Skip to main content

Some new fractional integral inequalities for exponentially m-convex functions via extended generalized Mittag-Leffler function

Abstract

In the article, we establish some new general fractional integral inequalities for exponentially m-convex functions involving an extended Mittag-Leffler function, provide several kinds of fractional integral operator inequalities and give certain special cases for our obtained results.

Introduction

Convex functions and their variant forms are being used to study a wide class of problems which arises in various branches of pure and applied mathematics [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]. This theory provides us a natural, unified and general framework to study a wide class of unrelated problems. Many applications, generalizations and other aspects of convex functions and their variant forms can be found in the recent literature [25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62].

An important class of convex functions, which is called the class of exponential convex functions, was introduced and studied by Antczak [63] and Dragomir et al. [64]. Alirezai and Mathar [65] investigated their mathematical properties along with their potential applications in statistics and information theory. Due to its significance, Awan et al. [66], and Jakšetić and Pečarić [67] defined another kind of exponential convex functions, which have shown that the class of exponential convex functions unifies various concepts in different manners.

In [68], Toader defined the m-convexity as an intermediate between the usual convexity and star shaped property. If we set \(m=0\), then we have the concept of star shaped functions on \([a,b]\). We recall that \(f:[a,b]\rightarrow \mathbb{R}\) is said to be star shaped if \(f(tx)\leq tf(x)\) for all \(t\in [0,1]\) and \(x\in [a,b]\).

We would like to emphasize that exponentially convex functions and m-convex functions are two distinct classes of convex functions. It is natural to introduce a new class of convex functions to unify these concepts. For this purpose, we need to recall some basic concepts as follows.

Definition 1.1

(See [68])

Let \(m\in [0, 1]\). Then the real number set \({I}\subseteq \mathbb{R}\) is said to be m-convex if

$$ (1-t)a+mtb\in {I} $$

for all \(a, b\in I\) and \(t\in [0, 1]\).

From Definition 1.1 we clearly see that the m-convex set I contains the line segment between points a and mb for every pair of points a and b of I.

Definition 1.2

(See [68])

Let \(m\in [0, 1]\) and \(I\subseteq \mathbb{R}\) be a m-convex set. Then A real-valued function \(f:{I}\subset \mathbb{R}\rightarrow \mathbb{R}\) is said to be a m-convex if

$$ f\bigl[(1-t)a+mtb\bigr]\leq (1-t)f(a)+mtf(b) $$

for all \(a, b\in I\) and \(t\in [0, 1]\).

Remark 1.3

From Definition 1.2 we clearly see that the 1-convex function is a convex function in the ordinary sense and the 0-convex function is the star shaped function. If we take \(m=1\), then we recapture the concept of convex functions. If we take \(t=1\), then we get

$$ f(mb)\leq mf(b) $$

for all \(a, b\in I\), which implies that the function f is sub-homogeneous.

Definition 1.4

(See [64])

A real-valued function \(f:{K}\subseteq \mathbb{R}\rightarrow \mathbb{R}\) is said to be an exponentially convex on K if

$$ e^{f[(1-t)a+tb]}\leq (1-t)e^{f(a)}+te^{f(b)} $$

for all \(a, b\in K\) and \(t\in [0, 1]\).

Definition 1.5

Let \(m\in [0, 1]\) and \(K\subseteq \mathbb{R}\) be a m-convex set. Then a real-valued function \(f:{K}\rightarrow \mathbb{R}\) is said to be exponentially m-convex if

$$ e^{f[(1-t)a+mtb]}\leq (1-t)e^{f(a)}+mte^{f(b)} $$

for all \(a, b\in K\) and \(t\in [0, 1]\).

Example 1.6

Let \(f(x)=2\log x\). Then \(f(x)\) is exponentially m-convex on \((0, \infty )\) for any \(m\in [0, 1]\). Indeed,

$$\begin{aligned}& e^{f[(1-t)a+mtb]}=\bigl[(1-t)a+mtb\bigr]^{2}, \\& (1-t)e^{f(a)}+mte^{f(b)}=(1-t)a ^{2}+mtb^{2}, \\& (1-t)e^{f(a)}+mte^{f(b)}-e^{f[(1-t)a+mtb]} \\& \quad =t \bigl[(1-t)a^{2}+m(1-mt)b^{2}-2m(1-t)ab \bigr] \\& \quad \geq 2t \bigl[\sqrt{m(1-t) (1-mt)}-m(1-t) \bigr]ab \\& \quad =2t\sqrt{m(1-t)} \bigl[\sqrt{1-mt}-\sqrt{m(1-t)} \bigr]ab \geq 0 \end{aligned}$$

for all \(a, b\in (0, \infty )\) and \(m, t\in [0, 1]\).

Fractional analysis can be regarded as an expansion of classical analysis. Fractional analysis has been studied by many scientists and they have expressed the fractional derivative and integral in different ways with different notations. Although the expressions between these different definitions can be transformed into each other, but these different definitions and expressions have different physical meanings. It is well known that the first fractional integral operator is the Riemann–Liouville fractional integral operator. Recently, some new definitions of the fractional derivative were given by many mathematicians, which are the natural extensions of the classical derivative. These new definitions drew attention with their variability to classical derivative.

Let \(b>a\geq 0\), \(u>0\) and \(f\in L_{1}[a,b]\). Then the left and right sided Riemann–Liouville fractional integrals of order u are defined by

$$ I_{a^{+}}^{u}f(t)=\frac{1}{\varGamma (u)} \int _{a}^{t}(t-\xi )^{u-1}f( \xi )\,d\xi ,\quad t>a, $$

and

$$ I_{b^{-}}^{u}f(t)=\frac{1}{\varGamma (u)} \int _{t}^{b}(\xi -t)^{u-1}f( \xi )\,d\xi ,\quad t>a, $$

where \(\varGamma (u)=\int _{0}^{\infty } t^{u-1}e^{-t}\,dt\) denotes the Gamma function [69,70,71].

Definition 1.7

Let \(\mu ,\alpha ,\jmath ,\gamma ,c\in \mathbb{C}\), \(\mathfrak{R(\mu )}, \mathfrak{R(\alpha )},\mathfrak{R(\jmath )}>0\), \(\mathfrak{R(c)}> \mathfrak{R(\gamma )}>0\), \(p\geq 0\), \(\delta >0\) and \(0<\kappa \leq \delta +\mathfrak{R(\mu )}\). Then the extended generalized Mittag-Leffler function \(E_{\mu ,\alpha ,\jmath }^{\gamma ,\delta , \kappa ,c}(t;p)\) is defined by

$$ E_{\mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c}(t;p)=\sum_{n=0} ^{\infty }\frac{\beta _{p}(\gamma +n\kappa ,c-\gamma )(c)_{n\kappa }}{ \beta (\gamma ,c-\gamma )\varGamma (\mu n+\alpha )}\frac{t^{n}}{(\jmath )_{n\delta }}, $$

where \(\beta _{p}\) is the generalized beta function defined by

$$ \beta _{p}(x,y)= \int _{0}^{1}t^{x-1}(1-t)^{y-1}e^{- \frac{p}{t(1-t)}}\,dt $$

and \((c)_{n\kappa }\) is the Pochhammer symbol [72,73,74] defined as \((c)_{n\kappa }=\varGamma (c+n\kappa )/\varGamma (c)\).

In [75], several properties of the generalized Mittag-Leffler function are discussed, and it has been proved that \(E_{\mu ,\alpha ,\jmath } ^{\gamma ,\delta ,\kappa ,c}(t;p)\) is absolutely convergent for \(k<\delta +\mathfrak{\mu }\). If S is the sum of the series of absolute terms of the Mittag-Leffler function \(E_{\mu ,\alpha ,\jmath }^{ \gamma ,\delta ,\kappa ,c}(t;p)\), then we have \(|E_{\mu ,\alpha , \jmath }^{\gamma ,\delta ,\kappa ,c}(\cdot ;p)|\leq S\). This property will be used to prove our main results.

The corresponding left and right sided extended generalized fractional integral operators are defined by Theorem 1.8 as follows.

Theorem 1.8

(See [75])

Let \(\mu ,\alpha ,\jmath ,\gamma ,c\in \mathbb{C}\), \(\mathfrak{R(\mu )}, \mathfrak{R(\alpha )},\mathfrak{R(\jmath )}>0\), \(\mathfrak{R(c)}>\mathfrak{R(\gamma )}>0\), \(p\geq 0\), \(\delta >0\), \(0<\kappa \leq \delta +\mathfrak{R(\mu )}\), \(f\in L_{1}[a,b]\)and \(x\in {}[ a,b]\). Then the extended generalized fractional integral operators \(\epsilon _{\mu ,\alpha ,\jmath ,w,a^{+}}^{\gamma ,\delta , \kappa ,c}f\)and \(\epsilon _{\mu ,\alpha ,\jmath ,w,b^{-}}^{\gamma , \delta ,\kappa ,c}f\)can be defined by

$$ \epsilon _{\mu ,\alpha ,\jmath ,w,a^{+}}^{\gamma ,\delta ,\kappa ,c}f(x;p)= \int _{a}^{x}(x-t)^{\alpha -1}E_{\mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl(w(x-t)^{\mu };p\bigr)f(t)\,dt $$

and

$$ \epsilon _{\mu ,\alpha ,\jmath ,w,b^{-}}^{\gamma ,\delta ,\kappa ,c}f(x;p)= \int _{x}^{b}(t-x)^{\alpha -1}E_{\mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl(w(t-x)^{\mu };p\bigr)f(t)\,dt. $$

From the extended generalized fractional integral operators, we have

$$\begin{aligned} \bigl(\epsilon _{\mu ,\alpha ,\jmath ,w,a^{+}}^{\gamma ,\delta ,\kappa ,c}1 \bigr) (x;p) =& \int _{a}^{x}(x-t)^{\alpha -1}E_{\mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl(w(x-t)^{\mu };p\bigr)\,dt \\ =& \int _{a}^{x}(x-t)^{\alpha -1}\sum _{n=0}^{\infty }\frac{\beta _{p}( \gamma +n\kappa ,c-\gamma )(c)_{n\kappa }}{\beta (\gamma ,c-\gamma ) \varGamma (\mu n+\alpha )} \frac{w^{n}(x-t)^{\mu n}}{(\jmath )_{n\delta }}\,dt \\ =&\sum_{n=0}^{\infty }\frac{\beta _{p}(\gamma +n\kappa ,c-\gamma )(c)_{n \kappa }}{\beta (\gamma ,c-\gamma )\varGamma (\mu n+\alpha )} \frac{w^{n}}{( \jmath )_{n\delta }} \int _{a}^{x}(x-t)^{\mu n+\alpha -1}\,dt \\ =&(x-a)^{\alpha }\sum_{n=0}^{\infty } \frac{\beta _{p}(\gamma +n\kappa ,c-\gamma )(c)_{n\kappa }}{\beta (\gamma ,c-\gamma )\varGamma (\mu n+ \alpha )}\frac{w^{n}}{(\jmath )_{n\delta }}(x-a)^{\mu n} \frac{1}{ \mu n+\alpha }. \end{aligned}$$

Hence

$$ \bigl(\epsilon _{\mu ,\alpha ,\jmath ,w,a^{+}}^{\gamma ,\delta ,\kappa ,c}1 \bigr) (x;p)=(x-a)^{\alpha }E_{\mu ,\alpha ,\jmath }^{\gamma , \delta ,\kappa ,c} \bigl(w(x-a)^{\mu };p\bigr), $$

and similarly

$$ \bigl(\epsilon _{\mu ,\alpha ,\jmath ,w,b^{-}}^{\gamma ,\delta ,\kappa ,c}1 \bigr) (x;p)=(x-a)^{\alpha }E_{\mu ,\alpha ,\jmath }^{\gamma , \delta ,\kappa ,c} \bigl(w(b-x)^{\mu };p\bigr). $$

We will use the following notations in the article:

$$ \zeta _{\alpha ,a^{+}}(x;p)= \bigl(\epsilon _{\mu ,\alpha ,\jmath ,w,a ^{+}}^{\gamma ,\delta ,\kappa ,c}1 \bigr) (x;p) $$

and

$$ \zeta _{\alpha ,b^{-}}(x;p)= \bigl(\epsilon _{\mu ,\alpha ,\jmath ,w,b ^{-}}^{\gamma ,\delta ,\kappa ,c}1 \bigr) (x;p). $$

More information related to the Mittag-Leffler functions and the corresponding fractional integral operators can be found in the literature [76,77,78].

The main purpose of the article is to establish a Hadamard type inequality and several general fractional integral inequalities for the exponentially m-convex functions involving an extended Mittag-Leffler function, and deduce some new results which are quite general.

Main results

In order to establish our main results we need a lemma, which we present in this section.

Lemma 2.1

Let \(0\leq a< mb\)and \(f:[a, mb]\rightarrow \mathbb{R}\)be a differentiable function such that \((e^{f} )^{\prime }\in L_{1}[a,mb]\). Then one has

$$\begin{aligned}& \biggl( \int _{a}^{mb}w(s)E_{\mu ,\alpha ,\jmath }^{\gamma , \delta ,\kappa ,c} \bigl(\eta s^{\mu };p\bigr) \biggr)^{\alpha } \bigl[e^{f(a)}+e ^{f(mb)} \bigr] \\& \qquad {}-\alpha \int _{a}^{mb} \biggl( \int _{a}^{t}w(s)E_{\mu , \alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl(\eta s^{\mu };p\bigr) \biggr)^{ \alpha -1}w(t)E_{\mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl( \eta t^{\mu };p\bigr)e^{f(t)}\,dt \\& \qquad {}-\alpha \int _{a}^{mb} \biggl( \int _{t}^{mb}w(s)E_{\mu , \alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl(\eta s^{\mu };p\bigr) \biggr)^{ \alpha -1}w(t)E_{\mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl( \eta t^{\mu };p\bigr)e^{f(t)}\,dt \\& \quad = \int _{a}^{mb} \biggl( \int _{a}^{t}w(s)E_{\mu ,\alpha , \jmath }^{\gamma ,\delta ,\kappa ,c} \bigl(\eta s^{\mu };p\bigr) \biggr)^{\alpha }e ^{f(t)}f^{\prime }(t)\,dt \\& \qquad {}- \int _{a}^{mb} \biggl( \int _{t}^{mb}w(s)E _{\mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl(\eta s^{\mu };p\bigr) \biggr)^{\alpha }e^{f(t)}f^{\prime }(t)\,dt. \end{aligned}$$
(2.1)

Proof

Integrating by parts gives

$$\begin{aligned}& \int _{a}^{mb} \biggl( \int _{a}^{t}w(s)E_{\mu ,\alpha , \jmath }^{\gamma ,\delta ,\kappa ,c} \bigl(\eta s^{\mu };p\bigr) \biggr)^{\alpha }e ^{f(t)}f^{\prime }(t)\,dt \\& \quad = \biggl( \int _{a}^{mb}w(s)E_{\mu ,\alpha ,\jmath }^{\gamma , \delta ,\kappa ,c} \bigl(\eta s^{\mu };p\bigr) \biggr)^{\alpha }e^{f(mb)} \\& \qquad {}-\alpha \int _{a}^{mb} \biggl( \int _{a}^{t}w(s)E_{\mu , \alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl(\eta s^{\mu };p\bigr) \biggr)^{ \alpha -1}w(s)E_{\mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl( \eta s^{\mu };p\bigr)e^{f(t)}\,dt \end{aligned}$$
(2.2)

and

$$\begin{aligned}& \int _{a}^{mb} \biggl( \int _{t}^{mb}w(s)E_{\mu ,\alpha , \jmath }^{\gamma ,\delta ,\kappa ,c} \bigl(\eta s^{\mu };p\bigr) \biggr)^{\alpha }e ^{f(t)}f^{\prime }(t)\,dt \\& \quad =- \biggl( \int _{a}^{mb}w(s)E_{\mu ,\alpha ,\jmath }^{\gamma , \delta ,\kappa ,c} \bigl(\eta s^{\mu };p\bigr) \biggr)^{\alpha }e^{f(a)} \\& \qquad {}+\alpha \int _{a}^{mb} \biggl( \int _{t}^{mb}w(s)E_{\mu , \alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl(\eta s^{\mu };p\bigr) \biggr)^{ \alpha -1}w(s)E_{\mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl( \eta s^{\mu };p\bigr)e^{f(t)}\,dt. \end{aligned}$$
(2.3)

Therefore, identity (2.1) follows from (2.2) and (2.3). □

Let \(m=1\). Then Lemma 2.1 leads to Corollary 2.2 immediately.

Corollary 2.2

Let \(0\leq a< b\)and \(f:[a,b]\rightarrow \mathbb{R}\)be a differentiable exponential function such that \((e^{f} )^{\prime }\in L_{1}[a,b]\). Then the identity for the extended generalized fractional integral operators

$$\begin{aligned}& \biggl( \int _{a}^{b}w(s)E_{\mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl(\eta s^{\mu };p\bigr) \biggr)^{\alpha } \bigl[e^{f(a)}+e^{f(b)} \bigr] \\& \qquad {}-\alpha \int _{a}^{b} \biggl( \int _{a}^{t}w(s)E_{ \mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl(\eta s^{\mu };p\bigr) \biggr)^{\alpha -1}w(t)E_{\mu ,\alpha ,\jmath }^{\gamma ,\delta , \kappa ,c} \bigl(\eta t^{\mu };p\bigr)e^{f(t)}\,dt \\& \qquad {}-\alpha \int _{a}^{b} \biggl( \int _{t}^{b}w(s)E_{\mu , \alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl(\eta s^{\mu };p\bigr) \biggr)^{ \alpha -1}w(t)E_{\mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl( \eta t^{\mu };p\bigr)e^{f(t)}\,dt \\& \quad = \int _{a}^{b} \biggl( \int _{a}^{t}w(s)E_{\mu ,\alpha , \jmath }^{\gamma ,\delta ,\kappa ,c} \bigl(\eta s^{\mu };p\bigr) \biggr)^{\alpha }e ^{f(t)}f^{\prime }(t)\,dt \\& \qquad {}- \int _{a}^{b} \biggl( \int _{t}^{b}w(s)E _{\mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl(\eta s^{\mu };p\bigr) \biggr)^{\alpha }e^{f(t)}f^{\prime }(t)\,dt \end{aligned}$$
(2.4)

holds.

Theorem 2.3

Let \(0\leq a< mb\)and \(f:[a, mb]\rightarrow \mathbb{R}\)be a differentiable function such that \((e^{f} )^{\prime }\in L_{1}[a,mb]\). Then the inequality

$$\begin{aligned}& \biggl\vert \biggl( \int _{a}^{mb}w(s)E_{\mu ,\alpha ,\jmath } ^{\gamma ,\delta ,\kappa ,c}\bigl(\eta s^{\mu };p\bigr) \biggr)^{\alpha } \bigl[e ^{f(a)}+e^{f(mb)} \bigr] \\& \quad \quad {}-\alpha \int _{a}^{mb} \biggl( \int _{a}^{t}w(s)E_{\mu , \alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl(\eta s^{\mu };p\bigr) \biggr)^{ \alpha -1}w(t)E_{\mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl( \eta t^{\mu };p\bigr)e^{f(t)}\,dt \\& \qquad {}-\alpha \int _{a}^{mb} \biggl( \int _{t}^{mb}w(s)E_{\mu , \alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl(\eta s^{\mu };p\bigr) \biggr)^{ \alpha -1}w(t)E_{\mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl( \eta t^{\mu };p\bigr)e^{f(t)}\,dt \biggr\vert \\& \quad \leq \frac{(mb-a)^{\alpha +1} \Vert w \Vert _{\infty }^{\alpha }S^{ \alpha }}{(\alpha +1)(\alpha +2)(\alpha +3)} \\& \qquad {}\times \bigl(\bigl(\alpha ^{2}+3\alpha \bigr) \bigl[ \bigl\vert e^{f(a)}f^{\prime }(a) \bigr\vert +m^{2} \bigl\vert e^{f(b)}f^{\prime }(b) \bigr\vert \bigr]+2m( \alpha +1)\Delta (a,b) \bigr) \end{aligned}$$
(2.5)

for the extended generalized fractional integral operators holds if \(|f^{\prime }|\)is an exponentiallym-convex function on \([a,mb]\), where

$$ \Delta (a,b)= \bigl\{ \bigl\vert e^{f(a)}f^{\prime }(b) \bigr\vert + \bigl\vert e^{f(b)}f^{\prime }(a) \bigr\vert \bigr\} . $$

Proof

It follows from Lemma 2.1 that

$$\begin{aligned}& \biggl\vert \biggl( \int _{a}^{mb}w(s)E_{\mu ,\alpha ,\jmath } ^{\gamma ,\delta ,\kappa ,c}\bigl(\eta s^{\mu };p\bigr) \biggr)^{\alpha } \bigl[e ^{f(a)}+e^{f(mb)} \bigr] \\& \qquad {}-\alpha \int _{a}^{mb} \biggl( \int _{a}^{t}w(s)E_{\mu , \alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl(\eta s^{\mu };p\bigr) \biggr)^{ \alpha -1}w(t)E_{\mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl( \eta t^{\mu };p\bigr)e^{f(t)}\,dt \\& \qquad {}-\alpha \int _{a}^{mb} \biggl( \int _{t}^{mb}w(s)E_{\mu , \alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl(\eta s^{\mu };p\bigr) \biggr)^{ \alpha -1}w(t)E_{\mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl( \eta t^{\mu };p\bigr)e^{f(t)}\,dt \biggr\vert \\& \quad \leq \int _{a}^{mb} \biggl\vert \int _{a}^{t}w(s)E_{\mu , \alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl(\eta s^{\mu };p\bigr) \biggr\vert ^{\alpha } \bigl\vert e^{f(t)}f^{\prime }(t) \bigr\vert \,dt \\& \qquad {}+ \int _{a}^{mb} \biggl\vert \int _{t}^{mb}w(s)E_{\mu , \alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl(\eta s^{\mu };p\bigr) \biggr\vert ^{\alpha } \bigl\vert e^{f(t)}f^{\prime }(t) \bigr\vert \,dt. \end{aligned}$$
(2.6)

By using absolute convergence property of the Mittag-Leffler function and \(\Vert w\Vert _{\infty }=\sup_{t\in {}[ a,b]}|w(t)|\), we have

$$\begin{aligned}& \biggl\vert \biggl( \int _{a}^{mb}w(s)E_{\mu ,\alpha ,\jmath } ^{\gamma ,\delta ,\kappa ,c}\bigl(\eta s^{\mu };p\bigr) \biggr)^{\alpha } \bigl[e ^{f(a)}+e^{f(mb)} \bigr] \\& \qquad {}-\alpha \int _{a}^{mb} \biggl( \int _{a}^{t}w(s)E_{\mu , \alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl(\eta s^{\mu };p\bigr) \biggr)^{ \alpha -1}w(t)E_{\mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl( \eta t^{\mu };p\bigr)e^{f(t)}\,dt \\& \qquad {}-\alpha \int _{a}^{mb} \biggl( \int _{t}^{mb}w(s)E_{\mu , \alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl(\eta s^{\mu };p\bigr) \biggr)^{ \alpha -1}w(t)E_{\mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl( \eta t^{\mu };p\bigr)e^{f(t)}\,dt \biggr\vert \\& \quad \leq \Vert w \Vert _{\infty }^{\alpha }S^{\alpha } \biggl( \int _{a} ^{mb}(t-a)^{\alpha } \bigl\vert e^{f(t)}f^{\prime }(t) \bigr\vert \,dt+ \int _{a}^{mb}(mb-t)^{ \alpha } \bigl\vert e^{f(t)}f^{\prime }(t) \bigr\vert \,dt \biggr). \end{aligned}$$
(2.7)

Since \(|e^{f(t)}f^{\prime }(t)|\) is an exponentially m-convex function, we get

$$\begin{aligned}& \bigl\vert e^{f(t)}f^{\prime }(t) \bigr\vert \\& \quad \leq \biggl[\frac{mb-t}{mb-a} \bigl\vert e^{f(a)} \bigr\vert +m \frac{t-a}{mb-a} \bigl\vert e^{f(b)} \bigr\vert \biggr] \biggl[ \frac{mb-t}{mb-a} \bigl\vert f^{\prime }(a) \bigr\vert +m \frac{t-a}{mb-a} \bigl\vert {f(b)} \bigr\vert \biggr] \\& \quad = \biggl(\frac{mb-t}{mb-a} \biggr)^{2} \bigl\vert e^{f(a)}f^{\prime }(a) \bigr\vert +m^{2} \biggl( \frac{t-a}{mb-a} \biggr)^{2} \bigl\vert e^{f(b)}f^{\prime }(b) \bigr\vert \\& \qquad {}+m \biggl(\frac{mb-t}{mb-a} \biggr) \biggl(\frac{t-a}{mb-a} \biggr) \bigl\{ \bigl\vert e ^{f(a)}f^{\prime }(b) \bigr\vert + \bigl\vert e^{f(b)}f^{\prime }(a) \bigr\vert \bigr\} \\& \quad = \biggl(\frac{mb-t}{mb-a} \biggr)^{2} \bigl\vert e^{f(a)}f^{\prime }(a) \bigr\vert +m^{2} \biggl( \frac{t-a}{mb-a} \biggr)^{2} \bigl\vert e^{f(b)}f^{\prime }(b) \bigr\vert \\& \qquad {}+m \biggl(\frac{mb-t}{mb-a} \biggr) \biggl(\frac{t-a}{mb-a} \biggr) \Delta (a,b). \end{aligned}$$
(2.8)

By taking into account the inequalities (2.7) and (2.8), we deduce

$$\begin{aligned}& \biggl\vert \biggl( \int _{a}^{mb}w(s)E_{\mu ,\alpha ,\jmath } ^{\gamma ,\delta ,\kappa ,c}\bigl(\eta s^{\mu };p\bigr) \biggr)^{\alpha } \bigl[e ^{f(a)}+e^{f(mb)} \bigr] \\& \qquad {}-\alpha \int _{a}^{mb} \biggl( \int _{a}^{t}w(s)E_{\mu , \alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl(\eta s^{\mu };p\bigr) \biggr)^{ \alpha -1}w(t)E_{\mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl( \eta t^{\mu };p\bigr)e^{f(t)}\,dt \\& \qquad {}-\alpha \int _{a}^{mb} \biggl( \int _{t}^{mb}w(s)E_{\mu , \alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl(\eta s^{\mu };p\bigr) \biggr)^{ \alpha -1}w(t)E_{\mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl( \eta t^{\mu };p\bigr)e^{f(t)}\,dt \biggr\vert \\& \quad \leq \Vert w \Vert _{\infty }^{\alpha }S^{\alpha } \biggl( \int _{a} ^{mb}(t-a)^{\alpha } \biggl[ \biggl(\frac{mb-t}{mb-a} \biggr) ^{2} \bigl\vert e^{f(a)}f ^{\prime }(a) \bigr\vert +m^{2} \biggl( \frac{t-a}{mb-a} \biggr) ^{2} \bigl\vert e^{f(b)}f^{ \prime }(b) \bigr\vert \\& \qquad {}+m \biggl(\frac{mb-t}{mb-a} \biggr) \biggl(\frac{t-a}{mb-a} \biggr) \Delta (a,b) \biggr]\,dt+ \int _{a}^{mb}(mb-t)^{\alpha } \biggl[ \biggl( \frac{mb-t}{mb-a} \biggr)^{2} \bigl\vert e^{f(a)}f^{\prime }(a) \bigr\vert \\& \qquad {}+m^{2} \biggl(\frac{t-a}{mb-a} \biggr)^{2} \bigl\vert e^{f(b)}f^{\prime }(b) \bigr\vert + m \biggl( \frac{mb-t}{mb-a} \biggr) \biggl(\frac{t-a}{mb-a} \biggr)\Delta (a,b) \biggr]\,dt \biggr) \\& \quad =\frac{(mb-a)^{\alpha +1} \Vert w \Vert _{\infty }^{\alpha }S^{\alpha }}{( \alpha +1)(\alpha +2)(\alpha +3)} \bigl(\bigl(\alpha ^{2}+3\alpha \bigr) \bigl[ \bigl\vert e ^{f(a)}f^{\prime }(a) \bigr\vert +m^{2} \bigl\vert e^{f(b)}f^{\prime }(b) \bigr\vert \bigr]+2m(\alpha +1) \Delta (a,b) \bigr). \end{aligned}$$

This completes the proof. □

Let \(m=1\). Then Theorem 2.3 leads to Corollary 2.4 immediately.

Corollary 2.4

Let \(0\leq a< b\)and \(f:[a,b]\rightarrow \mathbb{R}\)be a differentiable function such that \((e^{f} )^{\prime }\in L_{1}[a,b]\). Then the inequality

$$\begin{aligned}& \biggl\vert \biggl( \int _{a}^{b}w(s)E_{\mu ,\alpha ,\jmath }^{ \gamma ,\delta ,\kappa ,c} \bigl(\eta s^{\mu };p\bigr) \biggr)^{\alpha } \bigl[e ^{f(a)}+e^{f(b)} \bigr] \\& \qquad {}-\alpha \int _{a}^{b} \biggl( \int _{a}^{t}w(s)E_{\mu , \alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl(\eta s^{\mu };p\bigr) \biggr)^{ \alpha -1}w(t)E_{\mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl( \eta t^{\mu };p\bigr)e^{f(t)}\,dt \\& \qquad {}-\alpha \int _{a}^{mb} \biggl( \int _{t}^{b}w(s)E_{\mu , \alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl(\eta s^{\mu };p\bigr) \biggr)^{ \alpha -1}w(t)E_{\mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl( \eta t^{\mu };p\bigr)e^{f(t)}\,dt \biggr\vert \\& \quad \leq \frac{(b-a)^{\alpha +1} \Vert w \Vert _{\infty }^{\alpha }S^{\alpha }}{(\alpha +1)(\alpha +2)(\alpha +3)} \bigl(\bigl(\alpha ^{2}+3\alpha \bigr) \bigl[ \bigl\vert e^{f(a)}f^{\prime }(a) \bigr\vert + \bigl\vert e^{f(b)}f^{\prime }(b) \bigr\vert \bigr]+2(\alpha +1) \Delta (a,b) \bigr) \end{aligned}$$

for extended generalized fractional integral operators holds if \(|f^{\prime }|\)is a convex function on \([a,b]\)and \(k<\delta + \mathfrak{R(\mu )}\), where \(\Vert w\Vert _{\infty }^{\alpha }= \sup_{t\in {}[ a,b]}|w(t)|\)and \(\Delta (a,b)\)are given in Theorem 2.3.

Corollary 2.5

If \(p=0\)and all the assumptions of Theorem 2.3are satisfied, then one has

$$\begin{aligned}& \biggl\vert \biggl( \int _{a}^{mb}w(s)E_{\mu ,\alpha ,\jmath } ^{\gamma ,\delta ,\kappa ,c}\bigl(\eta s^{\mu }\bigr) \biggr)^{\alpha } \bigl[e^{f(a)}+e ^{f(mb)}\bigr] \\& \qquad {}-\alpha \int _{a}^{mb} \biggl( \int _{a}^{t}w(s)E_{\mu , \alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl(\eta s^{\mu }\bigr) \biggr)^{ \alpha -1}w(t)E_{\mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl( \eta t^{\mu }\bigr)e^{f(t)}\,dt \\& \qquad {}\alpha \int _{a}^{mb} \biggl( \int _{t}^{mb}w(s)E_{\mu , \alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl(\eta s^{\mu }\bigr) \biggr)^{ \alpha -1}w(t)E_{\mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl( \eta t^{\mu }\bigr)e^{f(t)}\,dt \biggr\vert \\& \quad \leq \frac{(mb-a)^{\alpha +1} \Vert w \Vert _{\infty }^{\alpha }S^{ \alpha }}{(\alpha +1)(\alpha +2)(\alpha +3)} \\& \qquad {}\times \bigl(\bigl(\alpha ^{2}+3\alpha \bigr) \bigl[ \bigl\vert e^{f(a)}f^{\prime }(a) \bigr\vert +m^{2} \bigl\vert e^{f(b)}f^{\prime }(b) \bigr\vert \bigr]+2m( \alpha +1)\Delta (a,b) \bigr) \end{aligned}$$

for \(k<\delta +\mathfrak{R(\mu )}\), where \(\Vert w\Vert _{\infty }^{ \alpha }= \sup_{t\in {}[ a,b]}|w(t)|\)and \(\Delta (a,b)\)is given in Theorem 2.3.

Corollary 2.6

Let \(\jmath =p=0\), \(m=1\)and all the assumptions of Theorem 2.3are satisfied, then we get

$$\begin{aligned}& \biggl\vert \biggl( \int _{a}^{b}w(s)\,ds \biggr)^{\alpha } \bigl[e ^{f(a)}+e^{f(b)} \bigr]-\alpha \int _{a}^{b} \biggl( \int _{a}^{t}w(s)\,ds \biggr)^{\alpha -1}w(t)e^{f(t)}\,dt \\& \qquad {}-\alpha \int _{a}^{b} \biggl( \int _{a}^{t}w(s)\,ds \biggr) ^{ \alpha -1}w(t)e^{f(t)}\,dt \biggr\vert \\& \quad \leq \frac{(b-a)^{\alpha +1} \Vert w \Vert _{\infty }^{\alpha }}{(\alpha +1)( \alpha +2) (\alpha +3)} \bigl(\bigl(\alpha ^{2}+3\alpha \bigr) \bigl[ \bigl\vert e^{f(a)}f ^{\prime }(a) \bigr\vert + \bigl\vert e^{f(b)}f^{\prime }(b) \bigr\vert \bigr]+2(\alpha +1) \Delta (a,b) \bigr), \end{aligned}$$

where \(\alpha >0\), \(\|w\|_{\infty }^{\alpha }=\sup_{t\in [a,b]}\vert w(t)\vert \)and \(\Delta (a,b)\)is given in Theorem 2.3.

Corollary 2.7

If we choose \(\jmath =p=0\), \(m=1\), \(\alpha =\mu /k\)and \(w(s)=1\), then we have the new result

$$\begin{aligned}& \biggl\vert \frac{e^{f(a)}+e^{f(b)}}{2}-\frac{\varGamma _{k}(\mu +k)}{2(b-a)^{\frac{ \mu }{k}}} \biggr\vert \bigl[I_{a^{+}}^{\mu ,k}e^{f(b)}+I_{b^{-}}^{ \mu ,k}e^{f(a)} \bigr] \\& \quad \leq \frac{(b-a)}{(\frac{\mu }{k}+1)(\frac{\mu }{k}+2)(\frac{\mu }{k}+3)} \biggl(\biggl(\frac{\mu }{k} \biggr)^{2}+3\frac{\mu }{k}\biggr) \bigl[ \bigl\vert e^{f(a)}f^{\prime }(a) \bigr\vert + \bigl\vert e^{f(b)}f^{\prime }(b) \bigr\vert \bigr] \\& \qquad {}+2\biggl( \frac{\mu }{k}+1\biggr)\Delta (a,b) ) \end{aligned}$$

under the assumptions of Theorem 2.3.

Corollary 2.8

If \(\jmath =p=0\), \(m=1\), \(\alpha =\frac{\mu }{k}\), \(w(s)=1\)and \(\alpha =\mu \), then the inequality

$$\begin{aligned}& \biggl\vert \frac{e^{f(a)}+e^{f(b)}}{2}-\frac{\varGamma (\mu +1)}{2(b-a)^{ \mu }} \biggr\vert \bigl[I_{a^{+}}^{\mu }e^{f(b)}+I_{b^{-}}^{\mu }e ^{f(a)} \bigr] \\& \quad \leq \frac{(b-a)}{(\mu +1)(\frac{\mu }{k}+2)(\mu +3)} \bigl((\mu )^{2}+3 \mu \bigr) \bigl[ \bigl\vert e^{f(a)}f^{\prime }(a) \bigr\vert + \bigl\vert e^{f(b)}f^{\prime }(b) \bigr\vert \bigr]+2( \mu +1)\Delta (a,b) ) \end{aligned}$$

holds under the assumption of Theorem 2.3.

Theorem 2.9

Let \(0\leq a< mb\), \(q, r>1\)such that \(1/q+1/r=1\), and \(f:[a,mb]\rightarrow \mathbb{R}\)be a differentiable function such that \((e^{f} )^{ \prime }\in L_{1}[a,mb]\). Then the inequality

$$\begin{aligned}& \biggl\vert \biggl( \int _{a}^{mb}w(s)E_{\mu ,\alpha ,\jmath } ^{\gamma ,\delta ,\kappa ,c}\bigl(\eta s^{\mu };p\bigr) \biggr)^{\alpha } \bigl[e ^{f(a)}+e^{f(mb)} \bigr] \\& \qquad {}-\alpha \int _{a}^{mb} \biggl( \int _{a}^{t}w(s)E_{\mu , \alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl(\eta s^{\mu };p\bigr) \biggr)^{ \alpha -1}w(t)E_{\mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl( \eta t^{\mu };p\bigr)e^{f(t)}\,dt \\& \qquad {}-\alpha \int _{a}^{mb} \biggl( \int _{t}^{mb}w(s)E_{\mu , \alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl(\eta s^{\mu };p\bigr) \biggr)^{ \alpha -1}w(t)E_{\mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl( \eta t^{\mu };p\bigr)e^{f(t)}\,dt \biggr\vert \\& \quad \leq \frac{2(mb-a)^{\alpha +1} \Vert w \Vert _{\infty }^{\alpha }S^{ \alpha } }{(\alpha r+1)^{\frac{1}{r}}} \biggl(\frac{2 \{ \vert e^{f(a)}f ^{\prime }(a) \vert ^{q}+m^{2} \vert e^{f(b)}f^{\prime }(b) \vert ^{q} \}+m\Delta _{1}(a,b)}{6} \biggr) ^{\frac{1}{q}} \end{aligned}$$
(2.9)

for extended generalized fractional integral operators holds if \(|f^{\prime }|^{q}\)is an exponentiallym-convex function on \([a,mb]\)and \(k<\delta +\mathfrak{R(\mu )}\), where \(\Vert w \Vert _{\infty }=\sup_{t\in {}[ a,mb]}|w(t)|\)and

$$ \Delta _{1}(a,b)= \bigl\vert e^{f(a)}f^{\prime }(b) \bigr\vert ^{q}+ \bigl\vert e^{f(b)}f^{\prime }(a) \bigr\vert ^{q}. $$

Proof

From Lemma 2.1 and the Hölder inequality we get

$$\begin{aligned}& \biggl\vert \biggl( \int _{a}^{mb}w(s)E_{\mu ,\alpha ,\jmath } ^{\gamma ,\delta ,\kappa ,c} \bigl(\eta s^{\mu };p\bigr) \biggr)^{\alpha } \bigl[e ^{f(a)}+e^{f(mb)} \bigr] \\& \qquad {}-\alpha \int _{a}^{mb} \biggl( \int _{a}^{t}w(s) E_{\mu , \alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl(\eta s^{\mu };p\bigr) \biggr)^{ \alpha -1}w(t)E_{\mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl( \eta t^{\mu };p\bigr)e^{f(t)}\,dt \\& \qquad {}-\alpha \int _{a}^{mb} \biggl( \int _{t}^{mb}w(s)E_{\mu , \alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl(\eta s^{\mu };p\bigr) \biggr)^{ \alpha -1}w(t)E_{\mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl( \eta t^{\mu };p\bigr)e^{f(t)}\,dt \biggr\vert \\& \quad \leq \biggl( \int _{a}^{mb} \biggl\vert \int _{a}^{t}w(s)E _{\mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl(\eta s^{\mu };p\bigr)\,ds \biggr\vert ^{\alpha r} \biggr)^{\frac{1}{r}} \biggl( \int _{a}^{mb} \bigl\vert e ^{f(t)} {f^{\prime }(t)} \bigr\vert \,dt \biggr)^{\frac{1}{q}} \\& \qquad {}+ \biggl( \int _{a}^{mb} \biggl\vert \int _{t}^{mb}w(s)E_{ \mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl(\eta s^{\mu };p\bigr)\,ds \biggr\vert ^{\alpha r} \biggr)^{\frac{1}{r}} \biggl( \int _{a}^{mb} \bigl\vert e ^{f(t)} {f^{\prime }(t)} \bigr\vert \,dt \biggr)^{\frac{1}{q}}. \end{aligned}$$

It follows from the absolute convergence property of the Mittag-Leffler function and \(\Vert w\Vert _{\infty }=\sup_{t\in {}[ a,b]}|w(t)|\) that

$$\begin{aligned}& \biggl\vert \biggl( \int _{a}^{mb}w(s)E_{\mu ,\alpha ,\jmath } ^{\gamma ,\delta ,\kappa ,c} \bigl(\eta s^{\mu };p\bigr) \biggr)^{\alpha } \bigl[e ^{f(a)}+e^{f(mb)} \bigr] \\& \qquad {}-\alpha \int _{a}^{mb} \biggl( \int _{a}^{t}w(s)E_{\mu , \alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl(\eta s^{\mu };p\bigr) \biggr)^{ \alpha -1}w(t)E_{\mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl( \eta t^{\mu };p\bigr)e^{f(t)}\,dt \\& \qquad {}-\alpha \int _{a}^{mb} \biggl( \int _{t}^{mb}w(s)E_{\mu , \alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl(\eta s^{\mu };p\bigr) \biggr)^{ \alpha -1}w(t)E_{\mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl( \eta t^{\mu };p\bigr)e^{f(t)}\,dt \biggr\vert \\& \quad \leq \Vert w \Vert _{\infty }^{\alpha }S^{\alpha } \biggl( \int _{a} ^{mb} \vert t-a \vert ^{\alpha r}\,dt \biggr)^{\frac{1}{p}} \\& \qquad {} + \int _{a}^{mb} \vert mb-t \vert ^{ \alpha r}\,dt )^{\frac{1}{p}} \biggl( \int _{a}^{mb} \bigl\vert e^{f(t)}f ^{\prime }(t) \bigr\vert ^{q}\,dt \biggr)^{\frac{1}{q}}. \end{aligned}$$
(2.10)

Since \(|e^{f(t)}f^{\prime }(t)|^{q}\) is an exponentially m-convex function, we obtain

$$\begin{aligned}& \bigl\vert e^{f(t)}f^{\prime }(t) \bigr\vert ^{q} \\& \quad \leq \biggl[\frac{mb-t}{mb-a} \bigl\vert e ^{f(a)} \bigr\vert ^{q} +m\frac{t-a}{mb-a} \bigl\vert e^{f(b)} \bigr\vert ^{q} \biggr] \biggl[ \frac{mb-t}{mb-a} \bigl\vert f^{\prime }(a) \bigr\vert ^{q}+m\frac{t-a}{mb-a} \bigl\vert {f(b)} \bigr\vert ^{q} \biggr] \\& \quad = \biggl(\frac{mb-t}{mb-a} \biggr)^{2} \bigl\vert e^{f(a)}f^{\prime }(a) \bigr\vert ^{q} +m^{2} \biggl(\frac{t-a}{mb-a} \biggr)^{2} \bigl\vert e^{f(b)}f^{\prime }(b) \bigr\vert ^{q} \\& \qquad {}+m \biggl(\frac{mb-t}{mb-a} \biggr) \biggl(\frac{t-a}{mb-a} \biggr) \bigl\{ \bigl\vert e ^{f(a)}f^{\prime }(b) \bigr\vert ^{q} + \bigl\vert e^{f(b)}f^{\prime }(a) \bigr\vert ^{q} \bigr\} \\& \quad = \biggl(\frac{mb-t}{mb-a} \biggr)^{2} \bigl\vert e^{f(a)}f^{\prime }(a) \bigr\vert ^{q} +m^{2} \biggl(\frac{t-a}{mb-a} \biggr)^{2} \bigl\vert e^{f(b)}f^{\prime }(b) \bigr\vert ^{q} \\& \qquad {}+m \biggl(\frac{mb-t}{mb-a} \biggr) \biggl(\frac{t-a}{mb-a} \biggr) \Delta (a,b). \end{aligned}$$
(2.11)

Inequalities (2.10) and (2.11) lead to

$$\begin{aligned}& \biggl\vert \biggl( \int _{a}^{mb}w(s)E_{\mu ,\alpha ,\jmath } ^{\gamma ,\delta ,\kappa ,c}\bigl(\eta s^{\mu };p\bigr) \biggr)^{\alpha } \bigl[e ^{f(a)}+e^{f(mb)} \bigr] \\& \qquad {}-\alpha \int _{a}^{mb} \biggl( \int _{a}^{t}w(s) E_{\mu , \alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl(\eta s^{\mu };p\bigr) \biggr)^{ \alpha -1}w(t)E_{\mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl( \eta t^{\mu };p\bigr)e^{f(t)}\,dt \\& \qquad {}-\alpha \int _{a}^{mb} \biggl( \int _{t}^{mb}w(s) E_{\mu , \alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl(\eta s^{\mu };p\bigr) \biggr)^{ \alpha -1}w(t)E_{\mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl( \eta t^{\mu };p\bigr)e^{f(t)}\,dt \biggr\vert \\& \quad \leq \Vert w \Vert _{\infty }^{\alpha }S^{\alpha } \biggl[ \biggl( \int _{a}^{mb} \vert t-a \vert ^{\alpha r}\,dt \biggr)^{\frac{1}{r}} + \biggl( \int _{a}^{mb} \vert mb-t \vert ^{\alpha r}\,dt \biggr)^{\frac{1}{r}} \biggr] \\& \qquad {}\times \biggl( \int _{a}^{mb} \biggl\{ \biggl( \frac{mb-t}{mb-a} \biggr)^{2} \bigl\vert e ^{f(a)}f^{\prime }(a) \bigr\vert ^{q} \\& \qquad {}+m^{2} \biggl(\frac{t-a}{mb-a} \biggr)^{2} \bigl\vert e^{f(b)}f^{\prime }(b) \bigr\vert ^{q}+m \biggl(\frac{mb-t}{mb-a} \biggr) \biggl(\frac{t-a}{mb-a} \biggr)\Delta (a,b) \biggr\} \,dt \biggr)^{\frac{1}{q}} \\& \quad =\frac{2(mb-a)^{\alpha +1} \Vert w \Vert _{\infty }^{\alpha }S^{\alpha }}{( \alpha r+1)^{\frac{1}{r}}} \biggl(\frac{2 \{ \vert e^{f(a)}f^{\prime }(a) \vert ^{q}+m ^{2} \vert e^{f(b)}f^{\prime }(b) \vert ^{q} \} +m\Delta _{1}(a,b)}{6} \biggr)^{ \frac{1}{q}}, \end{aligned}$$

which is the required result. □

If we take \(m=1\) in (2.9), then we get the following result for an exponentially convex function.

Corollary 2.10

Let \(0\leq a< b\), \(p, q>1\)such that \(1/p+1/q=1\), and \(f:[a,b]\rightarrow \mathbb{R}\)be a differentiable exponentially convex function such that \(f^{\prime }\in L_{1}[a,b]\). Then

$$\begin{aligned}& \biggl\vert \biggl( \int _{a}^{b}w(s)E_{\mu ,\alpha ,\jmath }^{ \gamma ,\delta ,\kappa ,c} \bigl(\eta s^{\mu };p\bigr) \biggr)^{\alpha } \bigl[e ^{f(a)}+e^{f(b)} \bigr] \\ & \qquad {}-\alpha \int _{a}^{b} \biggl( \int _{a}^{t}w(s) E_{\mu , \alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl(\eta s^{\mu };p\bigr) \biggr)^{ \alpha -1}w(t)E_{\mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl( \eta t^{\mu };p\bigr)e^{f(t)}\,dt \\& \qquad {}-\alpha \int _{a}^{b} \biggl( \int _{t}^{b}w(s) E_{\mu , \alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl(\eta s^{\mu };p\bigr) \biggr)^{ \alpha -1}w(t)E_{\mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl( \eta t^{\mu };p\bigr)e^{f(t)}\,dt \biggr\vert \\& \quad \leq \frac{2(b-a)^{\alpha +1} \Vert w \Vert _{\infty }^{\alpha }S^{ \alpha }}{(\alpha r+1)^{\frac{1}{r}}} \biggl(\frac{2 \{ \vert e^{f(a)}f ^{\prime }(a) \vert ^{q} + \vert e^{f(b)}f^{\prime }(b) \vert ^{q} \}+\Delta _{1}(a,b)}{6} \biggr)^{\frac{1}{q}} \end{aligned}$$

for \(k<\delta +\mathfrak{R(\mu )}\)if \(|f^{\prime }|^{q}\)is a convex function on \([a,b]\), where \(\Vert w\Vert _{\infty }= \sup_{t\in {}[ a,b]}|w(t)|\).

Corollary 2.11

If we set \(p=0\), then we get the inequality

$$\begin{aligned}& \biggl\vert \biggl( \int _{a}^{b}w(s)E_{\mu ,\alpha ,\jmath }^{ \gamma ,\delta ,\kappa ,c} \bigl(\eta s^{\mu }\bigr) \biggr)^{\alpha } \bigl[e^{f(a)}+e ^{f(b)} \bigr] \\ & \qquad {}-\alpha \int _{a}^{b} \biggl( \int _{a}^{t}w(s) E_{\mu , \alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl(\eta s^{\mu }\bigr) \biggr)^{ \alpha -1}w(t)E_{\mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl( \eta t^{\mu };p\bigr)e^{f(t)}\,dt \\ & \qquad {}-\alpha \int _{a}^{b} \biggl( \int _{t}^{b}w(s) E_{\mu , \alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl(\eta s^{\mu }\bigr) \biggr)^{ \alpha -1}w(t)E_{\mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl( \eta t^{\mu }\bigr)e^{f(t)}\,dt \biggr\vert \\ & \quad \leq \frac{2(b-a)^{\alpha +1} \Vert w \Vert _{\infty }^{\alpha }S^{ \alpha }}{(\alpha r+1)^{\frac{1}{r}}} \biggl(\frac{2 \{ \vert e^{f(a)}f ^{\prime }(a) \vert ^{q} + \vert e^{f(b)}f^{\prime }(b) \vert ^{q} \}+\Delta _{1}(a,b)}{6} \biggr)^{\frac{1}{q}} \end{aligned}$$

for \(k<\delta +\mathfrak{R(\mu )}\), where \(\Vert w\Vert _{\infty }= \sup_{t\in {}[ a,b]}|w(t)|\).

Corollary 2.12

If we set \(\jmath =p=0\)and \(m=1\), then one has

$$\begin{aligned}& \biggl\vert \biggl( \int _{a}^{b}w(s)\,ds \biggr)^{\alpha } \bigl[e^{f(a)}+e ^{f(b)}\bigr]-2\alpha \int _{a}^{b} \biggl( \int _{a}^{t}w(s)\,ds \biggr)^{\alpha -1}w(t)e^{f(t)}\,dt \biggr\vert \\& \quad \leq \frac{2(b-a)^{\alpha +1} \Vert w \Vert _{\infty }^{\alpha }}{( \alpha r+1)^{\frac{1}{r}}} \biggl(\frac{2 \{ \vert e^{f(a)}f^{\prime }(a) \vert ^{q} + \vert e^{f(b)}f^{\prime }(b) \vert ^{q} \}+\Delta _{1}(a,b)}{6} \biggr)^{ \frac{1}{q}} \end{aligned}$$

for \(k<\delta +\mathfrak{R(\mu )}\), where \(\Vert w\Vert _{\infty }= \sup_{t\in {}[ a,b]}|w(t)|\).

Corollary 2.13

If we set \(\jmath =p=0\), \(m=1\)and \(\alpha =\frac{\mu }{k}\), then we have

$$\begin{aligned}& \biggl\vert \biggl( \int _{a}^{b}w(s)\,ds \biggr)^{\frac{\mu }{k}} \bigl[e ^{f(a)}+e^{f(b)}\bigr]-\frac{\mu }{k} \int _{a}^{b} \biggl( \int _{a}^{t}w(s)\,ds \biggr)^{\frac{\mu }{k}-1}w(t)e^{f(t)}\,dt \biggr\vert \\& \quad \leq \frac{2(b-a)^{\frac{\mu }{k}+1} \Vert w \Vert _{\infty }^{\frac{ \mu }{k}}}{(\frac{\mu }{k}r+1)^{\frac{1}{r}}} \biggl(\frac{2 \{ \vert e ^{f(a)}f^{\prime }(a) \vert ^{q} + \vert e^{f(b)}f^{\prime }(b) \vert ^{q} \}+\Delta _{1}(a,b)}{6} \biggr)^{\frac{1}{q}} \end{aligned}$$

for \(k<\delta +\mathfrak{R(\mu )}\), where \(\Vert w\Vert _{\infty }= \sup_{t\in {}[ a,b]}|w(t)|\).

Corollary 2.14

If we set \(\jmath =p=0\), \(m=1\)and \(w(s)=1\), then

$$\begin{aligned}& \biggl\vert \frac{e^{f(a)}+e^{f(b)}}{2}-\frac{1}{2(b-a)^{\frac{\alpha }{k}}} \bigl[I_{a^{+}}^{\alpha ,k}e^{f(b)}+I_{b^{-}}^{\alpha ,k}e^{f(a)} \bigr] \biggr\vert \\& \quad \leq \frac{(b-a)}{(\frac{\mu }{k}r+1)^{\frac{1}{r}}} \biggl(\frac{2 \{ \vert e^{f(a)}f^{\prime }(a) \vert ^{q}+ \vert e^{f(b)}f^{\prime }(b) \vert ^{q} \} +\Delta _{1}(a,b)}{6} \biggr)^{\frac{1}{q}}. \end{aligned}$$

Corollary 2.15

If we set \(\jmath =p=0\), \(m=1\), \(w(s)=1\)and \(\alpha =1\), then

$$\begin{aligned}& \biggl\vert \frac{e^{f(a)}+e^{f(b)}}{2}-\frac{1}{b-a} \int _{a} ^{b}e^{f(x)}\,dx \biggr\vert \\& \quad \leq \frac{(b-a)}{(r+1)^{\frac{1}{r}}} \biggl(\frac{2 \{ \vert e^{f(a)}f ^{\prime }(a) \vert ^{q}+ \vert e^{f(b)}f^{\prime }(b) \vert ^{q} \} +\Delta _{1}(a,b)}{6} \biggr)^{\frac{1}{q}}. \end{aligned}$$

Next, we establish the Hermite–Hadamard type inequalities for exponentially m-convex functions via an extended Mittag-Leffler function.

Theorem 2.16

Let \(0\leq a< mb\), \(f:[a,mb]\longrightarrow \mathbb{R}\)be an exponentiallym-convex such that \(f\in L_{1}[a,mb]\). Then the inequalities for extended generalized fractional integral operators

$$\begin{aligned}& 2e^{f (\frac{a+mb}{2} )}\zeta _{{\alpha ,(\frac{a+mb}{2})}^{+}}(mb;p) \\& \quad \leq \bigl( \varepsilon _{\mu ,\alpha ,\jmath ,w^{\prime },(\frac{a+mb}{2})^{+}} ^{\gamma ,\delta ,\kappa ,c}e^{f} \bigr) (mb;p) +m^{\alpha +1} \bigl( \varepsilon _{\mu ,\alpha ,\jmath ,w^{\prime }, (\frac{a+mb}{2m})^{+}} ^{\gamma ,\delta ,\kappa ,c}e^{f} \bigr) \biggl(\frac{a}{m};p \biggr) \\& \quad \leq \frac{a}{mb-a} \bigl(e^{f(a)}-m^{2}e^{(\frac{a}{m^{2}})} \bigr) \zeta _{{\alpha +1,(\frac{a+mb}{2})}^{+}}(mb;p)+m^{\alpha +1} \bigl(e ^{f(b)} +me^{f\frac{a}{m^{2}}} \bigr) \zeta _{{\alpha ,(\frac{a+mb}{2m})}^{-}} \end{aligned}$$
(2.12)

hold, where \(w^{\prime }=\frac{2^{\mu }w}{(mb-u)^{\mu }}\).

Proof

Since f is an exponentially m-convex function, we get

$$ 2e^{f (\frac{a+mb}{2} )}\leq e^{f (\frac{t}{2}a+ \frac{2-t}{2}mb )} +me^{f (\frac{2-t}{2m}a+\frac{t}{2}b )}. $$
(2.13)

It follows from the definition of exponentially m-convexity that

$$\begin{aligned}& e^{f (\frac{t}{2}a+\frac{2-t}{2}mb )} +me^{f (\frac{2-t}{2m}a+ \frac{t}{2}b )} \\& \quad \leq \frac{t}{2} \bigl(e^{f(a)}-m^{2}e^{\frac{a}{m^{2}}} \bigr) +m \bigl(e^{f(b)}+me^{f(\frac{a}{m^{2}})} \bigr). \end{aligned}$$
(2.14)

Multiplying (2.13) by \(t^{\alpha -1}E_{\mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c}(wt^{\mu };p)\) on both sides and then integrating over \([0,1]\), we get

$$\begin{aligned}& 2e^{f (\frac{a+mb}{2} )} \int _{0}^{1}t^{\alpha -1} E_{ \mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c}\bigl(wt^{\mu };p\bigr)\,dt \\& \quad \leq t^{\alpha -1}E_{\mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl(wt^{\mu };p \bigr)e^{f (\frac{t}{2}a+\frac{2-t}{2}mb )}\,dt+mt^{\alpha -1} E_{\mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \bigl(wt^{\mu };p\bigr) e^{f (\frac{2-t}{2m}a+\frac{t}{2}b )}\,dt. \end{aligned}$$
(2.15)

Let \(u=\frac{t}{2}a+\frac{2-t}{2}mb\) and \(v=\frac{2-t}{2m}a+ \frac{t}{2}b\). Then (2.15) gives

$$\begin{aligned}& 2e^{f (\frac{a+mb}{2} )} \int _{a}^{mb}(mb-u)^{\alpha -1} E _{\mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c}\bigl(w(mb-u)^{\mu };p\bigr)\,du \\& \quad \leq (mb-u)^{\alpha -1}E_{\mu ,\alpha ,\jmath }^{\gamma ,\delta , \kappa ,c} \bigl(w(mb-u)^{\mu };p\bigr)e^{f(u)}\,du \\& \qquad {}+m^{\alpha +1} \int _{\frac{a}{m}}^{\frac{a+mb}{2m}} \biggl(v- \frac{a}{m} \biggr)^{\alpha -1} E_{\mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c}\biggl(w \biggl(v- \frac{a}{m} \biggr)^{\mu };p\biggr)e^{f(v)}\,dv. \end{aligned}$$
(2.16)

By using (2.13), (2.15) and (2.16), we get the first inequality of (2.12).

Now multiplying (2.14) by \(t^{\alpha -1}E_{\mu ,\alpha ,\jmath }^{ \gamma ,\delta ,\kappa ,c}(wt^{\mu };p)\) on both sides and then integrating over \([0,1]\), we have

$$\begin{aligned}& \int _{0}^{1}t^{\alpha -1}E_{\mu ,\alpha ,\jmath }^{\gamma , \delta ,\kappa ,c} \bigl(wt^{\mu };p\bigr) e^{f (\frac{t}{2}a+\frac{2-t}{2}mb )}\,dt \\& \qquad {}+m \int _{0}^{1}t^{\alpha -1}E_{\mu ,\alpha ,\jmath }^{\gamma , \delta ,\kappa ,c} \bigl(wt^{\mu };p\bigr) e^{f (\frac{2-t}{2m}a+\frac{t}{2}b )}\,dt \\& \quad \leq \int _{0}^{1}t^{\alpha }E_{\mu ,\alpha ,\jmath }^{\gamma , \delta ,\kappa ,c} \bigl(wt^{\mu };p\bigr)\,dt\frac{t}{2} \bigl(e^{f(a)}-m^{2}e ^{\frac{a}{m^{2}}} \bigr)\,dt \\& \qquad {}+m \int _{0}^{1}t^{\alpha -1}E_{\mu ,\alpha ,\jmath }^{\gamma , \delta ,\kappa ,c} \bigl(wt^{\mu };p\bigr) \bigl(e^{f(b)}+me^{f(\frac{a}{m^{2}})} \bigr)\,dt. \end{aligned}$$
(2.17)

By changing of the variables \(u=\frac{t}{2}a+\frac{2-t}{2}mb\) and \(v=\frac{2-t}{2m}a+\frac{t}{2}b\) in (2.17), we get

$$\begin{aligned}& \int _{\frac{a+mb}{2}}^{mb}(mb-u)^{\alpha -1} E_{\mu ,\alpha , \jmath }^{\gamma ,\delta ,\kappa ,c}\bigl(w(mb-u)^{\mu };p \bigr)e^{f(u)}\,du \\& \qquad {}+m \int _{\frac{a}{m}}^{\frac{a+mb}{2m}} \biggl(v-\frac{a}{m} \biggr) ^{\alpha -1}E_{\mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \biggl(w \biggl(v- \frac{a}{m} \biggr)^{\mu };p\biggr)e^{f(v)}\,dv \\& \quad \leq \frac{1}{2} \bigl(e^{f(a)}-m^{2}e^{\frac{a}{m^{2}}} \bigr) \int _{\frac{a+mb}{2m}}^{mb}(mb-u)^{\alpha } E_{\mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c}\bigl(w(mb-u)^{\mu };p\bigr)\,dt \frac{t}{2}\,du \\& \qquad {}+m^{\alpha +1} \int _{\frac{a}{m}}^{\frac{a+mb}{2m}} \biggl(v- \frac{a}{m} \biggr)^{\alpha -1} E_{\mu ,\alpha ,\jmath }^{\gamma ,\delta ,\kappa ,c} \biggl(m^{\mu }w \biggl(v-\frac{a}{m} \biggr)^{\mu };p\biggr) )\,dv . \end{aligned}$$
(2.18)

By using (2.14), (2.17) and (2.18), we obtain the second inequality of (2.12). □

Let \(m=1\). Then (2.12) leads to the Hermite–Hadamard type inequality for exponentially convex function.

Corollary 2.17

Let \(0\leq a< b\)and \(f:[a,b]\rightarrow \mathbb{R}\)be an exponentially convex function such that \(f\in L_{1}[a,b]\). Then the inequalities for extended generalized fractional integral operators

$$\begin{aligned}& 2e^{f (\frac{a+b}{2} )}\zeta _{{\alpha ,(\frac{a+b}{2})}^{+}}(b;p) \\& \quad \leq \bigl( \varepsilon _{\mu ,\alpha ,\jmath ,w^{\prime },(\frac{a+b}{2})^{+}} ^{\gamma ,\delta ,\kappa ,c} e^{f} \bigr) (b;p) + \bigl( \varepsilon _{\mu ,\alpha ,\jmath ,w^{\prime },(\frac{a+b}{2})^{+}} ^{\gamma ,\delta ,\kappa ,c}e^{f} \bigr) (a;p ) \\& \quad \leq \frac{e^{f(a)}+e^{f(b)}}{2}\zeta _{\alpha ,(\frac{a+b}{2})^{-}}(a;p) \end{aligned}$$

hold, where \(w^{\prime }=\frac{2^{\mu }w}{(mb-u)^{\mu }}\).

Corollary 2.18

If we set \(p=0\), then we have the following inequalities:

$$\begin{aligned}& 2e^{f (\frac{a+mb}{2} )} \bigl( \varepsilon _{\mu ,\alpha ,\jmath ,w^{\prime },(\frac{a+mb}{2})^{+}} ^{\gamma ,\delta ,\kappa }1 \bigr) (mb) \\& \quad \leq \bigl( \varepsilon _{\mu ,\alpha ,\jmath ,w^{\prime },(\frac{a+mb}{2})^{+}} ^{\gamma ,\delta ,\kappa }e^{f} \bigr) (mb) +m^{\alpha +1} \bigl( \varepsilon _{\mu ,\alpha ,\jmath ,w^{\prime },(\frac{a+mb}{2m})^{-}} ^{\gamma ,\delta ,\kappa }e^{f} \bigr) \biggl(\frac{a}{m} \biggr) \\& \quad \leq \frac{1}{mb-a} \bigl(e^{f(a)}-me^{f(\frac{a}{m^{2}})} \bigr) \bigl( \varepsilon _{\mu ,\alpha +1,\jmath ,w^{\prime }, (\frac{a+mb}{2})^{+}} ^{\gamma ,\delta ,\kappa }1 \bigr) (mb) \\& \qquad {}+m^{\alpha +1} \bigl(e^{f(b)}+me^{f(\frac{a}{m^{2}})} \bigr) \bigl( \varepsilon _{\mu ,\alpha ,\jmath ,w^{\prime }, (\frac{a+mb}{2m})^{+}} ^{\gamma ,\delta ,\kappa }1 \bigr) \biggl( \frac{a}{m} \biggr), \end{aligned}$$

where \(w^{\prime }=\frac{2^{\mu }w}{(mb-u)^{\mu }}\).

Conclusion

We have investigated more general fractional integral inequalities. By selecting specific values of parameters quite interesting results can be obtained. The idea can be extended for more diversified classes for convex and exponentially convex functions.

References

  1. 1.

    Huang, C.-X., Yang, Z.-C., Yi, T.-S., Zou, X.-F.: On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities. J. Differ. Equ. 256(7), 2101–2114 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    Tang, W.-S., Sun, Y.-J.: Construction of Runge–Kutta type methods for solving ordinary differential equations. Appl. Math. Comput. 234, 179–191 (2014)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Huang, C.-X., Guo, S., Liu, L.-Z.: Boundedness on Morrey space for Toeplitz type operator associated to singular integral operator with variable Calderón–Zygmund kernel. J. Math. Inequal. 8(3), 453–464 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Xie, D.-X., Li, J.: A new analysis of electrostatic free energy minimization and Poisson–Boltzmann equation for protein in ionic solvent. Nonlinear Anal., Real World Appl. 21, 185–196 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Dai, Z.-F., Chen, X.-H., Wen, F.-H.: A modified Perry’s conjugate gradient method-based derivative-free method for solving large-scale nonlinear monotone equations. Appl. Math. Comput. 270, 378–386 (2015)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Tan, Y.-X., Jing, K.: Existence and global exponential stability of almost periodic solution for delayed competitive neural networks with discontinuous activations. Math. Methods Appl. Sci. 39(11), 2821–2839 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    Duan, L., Huang, C.-X.: Existence and global attractivity of almost periodic solutions for a delayed differential neoclassical growth model. Math. Methods Appl. Sci. 40(3), 814–822 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    Duan, L., Huang, L.-H., Guo, Z.-Y., Fang, X.-W.: Periodic attractor for reaction-diffusion high-order Hopfield neural networks with time-varying delays. Comput. Math. Appl. 73(2), 233–245 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    Huang, C.-X., Liu, L.-Z.: Boundedness of multilinear singular integral operator with a non-smooth kernel and mean oscillation. Quaest. Math. 40(3), 295–312 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    Cai, Z.-W., Huang, J.-H., Huang, L.-H.: Generalized Lyapunov–Razumikhin method for retarded differential inclusions: applications to discontinuous neural networks. Discrete Contin. Dyn. Syst. 22B(9), 3591–3614 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Hu, H.-J., Zou, X.-F.: Existence of an extinction wave in the Fisher equation with a shifting habitat. Proc. Am. Math. Soc. 145(11), 4763–4771 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Tan, Y.-X., Huang, C.-X., Sun, B., Wang, T.: Dynamics of a class of delayed reaction-diffusion systems with Neumann boundary condition. J. Math. Anal. Appl. 458(2), 1115–1130 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    Tang, W.-S., Zhang, J.-J.: Symplecticity-preserving continuous-stage Runge–Kutta–Nyström methods. Appl. Math. Comput. 323, 204–219 (2018)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Duan, L., Fang, X.-W., Huang, C.-X.: Global exponential convergence in a delayed almost periodic Nicholson’s blowflies model with discontinuous harvesting. Math. Methods Appl. Sci. 41(5), 1954–1965 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    Liu, Z.-Y., Wu, N.-C., Qin, X.-R., Zhang, Y.-L.: Trigonometric transform splitting methods for real symmetric Toeplitz systems. Comput. Math. Appl. 75(8), 2782–2794 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Huang, C.-X., Qiao, Y.-C., Huang, L.-H., Agarwal, R.P.: Dynamical behaviors of a food-chain model with stage structure and time delays. Adv. Differ. Equ. 2018, Article ID 186 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    Cai, Z.-W., Huang, J.-H., Huang, L.-H.: Periodic orbit analysis for the delayed Filippov system. Proc. Am. Math. Soc. 146(11), 4667–4682 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Wang, J.-F., Chen, X.-Y., Huang, L.-H.: The number and stability of limit cycles for planar piecewise linear systems of node-saddle type. J. Math. Anal. Appl. 469(1), 405–427 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Li, J., Ying, J.-Y., Xie, D.-X.: On the analysis and application of an ion size-modified Poisson–Boltzmann equation. Nonlinear Anal., Real World Appl. 47, 188–203 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Wang, J.-F., Huang, C.-X., Huang, L.-H.: Discontinuity-induced limit cycles in a general planar piecewise linear system of saddle-focus type. Nonlinear Anal. Hybrid Syst. 33, 162–178 (2019)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Jiang, Y.-J., Xu, X.-J.: A monotone finite volume method for time fractional Fokker–Planck equations. Sci. China Math. 62(4), 783–794 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  22. 22.

    Peng, J., Zhang, Y.: Heron triangles with figurate number sides. Acta Math. Hung. 157(2), 478–488 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  23. 23.

    Tian, Z.-L., Liu, Y., Zhang, Y., Liu, Z.-Y., Tian, M.-Y.: The general inner-outer iteration method based on regular splittings for the PageRank problem. Appl. Math. Comput. 356, 479–501 (2019)

    MathSciNet  Google Scholar 

  24. 24.

    Wang, M.-K., Chu, H.-H., Chu, Y.-M.: Precise bounds for the weighted Hölder mean of the complete p-elliptic integrals. J. Math. Anal. Appl. https://doi.org/10.1016/j.jmaa.2019.123388

    MathSciNet  MATH  Article  Google Scholar 

  25. 25.

    Chu, Y.-M., Wang, M.-K., Qiu, S.-L.: Optimal combinations bounds of root-square and arithmetic means for Toader mean. Proc. Indian Acad. Sci. Math. Sci. 122(1), 41–51 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    Xu, H.-Z., Chu, Y.-M., Qian, W.-M.: Sharp bounds for the Sándor–Yang means in terms of arithmetic and contra-harmonic means. J. Inequal. Appl. 2018, Article ID 127 (2018)

    Article  Google Scholar 

  27. 27.

    Wu, S.-H., Chu, Y.-M.: Schur m-power convexity of generalized geometric Bonferroni mean involving three parameters. J. Inequal. Appl. 2019, Article ID 57 (2019)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Wang, J.-L., Qian, W.-M., He, Z.-Y., Chu, Y.-M.: On approximating the Toader mean by other bivariate means. J. Funct. Spaces 2019, Article ID 6082413 (2019)

    MathSciNet  MATH  Google Scholar 

  29. 29.

    Qian, W.-M., Xu, H.-Z., Chu, Y.-M.: Improvements of bounds for the Sándor–Yang means. J. Inequal. Appl. 2019, Article ID 73 (2019)

    Article  Google Scholar 

  30. 30.

    Qian, W.-M., He, Z.-Y., Zhang, H.-W., Chu, Y.-M.: Sharp bounds for Neuman means in terms of two-parameter contraharmonic and arithmetic mean. J. Inequal. Appl. 2019, Article ID 168 (2019)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Yang, Z.-H., Qian, W.-M., Chu, Y.-M., Zhang, W.: Monotonicity rule for the quotient of two functions and its application. J. Inequal. Appl. 2017, Article ID 106 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  32. 32.

    Khan, M.A., Chu, Y.-M., 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 

  33. 33.

    Khan, M.A., Begum, S., Khurshid, Y., Chu, Y.-M.: Ostrowski type inequalities involving conformable fractional integrals. J. Inequal. Appl. 2018, Article ID 70 (2018)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Khan, M.A., Chu, Y.-M., Kashuri, A., Liko, R., Ali, G.: Conformable fractional integrals versions of Hermite–Hadamard inequalities and their generalizations. J. Funct. Spaces 2018, Article ID 6928130 (2018)

    MathSciNet  MATH  Google Scholar 

  35. 35.

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

    MathSciNet  MATH  Google Scholar 

  36. 36.

    Khan, M.A., 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 

  37. 37.

    Huang, T.-R., Tan, S.-Y., Ma, X.-Y., Chu, Y.-M.: Monotonicity properties and bounds for the complete p-elliptic integrals. J. Inequal. Appl. 2018, Article ID 239 (2018)

    MathSciNet  Article  Google Scholar 

  38. 38.

    Khurshid, Y., Khan, M.A., Chu, Y.-M.: Conformable integral inequalities of the Hermite–Hadamard type in terms of GG- and GA-convexities. J. Funct. Spaces 2019, Article ID 6926107 (2019)

    MathSciNet  MATH  Google Scholar 

  39. 39.

    Khan, M.A., Wu, S.-H., Ullah, H., Chu, Y.-M.: Discrete majorization type inequalities for convex functions on rectangles. J. Inequal. Appl. 2019, Article ID 16 (2019)

    MathSciNet  Article  Google Scholar 

  40. 40.

    Wang, M.-K., Zhang, W., Chu, Y.-M.: Monotonicity, convexity and inequalities involving the generalized elliptic integrals. Acta Math. Sci. 39B(5), 1440–1450 (2019)

    Google Scholar 

  41. 41.

    Wang, W.-S., Chen, Y.-Z., Fang, H.: On the variable two-step IMEX BDF method for parabolic integro-differential equations with nonsmooth initial data arising in finance. SIAM J. Numer. Anal. 57(3), 1289–1317 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  42. 42.

    Shi, H.-P., Zhang, H.-Q.: Existence of gap solitons in periodic discrete nonlinear Schrödinger equations. J. Math. Anal. Appl. 361(2), 411–419 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  43. 43.

    Zhou, W.-J., Zhang, L.: Global convergence of a regularized factorized quasi-Newton method for nonlinear least squares problems. Comput. Appl. Math. 29(2), 195–214 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  44. 44.

    Li, J., Xu, Y.-J.: An inverse coefficient problem with nonlinear parabolic equation. J. Appl. Math. Comput. 134(1–2), 195–206 (2010)

    MathSciNet  MATH  Google Scholar 

  45. 45.

    Yang, X.-S., Zhu, Q.-X., Huang, C.-X.: Generalized lag-synchronization of chaotic mix-delayed systems with uncertain parameters and unknown perturbations. Nonlinear Anal., Real World Appl. 12(1), 93–105 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  46. 46.

    Zhu, Q.-X., Huang, C.-X., Yang, X.-S.: Exponential stability for stochastic jumping BAM neural networks with time-varying and distributed delays. Nonlinear Anal. Hybrid Syst. 5(1), 52–77 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  47. 47.

    Dai, Z.-F., Wen, F.-H.: A modified CG-DESCENT method for unconstrained optimization. J. Comput. Appl. Math. 235(11), 3332–3341 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  48. 48.

    Guo, K., Sun, B.: Numerical solution of the Goursat problem on a triangular domain with mixed boundary conditions. Appl. Math. Comput. 217(21), 8765–8777 (2011)

    MathSciNet  MATH  Google Scholar 

  49. 49.

    Lin, L., Liu, Z.-Y.: An alternating projected gradient algorithm for nonnegative matrix factorization. Appl. Math. Comput. 217(24), 9997–10002 (2011)

    MathSciNet  MATH  Google Scholar 

  50. 50.

    Xiao, C.-E., Liu, J.-B., Liu, Y.-L.: An inverse pollution problem in porous media. Appl. Math. Comput. 218(7), 3649–3653 (2011)

    MathSciNet  MATH  Google Scholar 

  51. 51.

    Dai, Z.-F., Wen, F.-H.: Another improved Wei–Yao–Liu nonlinear conjugate gradient method with sufficient descent property. Appl. Math. Comput. 218(14), 7421–7430 (2012)

    MathSciNet  MATH  Google Scholar 

  52. 52.

    Liu, Z.-Y., Zhang, Y.-L., Santos, J., Ralha, R.: On computing complex square roots of real matrices. Appl. Math. Lett. 25(10), 1565–1568 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  53. 53.

    Huang, C.-X., Liu, L.-Z.: Sharp function inequalities and boundness for Toeplitz type operator related to general fractional singular integral operator. Publ. Inst. Math. 92(106), 165–176 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  54. 54.

    Zhou, W.-J.: On the convergence of the modified Levenberg–Marquardt method with a nonmonotone second order Armijo type line search. J. Comput. Appl. Math. 239, 152–161 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  55. 55.

    Zhou, W.-J., Chen, X.-L.: On the convergence of a modified regularized Newton method for convex optimization with singular solutions. J. Comput. Appl. Math. 239, 179–188 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  56. 56.

    Jiang, Y.-J., Ma, J.-T.: Spectral collocation methods for Volterra-integro differential equations with noncompact kernels. J. Comput. Appl. Math. 244, 115–124 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  57. 57.

    Zhang, L., Jian, S.-Y.: Further studies on the Wei–Yao–Liu nonlinear conjugate gradient method. Appl. Math. Comput. 219(14), 7616–7621 (2013)

    MathSciNet  MATH  Google Scholar 

  58. 58.

    Li, X.-F., Tang, G.-J., Tang, B.-Q.: Stress field around a strike-slip fault in orthotropic elastic layers via a hypersingular integral equation. Comput. Math. Appl. 66(11), 2317–2326 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  59. 59.

    Qin, G.-X., Huang, C.-X., Xie, Y.-Q., Wen, F.-H.: Asymptotic behavior for third-order quasi-linear differential equations. Adv. Differ. Equ. 2013, Article ID 305 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  60. 60.

    Zhang, L., Li, J.-L.: A new globalization technique for nonlinear conjugate gradient methods for nonconvex minimization. Appl. Math. Comput. 217(24), 10295–10304 (2011)

    MathSciNet  MATH  Google Scholar 

  61. 61.

    Huang, C.-X., Zhang, H., Huang, L.-H.: Almost periodicity analysis for a delayed Nicholson’s blowflies model with nonlinear density-dependent mortality term. Commun. Pure Appl. Anal. 18(6), 3337–3349 (2019)

    MathSciNet  Article  Google Scholar 

  62. 62.

    Liu, F.-W., Feng, L.-B., Anh, V., Li, J.: Unstructured-mesh Galerkin finite element method for the two-dimensional multi-term time-space fractional Bloc–Torrey equations on irregular convex domains. Comput. Math. Appl. 78(5), 1637–1650 (2019)

    MathSciNet  Article  Google Scholar 

  63. 63.

    Antczak, T.: \((p,r)\)-invex sets and functions. J. Math. Anal. Appl. 263(2), 355–379 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  64. 64.

    Dragomir, S.S., Gomm, I.: Some Hermite–Hadamard type inequalities for functions whose exponentials are convex. Stud. Univ. Babeş–Bolyai, Math. 60(4), 527–534 (2015)

    MathSciNet  MATH  Google Scholar 

  65. 65.

    Alirezaei, G., Mathar, R.: On exponentially concave functions and their impact in information theory. J. Inf. Theory Appl. 9(5), 265–274 (2018)

    Google Scholar 

  66. 66.

    Awan, M.U., Noor, M.A., Noor, K.I.: Hermite–Hadamard inequalities for exponential convex functions. Appl. Math. Inf. Sci. 12(2), 405–409 (2018)

    MathSciNet  Article  Google Scholar 

  67. 67.

    Jakšetić, J., Pečarić, J.: Exponential convexity, Euler–Radau expansions and Stolarsky means. Rad Hrvat. Akad. Znan. Umjet. Mat. Znan. 17(515), 81–94 (2013)

    MathSciNet  MATH  Google Scholar 

  68. 68.

    Toader, G.: The order of starlike convex function. Bull. Appl. Comp. Math. 85, 347–350 (1998)

    Google Scholar 

  69. 69.

    Yang, Z.-H., Qian, W.-M., Chu, Y.-M., Zhang, W.: On rational bounds for the gamma function. J. Inequal. Appl. 2017, Article ID 210 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  70. 70.

    Huang, T.-R., Han, B.-W., Ma, X.-Y., Chu, Y.-M.: Optimal bounds for the generalized Euler–Mascheroni constant. J. Inequal. Appl. 2018, Article ID 118 (2018)

    MathSciNet  Article  Google Scholar 

  71. 71.

    Yang, Z.-H., Chu, Y.-M., Zhang, W.: High accuracy asymptotic bounds for the complete elliptic integral of the second kind. Appl. Math. Comput. 348, 552–564 (2019)

    MathSciNet  Article  Google Scholar 

  72. 72.

    He, X.-H., Qian, W.-M., Xu, H.-Z., Chu, Y.-M.: Sharp power mean bounds for two Sándor–Yang means. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 113(3), 2627–2638 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  73. 73.

    Wang, M.-K., Chu, Y.-M., Zhang, W.: Monotonicity and inequalities involving zero-balanced hypergeometric function. Math. Inequal. Appl. 22(2), 601–617 (2019)

    MathSciNet  MATH  Google Scholar 

  74. 74.

    Wang, M.-K., Chu, Y.-M., Zhang, W.: Precise estimates for the solution of Ramanujan’s generalized modular equation. Ramanujan J. 49(3), 653–668 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  75. 75.

    Andrić, M., Farid, G., Pečarić, J.: A generalization of Mittag-Leffer function associated with Opial type inequalities due to Mitrinovic and Pecaric. Preprint (2019)

  76. 76.

    Kilbas, A.A., Saigo, M., Saxena, R.K.: Generalized Mittag-Leffler function and generalized fractional calculus operators. Integral Transforms Spec. Funct. 15(1), 31–49 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  77. 77.

    Mihai, M.V., Awan, M.U., Noor, M.A., Noor, K.I.: Fractional Hermite–Hadamard inequalities containing generalized Mittag-Leffler function. J. Inequal. Appl. 2017, Article ID 265 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  78. 78.

    Nieto, J.J.: Maximum principles for fractional differential equations derived from Mittag-Leffler functions. Appl. Math. Lett. 23(10), 1248–1251 (2010)

    MathSciNet  MATH  Article  Google Scholar 

Download references

Acknowledgements

The authors would like to express their sincere thanks to the editor and the anonymous reviewers for their helpful comments and suggestions.

Availability of data and materials

Not applicable.

Funding

Not applicable.

Author information

Affiliations

Authors

Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Corresponding authors

Correspondence to Saima Rashid or Ahmet Ocak Akdemir.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Rashid, S., Safdar, F., Akdemir, A.O. et al. Some new fractional integral inequalities for exponentially m-convex functions via extended generalized Mittag-Leffler function. J Inequal Appl 2019, 299 (2019). https://doi.org/10.1186/s13660-019-2248-7

Download citation

MSC

  • 26A51
  • 26D10
  • 26D15

Keywords

  • Convex function
  • Exponentially convex function
  • m-convex function
  • Mittag-Leffler function
  • Generalized fractional integral operator
  • Hermite–Hadamard inequality