0, 1]\), we have

$$ h\bigl(tb+(1-t)c\bigr)\leq t^{s}h(b)+(1-t)^{s}h(c) $$

and

$$ h\bigl((1-t)b+tc\bigr)\leq (1-t)^{s}h(b)+t^{s}h(c), $$

and then

$$\begin{aligned} h\bigl(tb+(1-t)c\bigr)+h\bigl((1-t)b+tc\bigr)\leq \bigl(t^{s}+(1-t)^{s}\bigr) \bigl(h(b)+h(c)\bigr). \end{aligned}$$
(8)

Multiplying both sides of (8) by \(t^{\alpha -1}\) and then integrating, we obtain

$$\begin{aligned} & \int _{0}^{1}t^{\alpha -1}h\bigl(tb+(1-t)c\bigr) \,dt+ \int _{0}^{1}t^{\alpha -1}h\bigl((1-t)b+tc\bigr) \,dt \\ &\quad\leq \int _{0}^{1}t^{\alpha -1}\bigl(t^{s}+(1-t)^{s} \bigr) \bigl(h(b)+h(c)\bigr)\,dt \\ &\quad= \biggl[ \int _{0}^{1}t^{\alpha +s-1}\,dt+ \int _{0}^{1}t^{\alpha -1}(1-t)^{s}\,dt \biggr]\bigl(h(b)+h(c)\bigr) \\ &\quad = \biggl[\frac{1}{\alpha +s}+ \int _{0}^{\frac{1}{2}}t^{\alpha -1}(1-t)^{s} \,dt+ \int _{\frac{1}{2}}^{1}t^{\alpha -1}(1-t)^{s}\,dt \biggr]\bigl(h(b)+h(c)\bigr) \\ &\quad\leq \biggl[\frac{1}{\alpha +s}+ \int _{0}^{\frac{1}{2}}(1-t)^{ \alpha +s-1}\,dt+ \int _{\frac{1}{2}}^{1}t^{\alpha +s-1}\,dt \biggr] \bigl(h(b)+h(c)\bigr) \\ &\quad\leq \biggl[\frac{3}{\alpha +s}+\frac{1}{(\alpha +s)2^{\alpha +s}} \biggr]\bigl(h(b)+h(c) \bigr). \end{aligned}$$

So then

$$\begin{aligned} &\frac{\varGamma (\alpha +1)}{2(c-b)^{\alpha }} \bigl[I_{\psi ^{-1}(b)^{+}}^{ \alpha : \psi }\bigl({h^{\circ } \psi }\bigr) \bigl( \psi ^{-1}(c)\bigr)+I_{\psi ^{-1}(c)^{-}}^{\alpha : \psi } \bigl( {h^{\circ }\psi }\bigr) \bigl(\psi ^{-1}(b)\bigr) \bigr] \\ &\quad= \frac{\alpha }{2} \biggl[ \int _{0}^{1}t^{\alpha -1}h\bigl(tb+(1-t)c\bigr) \,dt+ \int _{0}^{1}t^{\alpha -1}h\bigl((1-t)b+tc\bigr) \,dt \biggr] \\ &\quad\leq \frac{\alpha }{2}\times \biggl[\frac{3}{\alpha +s}- \frac{1}{(\alpha +s)2^{\alpha +s}} \biggr]\bigl(h(b)+h(c)\bigr) \\ &\quad= \biggl[\frac{3\alpha }{\alpha +s}- \frac{\alpha }{(\alpha +s)2^{\alpha +s}} \biggr]\frac{h(b)+h(c)}{2}. \end{aligned}$$

The proof is completed. □

Theorem 2.2

Let\(h: [b, c]\rightarrow R\)be a positive function with\(0 \leq b < c\)such that\(h\in L[b, c]\), and letψbe an increasing positive function on\([b, c]\)having a continuous derivative\(\psi '\)on\((b, c)\). If\(h'\)is ans-convex function on\([b, c]\)for some fixed\(s\in (0,1]\), then we have the following inequality for fractional integrals:

$$\begin{aligned} & \biggl\vert \frac{\varGamma (\alpha +1)}{2(c-b)^{\alpha }} \bigl[I_{\psi ^{-1}(b)^{+}}^{ \alpha : \psi }\bigl({h^{\circ }\psi }\bigr) \bigl( \psi ^{-1}(c)\bigr)+I_{\psi ^{-1}(c)^{-}}^{\alpha : \psi }\bigl( {h^{\circ }\psi }\bigr) \bigl(\psi ^{-1}(b)\bigr) \bigr]- {h \biggl(\frac{b+c}{2} \biggr)} \biggr\vert \\ &\quad\leq \frac{c-b}{2(s+1)}\bigl( \bigl\vert h'(b) \bigr\vert + \bigl\vert h'(c) \bigr\vert \bigr). \end{aligned}$$
(9)

Proof

Using Lemma 1.4 and the s-convexity of h, we have

$$\begin{aligned} & \biggl\vert \frac{\varGamma (\alpha +1)}{2(c-b)^{\alpha }} \bigl[I_{\psi ^{-1}(b)^{+}}^{ \alpha : \psi } \bigl({h^{\circ }\psi }\bigr) \bigl( \psi ^{-1}(c) \bigr)+I_{\psi ^{-1}(c)^{-}}^{\alpha : \psi }\bigl( {h^{\circ }\psi }\bigr) \bigl( \psi ^{-1}(b)\bigr) \bigr]- {h \biggl(\frac{b+c}{2} \biggr)} \biggr\vert \\ &\quad = \frac{c-b}{2} \biggl\vert \int _{0}^{1}\bigl(k+t^{\alpha }-(1-t)^{\alpha } \bigr)h'\bigl(tb+(1-t)c\bigr)\,dt \biggr\vert \\ &\quad\leq \frac{c-b}{2} \biggl\{ \int _{0}^{\frac{1}{2}}\bigl(1+t^{\alpha }-(1-t)^{\alpha } \bigr)\bigl[t^{s} \bigl\vert h'(b) \bigr\vert +(1-t)^{s} \bigl\vert h'(c) \bigr\vert \bigr]\,dt \\ &\qquad{}+ \int _{\frac{1}{2}}^{1}\bigl((1-t)^{\alpha }+1-t^{\alpha } \bigr)\bigl[t^{s} \bigl\vert h'(b) \bigr\vert +(1-t)^{s} \bigl\vert h'(c) \bigr\vert \bigr]\,dt \biggr\} \\ &\quad= \frac{c-b}{2} \biggl\{ \bigl\vert h'(b) \bigr\vert \int _{0}^{\frac{1}{2}}\bigl[t^{s}+t^{ \alpha +s}-t^{s}(1-t)^{\alpha } \bigr]\,dt \\ &\qquad{}+ \bigl\vert h'(c) \bigr\vert \int _{0}^{\frac{1}{2}}\bigl[(1-t)^{s}+t^{\alpha }(1-t)^{s}-(1-t)^{ \alpha +s} \bigr]\,dt \\ &\qquad{}+ \bigl\vert h'(b) \bigr\vert \int _{\frac{1}{2}}^{1}\bigl[t^{s}(1-t)^{\alpha }+t^{s}-t^{\alpha +s} \bigr]\,dt \\ &\qquad{}+ \bigl\vert h'(c) \bigr\vert \int _{\frac{1}{2}}^{1}\bigl[(1-t)^{\alpha +s}+(1-t)^{s}-t^{\alpha }(1-t)^{s} \bigr]\,dt \biggr\} \\ &\quad\leq \frac{c-b}{2} \biggl\{ \bigl\vert h'(b) \bigr\vert \int _{0}^{\frac{1}{2}}t^{s}\,dt+ \bigl\vert h'(c) \bigr\vert \int _{0}^{\frac{1}{2}}(1-t)^{s}\,dt+ \bigl\vert h'(b) \bigr\vert \int _{\frac{1}{2}}^{1}t^{s}\,dt\\ &\qquad{}+ \bigl\vert h'(c) \bigr\vert \int _{\frac{1}{2}}^{1}(1-t)^{s}\,dt \biggr\} \\ &\quad= \frac{c-b}{2} \biggl\{ \bigl\vert h'(b) \bigr\vert \int _{0}^{1}t^{s}\,dt+ \bigl\vert h'(c) \bigr\vert \int _{0}^{1}(1-t)^{s}\,dt \biggr\} \\ &\quad= \frac{c-b}{2(s+1)}\bigl( \bigl\vert h'(b) \bigr\vert + \bigl\vert h'(c) \bigr\vert \bigr). \end{aligned}$$

The proof is completed. □

Theorem 2.3

Let\(h: [b, c]\rightarrow R\)be a positive function with\(0 \leq b < c\)such that\(h\in L[b, c]\), and letψbe an increasing positive function on\([b, c]\)having a continuous derivative\(\psi '\)on\((b, c)\). If\(|h'|^{q}\ (q>1)\)is ans-convex function on\([b, c]\)for some fixed\(s\in (0,1]\), then we have the following inequality for fractional integrals:

$$\begin{aligned} & \biggl\vert \frac{\varGamma (\alpha +1)}{2(c-b)^{\alpha }} \bigl[I_{\psi ^{-1}(b)^{+}}^{ \alpha : \psi }\bigl({h^{\circ }\psi }\bigr) \bigl( \psi ^{-1}(c)\bigr)+I_{\psi ^{-1}(c)^{-}}^{\alpha : \psi }\bigl( {h^{\circ }\psi }\bigr) \bigl(\psi ^{-1}(b)\bigr) \bigr]- {h \biggl(\frac{b+c}{2} \biggr)} \biggr\vert \\ &\quad\leq (c-b) \biggl(\frac{1}{(\alpha p+1)2^{\alpha p+1}} \biggr)^{ \frac{1}{p}} \biggl( \frac{1}{(s+1)2^{s+1}} \biggr)^{\frac{1}{q}} \\ &\qquad{} \times \bigl[\bigl( \bigl\vert h'(b) \bigr\vert ^{q}+\bigl(2^{s+1}-1\bigr) \bigl\vert h'(c) \bigr\vert ^{q}\bigr)^{\frac{1}{q}}+\bigl(\bigl(2^{s+1}-1 \bigr) \bigl\vert h'(b) \bigr\vert ^{q}+ \bigl\vert h'(c) \bigr\vert ^{q}\bigr)^{ \frac{1}{q}}\bigr], \end{aligned}$$
(10)

where\(\frac{1}{p}=1-\frac{1}{q}\).

Proof

Using Lemma 1.4 and the Hölder inequality via the s-convexity of \(|h'|^{q}\ (q>1)\), we have

$$\begin{aligned} & \biggl\vert \frac{\varGamma (\alpha +1)}{2(c-b)^{\alpha }} \bigl[I_{\psi ^{-1}(b)^{+}}^{ \alpha : \psi } \bigl({h^{\circ }\psi }\bigr) \bigl( \psi ^{-1}(c) \bigr)+I_{\psi ^{-1}(c)^{-}}^{\alpha : \psi }\bigl( {h^{\circ }\psi }\bigr) \bigl( \psi ^{-1}(b)\bigr) \bigr]- {h \biggl(\frac{b+c}{2} \biggr)} \biggr\vert \\ &\quad\leq \frac{c-b}{2} \biggl\{ \int _{0}^{\frac{1}{2}}\bigl(1+t^{\alpha }-(1-t)^{\alpha } \bigr) \bigl\vert h'\bigl(tb+(1-t)c\bigr) \bigr\vert \,dt \\ &\qquad{}+ \int _{\frac{1}{2}}^{1}\bigl((1-t)^{\alpha }+1-t^{\alpha } \bigr) \bigl\vert h'\bigl(tb+(1-t)c\bigr) \bigr\vert \,dt \biggr\} \\ &\quad\leq \frac{c-b}{2} \biggl\{ \biggl( \int _{0}^{\frac{1}{2}}\bigl(1+t^{\alpha }-(1-t)^{\alpha } \bigr)^{p} \,dt \biggr)^{\frac{1}{p}} \biggl( \int _{0}^{\frac{1}{2}} \bigl\vert h' \bigl(tb+(1-t)c\bigr) \bigr\vert ^{q} \,dt \biggr)^{\frac{1}{q}} \\ &\qquad{}+ \biggl( \int _{\frac{1}{2}}^{1}\bigl((1-t)^{\alpha }+1-t^{\alpha } \bigr)^{p} \,dt \biggr)^{\frac{1}{p}} \biggl( \int _{\frac{1}{2}}^{1} \bigl\vert h' \bigl(tb+(1-t)c\bigr) \bigr\vert ^{q} \,dt \biggr)^{\frac{1}{q}} \biggr\} \\ &\quad\leq \frac{c-b}{2} \biggl( \int _{0}^{\frac{1}{2}}\bigl(1+t^{\alpha }-(1-t)^{\alpha } \bigr)^{p} \,dt \biggr)^{\frac{1}{p}} \biggl\{ \biggl( \int _{0}^{ \frac{1}{2}}\bigl[t^{s} \bigl\vert h'(b) \bigr\vert ^{q}+(1-t)^{s} \bigl\vert h'(c) \bigr\vert ^{q}\bigr] \,dt \biggr)^{ \frac{1}{q}} \\ &\qquad{}+ \biggl( \int _{\frac{1}{2}}^{1}\bigl[t^{s} \bigl\vert h'(b) \bigr\vert ^{q}+(1-t)^{s} \bigl\vert h'(c) \bigr\vert ^{q}\bigr] \,dt \biggr)^{\frac{1}{q}} \biggr\} \\ &\quad\leq \frac{c-b}{2} \biggl( \int _{0}^{\frac{1}{2}}\bigl(1+t^{\alpha }- \bigl(1-t^{\alpha }\bigr)\bigr)^{p} \,dt \biggr)^{\frac{1}{p}}\\ &\qquad{}\times \biggl\{ \biggl( \frac{1}{(s+1)2^{s+1}} \bigl\vert h'(b) \bigr\vert ^{q}+\frac{1}{s+1} \biggl(1- \frac{1}{2^{s+1}} \biggr) \bigl\vert h'(c) \bigr\vert ^{q} \biggr)^{\frac{1}{q}} \\ &\qquad{}+ \biggl(\frac{1}{s+1} \biggl(1-\frac{1}{2^{s+1}} \biggr) \bigl\vert h'(b) \bigr\vert ^{q}+ \frac{1}{(s+1)2^{s+1}} \bigl\vert h'(c) \bigr\vert ^{q} \biggr)^{\frac{1}{q}} \biggr\} \\ &\quad\leq \frac{c-b}{2} \biggl(2^{p} \int _{0}^{\frac{1}{2}}t^{\alpha p} \,dt \biggr)^{\frac{1}{p}} \biggl(\frac{1}{(s+1)2^{s+1}} \biggr)^{\frac{1}{q}} \\ &\qquad{} \times \bigl[\bigl( \bigl\vert h'(b) \bigr\vert ^{q}+\bigl(2^{s+1}-1\bigr) \bigl\vert h'(c) \bigr\vert ^{q}\bigr)^{\frac{1}{q}}+\bigl(\bigl(2^{s+1}-1 \bigr) \bigl\vert h'(b) \bigr\vert ^{q}+ \bigl\vert h'(c) \bigr\vert ^{q}\bigr)^{ \frac{1}{q}}\bigr] \\ &\quad\leq (c-b) \biggl(\frac{1}{(\alpha p+1)2^{\alpha p+1}} \biggr)^{ \frac{1}{p}} \biggl( \frac{1}{(s+1)2^{s+1}} \biggr)^{\frac{1}{q}} \\ &\qquad{} \times \bigl[\bigl( \bigl\vert h'(b) \bigr\vert ^{q}+\bigl(2^{s+1}-1\bigr) \bigl\vert h'(c) \bigr\vert ^{q}\bigr)^{\frac{1}{q}}+\bigl(\bigl(2^{s+1}-1 \bigr) \bigl\vert h'(b) \bigr\vert ^{q}+ \bigl\vert h'(c) \bigr\vert ^{q}\bigr)^{ \frac{1}{q}}\bigr]. \end{aligned}$$

The proof is completed. □

Corollary 2.4

Let\(h: [b, c]\rightarrow R\)be a positive function with\(0 \leq b < c\)such that\(h\in L[b, c]\)\(\psi (\cdot )\)is an increasing and positive monotone function on\([b, c]\), having a continuous derivative\(\psi '\)on\((b, c)\). If\(|h'|^{q}\ (q>1)\)is ans-convex function on\([b, c]\)for some fixed\(s\in (0,1]\), then we have the following inequality for fractional integrals:

$$\begin{aligned} & \biggl\vert \frac{\varGamma (\alpha +1)}{2(c-b)^{\alpha }} \bigl[I_{\psi ^{-1}(b)^{+}}^{ \alpha : \psi }\bigl({h^{\circ }\psi }\bigr) \bigl( \psi ^{-1}(c)\bigr)+I_{\psi ^{-1}(c)^{-}}^{\alpha : \psi }\bigl( {h^{\circ }\psi }\bigr) \bigl(\psi ^{-1}(b)\bigr) \bigr]- {h \biggl(\frac{b+c}{2} \biggr)} \biggr\vert \\ &\quad\leq (c-b) \biggl(\frac{1}{(\alpha p+1)2^{\alpha p+1}} \biggr)^{ \frac{1}{p}} \biggl( \frac{1}{(s+1)2^{s+1}} \biggr)^{\frac{1}{q}} \\ &\qquad{} \times \bigl(1+\bigl(2^{s+1}-1\bigr)^{\frac{1}{q}}\bigr) \bigl( \bigl\vert h'(b) \bigr\vert + \bigl\vert h'(c) \bigr\vert \bigr), \end{aligned}$$
(11)

where\(\frac{1}{p}=1-\frac{1}{q}\).

Proof

We consider inequality (10), and we let \(a_{1}=|h'(b)|^{q}\), \(b_{1}=(2^{s+1}-1)|h'(c)|^{q}\), \(a_{2}=(2^{s+1}-1)|h'(b)|^{q}\), \(b_{2}=|h'(c)|^{q}\). Here \(0<\frac{1}{q}<1\) for \(q>1\). Using the inequality \(\sum_{i=1}^{n}(a_{i}+b_{i})^{r}\leq \sum_{i=1}^{n}a_{i}^{r}+\sum_{i=1}^{n}b_{i}^{r}\) for \(0< r<1, a_{i}>0, b_{i}>0, i=1,2,\ldots ,n\), we obtain the required result. This completes the proof. □

Theorem 2.5

Let\(h: [b, c]\rightarrow R\)be a positive function with\(0 \leq b < c\)such that\(h\in L[b, c]\), and letψis an increasing positive function on\([b, c]\)having a continuous derivative\(\psi '\)on\((b, c)\). If\(|h'|^{q}\ (q>1)\)is ans-convex function on\([b, c]\)for some fixed\(s\in (0,1]\), then we have the following inequality for fractional integrals:

$$\begin{aligned} & \biggl\vert \frac{\varGamma (\alpha +1)}{2(c-b)^{\alpha }} \bigl[I_{\psi ^{-1}(b)^{+}}^{ \alpha : \psi }\bigl({h^{\circ }\psi }\bigr) \bigl( \psi ^{-1}(c)\bigr)+I_{\psi ^{-1}(c)^{-}}^{\alpha : \psi }\bigl( {h^{\circ }\psi }\bigr) \bigl(\psi ^{-1}(b)\bigr) \bigr]- {h \biggl(\frac{b+c}{2} \biggr)} \biggr\vert \\ &\quad\leq \frac{c-b}{2} \biggl(\frac{1}{\alpha +1} \biggr)^{1-\frac{1}{q}} \biggl(\frac{\alpha -1}{2}+\frac{1}{2^{\alpha }} \biggr)^{1-\frac{1}{q}} \biggl( \frac{1}{(s+1)2^{s+1}} \biggr)^{\frac{1}{q}} \\ &\qquad{} \times \bigl[\bigl( \bigl\vert h'(b) \bigr\vert ^{q}+\bigl(2^{s+1}-1\bigr) \bigl\vert h'(c) \bigr\vert ^{q}\bigr)^{\frac{1}{q}}+\bigl(\bigl(2^{s+1}-1 \bigr) \bigl\vert h'(b) \bigr\vert ^{q}+ \bigl\vert h'(c) \bigr\vert ^{q}\bigr)^{ \frac{1}{q}}\bigr]. \end{aligned}$$
(12)

Proof

Using Lemma 1.4 and the power mean inequality via the s-convexity of \(|h'|^{q}\ (q>1)\), we have

$$\begin{aligned} & \biggl\vert \frac{\varGamma (\alpha +1)}{2(c-b)^{\alpha }} \bigl[I_{\psi ^{-1}(b)^{+}}^{ \alpha : \psi } \bigl({h^{\circ }\psi }\bigr) \bigl( \psi ^{-1}(c) \bigr)+I_{\psi ^{-1}(c)^{-}}^{\alpha : \psi }\bigl( {h^{\circ }\psi }\bigr) \bigl( \psi ^{-1}(b)\bigr) \bigr]- {h \biggl(\frac{b+c}{2} \biggr)} \biggr\vert \\ &\quad\leq \frac{c-b}{2} \biggl\{ \int _{0}^{\frac{1}{2}}\bigl(1+t^{\alpha }-(1-t)^{\alpha } \bigr) \bigl\vert h'\bigl(tb+(1-t)c\bigr) \bigr\vert \,dt \\ &\qquad{}+ \int _{\frac{1}{2}}^{1}\bigl((1-t)^{\alpha }+1-t^{\alpha } \bigr) \bigl\vert h'\bigl(tb+(1-t)c\bigr) \bigr\vert \,dt \biggr\} \\ &\quad\leq \frac{c-b}{2} \biggl\{ \biggl( \int _{0}^{\frac{1}{2}}\bigl(1+t^{\alpha }-(1-t)^{\alpha } \bigr) \,dt \biggr)^{1-\frac{1}{q}}\\ &\qquad{}\times \biggl( \int _{0}^{\frac{1}{2}}\bigl(1+t^{\alpha }-(1-t)^{\alpha } \bigr) \bigl\vert h'\bigl(tb+(1-t)c\bigr) \bigr\vert ^{q} \,dt \biggr)^{\frac{1}{q}} \\ &\qquad{}+ \biggl( \int _{\frac{1}{2}}^{1}\bigl((1-t)^{\alpha }+1-t^{\alpha } \bigr) \,dt \biggr)^{1- \frac{1}{q}} \biggl( \int _{\frac{1}{2}}^{1}\bigl((1-t)^{\alpha }+1-t^{\alpha } \bigr) \bigl\vert h'\bigl(tb+(1-t)c\bigr) \bigr\vert ^{q} \,dt \biggr)^{\frac{1}{q}} \biggr\} \\ &\quad\leq \frac{c-b}{2} \biggl( \int _{0}^{\frac{1}{2}}\bigl(1+t^{\alpha }-(1-t)^{\alpha } \bigr) \,dt \biggr)^{1-\frac{1}{q}} \\ &\qquad{}\times \biggl\{ \biggl( \int _{0}^{\frac{1}{2}}\bigl(1+t^{\alpha }-(1-t)^{\alpha } \bigr)\bigl[t^{s} \bigl\vert h'(b) \bigr\vert ^{q}+(1-t)^{s} \bigl\vert h'(c) \bigr\vert ^{q}\bigr] \,dt \biggr)^{\frac{1}{q}} \\ &\qquad{}+ \biggl( \int _{\frac{1}{2}}^{1}\bigl((1-t)^{\alpha }+1-t^{\alpha } \bigr)\bigl[t^{s} \bigl\vert h'(b) \bigr\vert ^{q}+(1-t)^{s} \bigl\vert h'(c) \bigr\vert ^{q}\bigr] \,dt \biggr)^{\frac{1}{q}} \biggr\} \\ &\quad\leq \frac{c-b}{2} \biggl(\frac{1}{\alpha +1} \biggr)^{1-\frac{1}{q}} \biggl(\frac{\alpha -1}{2}+\frac{1}{2^{\alpha }} \biggr)^{1-\frac{1}{q}} \\ &\qquad{}\times \biggl\{ \biggl( \bigl\vert h'(b) \bigr\vert ^{q} \int _{0}^{\frac{1}{2}}\bigl[t^{s}+t^{ \alpha +s}-t^{s}(1-t)^{\alpha } \bigr]\,dt \\ &\qquad{}+ \bigl\vert h'(c) \bigr\vert ^{q} \int _{0}^{\frac{1}{2}}\bigl[(1-t)^{s}+t^{\alpha }(1-t)^{s}-(1-t)^{ \alpha +s} \bigr]\,dt \biggr)^{\frac{1}{q}} \\ &\qquad{}+ \biggl( \bigl\vert h'(b) \bigr\vert ^{q} \int _{\frac{1}{2}}^{1}\bigl[t^{s}(1-t)^{\alpha }+t^{s}-t^{ \alpha +s} \bigr]\,dt \\ &\qquad{}+ \bigl\vert h'(c) \bigr\vert ^{q} \int _{\frac{1}{2}}^{1}\bigl[(1-t)^{\alpha +s}+(1-t)^{s}-t^{\alpha }(1-t)^{s} \bigr]\,dt \biggr)^{\frac{1}{q}} \biggr\} \\ &\quad\leq \frac{c-b}{2} \biggl(\frac{1}{\alpha +1} \biggr)^{1-\frac{1}{q}} \biggl(\frac{\alpha -1}{2}+\frac{1}{2^{\alpha }} \biggr)^{1-\frac{1}{q}} \\ &\qquad{}\times \biggl\{ \biggl( \bigl\vert h'(b) \bigr\vert ^{q} \int _{0}^{\frac{1}{2}}t^{s}\,dt+ \bigl\vert h'(c) \bigr\vert ^{q} \int _{0}^{\frac{1}{2}}(1-t)^{s}\,dt \biggr)^{\frac{1}{q}} \\ &\qquad{}+ \biggl( \bigl\vert h'(b) \bigr\vert ^{q} \int _{\frac{1}{2}}^{1}t^{s}\,dt+ \bigl\vert h'(c) \bigr\vert ^{q} \int _{ \frac{1}{2}}^{1}(1-t)^{s}\,dt \biggr)^{\frac{1}{q}} \biggr\} \\ &\quad\leq \frac{c-b}{2} \biggl(\frac{1}{\alpha +1} \biggr)^{1-\frac{1}{q}} \biggl(\frac{\alpha -1}{2}+\frac{1}{2^{\alpha }} \biggr)^{1-\frac{1}{q}} \\ &\qquad{} \times \biggl\{ \biggl(\frac{1}{(s+1)2^{s+1}} \bigl\vert h'(b) \bigr\vert ^{q}+ \frac{1}{s+1} \biggl(1-\frac{1}{2^{s+1}} \biggr) \bigl\vert h'(c) \bigr\vert ^{q} \biggr)^{ \frac{1}{q}} \\ &\qquad{}+ \biggl(\frac{1}{s+1} \biggl(1-\frac{1}{2^{s+1}} \biggr) \bigl\vert h'(b) \bigr\vert ^{q}+ \frac{1}{(s+1)2^{s+1}} \bigl\vert h'(c) \bigr\vert ^{q} \biggr)^{\frac{1}{q}} \biggr\} \\ &\quad\leq \frac{c-b}{2} \biggl(\frac{1}{\alpha +1} \biggr)^{1-\frac{1}{q}} \biggl(\frac{\alpha -1}{2}+\frac{1}{2^{\alpha }} \biggr)^{1-\frac{1}{q}} \biggl( \frac{1}{(s+1)2^{s+1}} \biggr)^{\frac{1}{q}} \\ &\qquad{} \times \bigl[\bigl( \bigl\vert h'(b) \bigr\vert ^{q}+\bigl(2^{s+1}-1\bigr) \bigl\vert h'(c) \bigr\vert ^{q}\bigr)^{\frac{1}{q}}+\bigl(\bigl(2^{s+1}-1 \bigr) \bigl\vert h'(b) \bigr\vert ^{q}+ \bigl\vert h'(c) \bigr\vert ^{q}\bigr)^{ \frac{1}{q}}\bigr]. \end{aligned}$$

The proof is completed. □

Corollary 2.6

Let\(h: [b, c]\rightarrow R\)be a positive function with\(0 \leq b < c\)such that\(h\in L[b, c]\), and letψbe an increasing positive function on\([b, c]\)having a continuous derivative\(\psi '\)on\((b, c)\). If\(|h'|^{q}\ (q>1)\)is ans-convex function on\([b, c]\)for some fixed\(s\in (0,1]\), then we have the following inequality for fractional integrals:

$$\begin{aligned} & \biggl\vert \frac{\varGamma (\alpha +1)}{2(c-b)^{\alpha }} \bigl[I_{\psi ^{-1}(b)^{+}}^{ \alpha : \psi }\bigl({h^{\circ }\psi }\bigr) \bigl( \psi ^{-1}(c)\bigr)+I_{\psi ^{-1}(c)^{-}}^{\alpha : \psi }\bigl( {h^{\circ }\psi }\bigr) \bigl(\psi ^{-1}(b)\bigr) \bigr]- {h \biggl(\frac{b+c}{2} \biggr)} \biggr\vert \\ &\quad\leq \frac{c-b}{2} \biggl(\frac{1}{\alpha +1} \biggr)^{1-\frac{1}{q}} \biggl(\frac{\alpha -1}{2}+\frac{1}{2^{\alpha }} \biggr)^{1-\frac{1}{q}} \biggl( \frac{1}{(s+1)2^{s+1}} \biggr)^{\frac{1}{q}} \\ &\qquad{} \times \bigl(1+\bigl(2^{s+1}-1\bigr)^{\frac{1}{q}}\bigr) \bigl( \bigl\vert h'(b) \bigr\vert + \bigl\vert h'(c) \bigr\vert \bigr). \end{aligned}$$
(13)

Proof

We can obtain the result using the technique in the proof of Corollary 2.4 by considering inequality (13). □

Theorem 2.7

Let\(h: [b, c]\rightarrow R\)be a positive function with\(0 \leq b < c\)such that\(h\in L[b, c]\), and letψbe an increasing positive function on\([b, c]\)having a continuous derivative\(\psi '\)on\((b, c)\). If\(|h'|^{q}\ (q>1)\)is a concave function on\([b, c]\), then we have the following inequality for fractional integrals:

$$\begin{aligned} & \biggl\vert \frac{\varGamma (\alpha +1)}{2(c-b)^{\alpha }} \bigl[I_{\psi ^{-1}(b)^{+}}^{ \alpha : \psi }\bigl({h^{\circ }\psi }\bigr) \bigl( \psi ^{-1}(c)\bigr)+I_{\psi ^{-1}(c)^{-}}^{\alpha : \psi }\bigl( {h^{\circ }\psi }\bigr) \bigl(\psi ^{-1}(b)\bigr) \bigr]- {h \biggl(\frac{b+c}{2} \biggr)} \biggr\vert \\ &\quad \leq (c-b) \biggl(\frac{1}{(\alpha p+1)2^{\alpha p+1}} \biggr)^{ \frac{1}{p}} \biggl( \frac{1}{2} \biggr)^{\frac{1}{q}} \biggl( \biggl\vert h' \biggl( \frac{b+3c}{4} \biggr) \biggr\vert + \biggl\vert h' \biggl(\frac{3b+c}{4} \biggr) \biggr\vert \biggr), \end{aligned}$$
(14)

where\(\frac{1}{p}=1-\frac{1}{q}\).

Proof

Using Lemma 1.4 and the Hölder inequality, we have

$$\begin{aligned} & \biggl\vert \frac{\varGamma (\alpha +1)}{2(c-b)^{\alpha }} \bigl[I_{\psi ^{-1}(b)^{+}}^{ \alpha : \psi } \bigl({h^{\circ }\psi }\bigr) \bigl( \psi ^{-1}(c) \bigr)+I_{\psi ^{-1}(c)^{-}}^{\alpha : \psi }\bigl( {h^{\circ }\psi }\bigr) \bigl( \psi ^{-1}(b)\bigr) \bigr]- {h \biggl(\frac{b+c}{2} \biggr)} \biggr\vert \\ &\quad\leq \frac{c-b}{2} \biggl( \int _{0}^{\frac{1}{2}}\bigl(1+t^{\alpha }-(1-t)^{\alpha } \bigr)^{p} \,dt \biggr)^{\frac{1}{p}} \biggl\{ \biggl( \int _{0}^{ \frac{1}{2}} \bigl\vert h' \bigl(tb+(1-t)c\bigr) \bigr\vert ^{q} \,dt \biggr)^{\frac{1}{q}} \\ &\qquad{}+ \biggl( \int _{\frac{1}{2}}^{1} \bigl\vert h' \bigl(tb+(1-t)c\bigr) \bigr\vert ^{q} \,dt \biggr)^{ \frac{1}{q}} \biggr\} \\ &\quad= (c-b) \biggl(\frac{1}{(\alpha p+1)2^{\alpha p+1}} \biggr)^{ \frac{1}{p}} \biggl\{ \biggl( \int _{0}^{\frac{1}{2}} \bigl\vert h' \bigl(tb+(1-t)c\bigr) \bigr\vert ^{q} \,dt \biggr)^{\frac{1}{q}} \\ &\qquad{}+ \biggl( \int _{\frac{1}{2}}^{1} \bigl\vert h' \bigl(tb+(1-t)c\bigr) \bigr\vert ^{q} \,dt \biggr)^{ \frac{1}{q}} \biggr\} . \end{aligned}$$

Noting that \(|h'|^{q}\ (q>1)\) is concave on \([b, c]\) and using the Jensen integral inequality (5), we have

$$ \int _{0}^{\frac{1}{2}} \bigl\vert h' \bigl(tb+(1-t)c\bigr) \bigr\vert ^{q} \,dt\leq \biggl( \int _{0}^{ \frac{1}{2}}t^{*} \,dt \biggr) \biggl\vert h' \biggl( \frac{\int _{0}^{\frac{1}{2}}(tb+(1-t)c)\,dt}{\int _{0}^{\frac{1}{2}}t^{*} \,dt} \biggr) \biggr\vert ^{q} \leq \frac{1}{2} \biggl\vert h' \biggl( \frac{b+3c}{4} \biggr) \biggr\vert ^{q}. $$

Similarly,

$$ \int _{\frac{1}{2}}^{1} \bigl\vert h' \bigl(tb+(1-t)c\bigr) \bigr\vert ^{q} \,dt\leq \biggl( \int _{ \frac{1}{2}}^{1}t^{*} \,dt \biggr) \biggl\vert h' \biggl( \frac{\int _{\frac{1}{2}}^{1}(tb+(1-t)c)\,dt}{\int _{\frac{1}{2}}^{1}t^{*} \,dt} \biggr) \biggr\vert ^{q} \leq \frac{1}{2} \biggl\vert h' \biggl( \frac{3b+c}{4} \biggr) \biggr\vert ^{q}. $$

In this formula, \(t^{*}\) is an arbitrary constant independent of t. Combined with the previous inequality, we get the required results. The proof is completed. □

Applications to some special means

Bivariate means are with respect to two elements. Consider the following bivariate means (see [33]) for arbitrary \(m, n\in R\), \(m\neq n\):

the harmonic mean

$$\begin{aligned} H(m,n) = \frac{2}{\frac{1}{m}+\frac{1}{n}},\quad m,n\in R \setminus \{0 \}, \end{aligned}$$

the arithmetic mean

$$\begin{aligned} A(m,n) = \frac{m+n}{2},\quad m,n\in R, \end{aligned}$$

the logarithmic mean

$$\begin{aligned} L(m,n) = \frac{n-m}{\ln \vert n \vert -\ln \vert m \vert },\quad \vert m \vert \neq \vert n \vert ,mn\neq 0, \end{aligned}$$

the r-logarithmic mean

$$\begin{aligned} L_{r}(m,n) = \biggl[\frac{n^{r+1}-m^{r+1}}{(r+1)(n-m)} \biggr]^{ \frac{1}{r}},\quad r\in Z \setminus \{-1,0\}, m,n\in R, m\neq n. \end{aligned}$$

Now we give some applications to special means of a real number.

Proposition 3.1

Let\(m,n\in R_{+}\), \(m< n\), \(r\in Z\), \(|r|\geq 2\), \(s\in (0,1]\), and\(q>1\). Then

$$ \bigl\vert L_{r}^{r}(m,n)-A^{r}(m,n) \bigr\vert {\leq } \textstyle\begin{cases} \frac{({n-m}) \vert r \vert }{s+1}A( \vert m \vert ^{r-1}, \vert n \vert ^{r-1}), \\ 2({n-m}) \vert r \vert ( \frac{1}{( p+1)2^{ p+1}} )^{\frac{1}{p}} ( \frac{1}{(s+1)2^{s+1}} )^{\frac{1}{q}} \\ \quad{} \times (1+(2^{s+1}-1)^{\frac{1}{q}})A( \vert m \vert ^{r-1}, \vert n \vert ^{r-1}), \\ ({n-m}) \vert r \vert (\frac{1}{4} )^{ \frac{1}{p}} (\frac{1}{(s+1)2^{s+1}} )^{\frac{1}{q}} \\ \quad{} \times (1+(2^{s+1}-1)^{\frac{1}{q}})A( \vert m \vert ^{r-1}, \vert n \vert ^{r-1}), \end{cases} $$

where\(\frac{1}{p}=1-\frac{1}{q}\).

Proof

Applying Theorem 2.2, Corollary 2.4, and Corollary 2.6, respectively, for \(h(x)=x^{r}\), \(\psi (x)=x\), and \(\alpha =1\), we immediately obtain the result. □

Proposition 3.2

Let\(m,n\in R_{+}\), \(m< n\), \(r\in Z\), \(s\in (0,1]\), and\(q>1\). Then

$$ \bigl\vert L^{-1}(m,n)-H\bigl(m^{-1},n^{-1} \bigr) \bigr\vert {\leq } \textstyle\begin{cases} \frac{{n-m}}{s+1}A( \vert m \vert ^{-2}, \vert n \vert ^{-2}), \\ 2({n-m}) (\frac{1}{( p+1)2^{ p+1}} )^{\frac{1}{p}} (\frac{1}{(s+1)2^{s+1}} )^{\frac{1}{q}} \\ \quad{}\times (1+(2^{s+1}-1)^{\frac{1}{q}})A( \vert m \vert ^{-2}, \vert n \vert ^{-2}), \\ ({n-m}) (\frac{1}{4} )^{ \frac{1}{p}} (\frac{1}{(s+1)2^{s+1}} )^{\frac{1}{q}} \\ \quad{} \times (1+(2^{s+1}-1)^{\frac{1}{q}})A( \vert m \vert ^{-2}, \vert n \vert ^{-2}). \end{cases} $$

where\(\frac{1}{p}=1-\frac{1}{q}\).

Proof

Applying Theorem 2.2, Corollary 2.4, and Corollary 2.6 respectively, for \(h(x)=\frac{1}{x}\), \(\psi (x)=x\), and \(\alpha =1\), we immediately obtain the result. □

References

  1. 1.

    Mitrinović, D., Lacković, I.: Hermite and convexity. Aequ. Math. 28(1), 229–232 (1985)

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    Abbas Baloch, I., Chu, Y.-M.: Petrović-type inequalities for harmonic-convex functions. J. Funct. Spaces 2020, Article ID 3075390 (2020)

    MATH  Google Scholar 

  3. 3.

    Ullah, S.Z., Khan, M.A., Chu, Y.-M.: A note on generalized convex functions. J. Inequal. Appl. 2019(1), 1 (2019)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Khan, M.A., Hanif, M., Khan, Z.A.H., Ahmad, K., Chu, Y.-M.: Association of Jensen’s inequality for s-convex function with Csiszár divergence. J. Inequal. Appl. 2019(1), 1 (2019)

    Article  Google Scholar 

  5. 5.

    Khan, M.A., Ullah, S.Z., Chu, Y.-M.: The concept of coordinate strongly convex functions and related inequalities. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 113(3), 2235–2251 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Ullah, S.Z., Khan, M.A., Chu, Y.-M.: Majorization theorems for strongly convex functions. J. Inequal. Appl. 2019(1), 58 (2019)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Khan, M.A., Wu, S.-H., Ullah, H., Chu, Y.-M.: Discrete majorization type inequalities for convex functions on rectangles. J. Inequal. Appl. 2019(1), 1 (2019)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Song, Y.-Q., Adil Khan, M., Zaheer Ullah, S., Chu, Y.-M.: Integral inequalities involving strongly convex functions. J. Funct. Spaces 2018, Article ID 6595921 (2018)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Khan, M.A., Chu, Y., Khan, T.U., Khan, J.: Some new inequalities of Hermite–Hadamard type for s-convex functions with applications. Open Math. 15(1), 1414–1430 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    Zaheer Ullah, S., Khan, M.A., Khan, Z.A., Chu, Y.-M.: Integral majorization type inequalities for the functions in the sense of strong convexity. J. Funct. Spaces 2019, Article ID 948782 (2019)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Adil Khan, M., Khurshid, Y., Du, T.-S., Chu, Y.-M.: Generalization of Hermite–Hadamard type inequalities via conformable fractional integrals. J. Funct. Spaces 2018, Article ID 5357463 (2018)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Xiao, Z.-G., Zhang, Z.-H., Wu, Y.-D.: On weighted Hermite–Hadamard inequalities. Appl. Math. Comput. 218(3), 1147–1152 (2011)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Dragomir, S.S., Pearce, C.: Selected topics on Hermite–Hadamard inequalities and applications. Math. Preprint Archiv. 2003(3), 463–817 (2003)

    Google Scholar 

  14. 14.

    Set, E., Özdemir, M., Dragomir, S.: On Hadamard-type inequalities involving several kinds of convexity. J. Inequal. Appl. 2010(1), 286845 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    Özdemir, M.E., Yıldız, Ç., Akdemir, A.O., Set, E.: On some inequalities for s-convex functions and applications. J. Inequal. Appl. 2013(1), 333 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Latif, M.: On some new inequalities of Hermite–Hadamard type for functions whose derivatives are s-convex in the second sense in the absolute value. Ukr. Math. J. 67(10), 1552–1571 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    Wu, X., Wang, J., Zhang, J.: Hermite–Hadamard-type inequalities for convex functions via the fractional integrals with exponential kernel. Mathematics 7(9), 845 (2019)

    Article  Google Scholar 

  18. 18.

    Yin, H.-P., Wang, J.-Y.: Some integral inequalities of Hermite–Hadamard type for s-geometrically convex functions. Miskolc Math. Notes 19(1), 699–705 (2018)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Sarikaya, M.Z., Set, E., Yaldiz, H., Başak, N.: Hermite–Hadamard’s inequalities for fractional integrals and related fractional inequalities. Math. Comput. Model. 57(9–10), 2403–2407 (2013)

    MATH  Article  Google Scholar 

  20. 20.

    İşcan, İ., Wu, S.: Hermite–Hadamard type inequalities for harmonically convex functions via fractional integrals. Appl. Math. Comput. 238, 237–244 (2014)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Jleli, M., O’Regan, D., Samet, B.: On Hermite–Hadamard type inequalities via generalized fractional integrals. Turk. J. Math. 40(6), 1221–1230 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  22. 22.

    Noor, M.A., Noor, K.I., Awan, M.U., Khan, S.: Fractional Hermite–Hadamard inequalities for some new classes of Godunova–Levin functions. Appl. Math. Inf. Sci. 8(6), 2865 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  23. 23.

    Gozpinar, A., Set, E., Dragomir, S.S.: Some generalized Hermite–Hadamard type inequalities involving fractional integral operator for functions whose second derivatives in absolute value are s-convex. Acta Math. Univ. Comen. 88(1), 87–100 (2019)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Ahmad, B., Alsaedi, A., Kirane, M., Torebek, B.T.: Hermite–Hadamard, Hermite–Hadamard–Fejér, Dragomir–Agarwal and Pachpatte type inequalities for convex functions via new fractional integrals. J. Comput. Appl. Math. 353, 120–129 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  25. 25.

    Chen, H., Katugampola, U.N.: Hermite–Hadamard and Hermite–Hadamard–Fejér type inequalities for generalized fractional integrals. J. Math. Anal. Appl. 446(2), 1274–1291 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    Wang, J., Zhu, C., Zhou, Y.: New generalized Hermite–Hadamard type inequalities and applications to special means. J. Inequal. Appl. 2013(1), 325 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  27. 27.

    Mehreen, N., Anwar, M.: Some inequalities via ψ-Riemann–Liouville fractional integrals. AIMS Math. 4, 1403–1415 (2019)

    Article  Google Scholar 

  28. 28.

    da Sousa, J.V., de Oliveira, E.C.: On the ψ-Hilfer fractional derivative. Commun. Nonlinear Sci. Numer. Simul. 60, 72–91 (2018)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Hudzik, H., Maligranda, L.: Some remarks ons-convex functions. Aequ. Math. 48(1), 100–111 (1994)

    MathSciNet  MATH  Article  Google Scholar 

  30. 30.

    Wang, J., Li, X., Zhou, Y.: Hermite–Hadamard inequalities involving Riemann–Liouville fractional integrals via s-convex functions and applications to special means. Filomat 30(5), 1143–1150 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  31. 31.

    Liu, K., Wang, J., O’Regan, D.: On the Hermite–Hadamard type inequality for ψ-Riemann–Liouville fractional integrals via convex functions. J. Inequal. Appl. 2019, 27 (2019)

    MathSciNet  Google Scholar 

  32. 32.

    Dragomir, S.S., Khan, M.A., Abathun, A.: Refinement of the Jensen integral inequality. Open Math. 14(1), 221–228 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  33. 33.

    Pearce, C.E., Pečarić, J.: Inequalities for differentiable mappings with application to special means and quadrature formulae. Appl. Math. Lett. 13(2), 51–55 (2000)

    MathSciNet  MATH  Article  Google Scholar 

Download references

Acknowledgements

Not applicable.

Availability of data and materials

All data generated or analyzed during this study are included in this published paper.

Funding

This work was supported by the Key Disciplines of Guizhou Province Computer Science and technology (ZDXK [2018]007, the Key Supported Disciplines of Guizhou Province Computer application technology (No. QianXueWeiHeZi ZDXK[2016]20), Specialized Fund for Science and technology Platform and Talent Team Project of Guizhou Province (No. QianKeHePingTaiRenCai [2016]5609) and the Major Research Projects of Innovation Group of Guizhou Provincial Department of Education (No. QianJiaoHeKY [2016]040).

Author information

Affiliations

Authors

Contributions

Using the second sense of s-convex function, some new Hermite–Hadamard-type inequalities are established, involving the fractional integration of ψ-Riemann–Liouville. At the same time the authors gave many useful estimates for these new Hermite–Hadamard-type inequalities. The main idea of this paper was proposed by YZ and HS. ZC and WX prepared the manuscript initially and performed all the steps of the proofs in this research. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Yong Zhao.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Zhao, Y., Sang, H., Xiong, W. et al. Hermite–Hadamard-type inequalities involving ψ-Riemann–Liouville fractional integrals via s-convex functions. J Inequal Appl 2020, 128 (2020). https://doi.org/10.1186/s13660-020-02389-7

Download citation

Keywords

  • Hermite–Hadamard inequalities
  • ψ-Riemann–Liouville fractional integrals
  • s-convex functions