Modification of certain fractional integral inequalities for convex functions

Abstract

We consider the modified Hermite–Hadamard inequality and related results on integral inequalities, in the context of fractional calculus using the Riemann–Liouville fractional integrals. Our results generalize and modify some existing results. Finally, some applications to special means of real numbers are given. Moreover, some error estimates for the midpoint formula are pointed out.

1 Introduction

The generalization of certain integral inequalities to the fractional scope, in both continuous and discrete versions, have attracted many researchers in the recent few years and before [1, 19, 20]. In this article, our work is devoted to Hadamard–Hermite type for convex functions in the framework of Riemann–Liouville fractional type integrals.

A function $g:\mathcal{I}\subseteq \mathbb{R}\to \mathbb{R}$ is said to be convex on the interval $\mathcal{I}$, if the inequality

$$\begin{aligned} g\bigl(\ell x+(1-\ell )y\bigr)\leq \ell g(x)+(1-\ell )g(y) \end{aligned}$$

(1)

holds for all $x,y\in \mathcal{I}$ and $\ell \in [0,1]$. We say that g is concave if −g is convex.

For convex functions (1), many equalities and inequalities have been established by many authors; such as the Hardy type inequality [3], Ostrowski type inequality [7], Olsen type inequality [8], Gagliardo–Nirenberg type inequality [22], midpoint type inequality [10] and trapezoidal type inequality [14]. But the most important inequality is the Hermite–Hadamard type inequality [6], which is defined by

$$\begin{aligned} g \biggl(\frac{u+v}{2} \biggr) &\leq \frac{1}{v-u} \int _{u}^{v} g(x)\,dx \leq \frac{g(u)+g(v)}{2}, \end{aligned}$$

(2)

where $g:\mathcal{I}\subseteq \mathbb{R}\to \mathbb{R}$ is assumed to be a convex function on $\mathcal{I}$ where $u,v\in \mathcal{I}$ with $u< v$.

A number of mathematicians in the field of applied and pure mathematics have devoted their efforts to generalizing, refining, finding counterparts of, and extending the Hermite–Hadamard inequality (2) for different classes of convex functions and mappings. For more recent results obtained in view of inequality (2), we refer the reader to [2, 4, 6, 13, 16, 18].

In [21], Sarikaya et al. obtained the Hermite–Hadamard inequalities in fractional integral form:

$$\begin{aligned} g \biggl(\frac{u+v}{2} \biggr) &\leq \frac{\varGamma (\vartheta +1)}{2(v-u)^{ \vartheta }} \bigl[\mathfrak{J}^{\vartheta }_{u^{+}}g(v)+ \mathfrak{J}^{\vartheta }_{v^{-}}g(u) \bigr]\leq \frac{g(u)+g(v)}{2}, \end{aligned}$$

(3)

where $g:[u,v]\subseteq \mathbb{R}\to \mathbb{R}$ is assumed to be a positive convex function on $[u,v]$, $g\in L_{1}[u,v]$ with $u< v$, and $\mathfrak{J}^{\vartheta }_{u^{+}}$ and $\mathfrak{J}^{\vartheta } _{v^{-}}$ are the left-sided and right-sided Riemann–Liouville fractional integrals of order $\vartheta >0$, which, respectively, are defined by [9]

$$\begin{aligned}& \mathfrak{J}^{\vartheta }_{u^{+}}g(x)=\frac{1}{\varGamma (\vartheta )} \int _{u}^{x} (x-\ell )^{\vartheta -1}g(t)\,d\ell , \quad x>u, \\& \mathfrak{J}^{\vartheta }_{v^{-}}g(x)=\frac{1}{\varGamma (\vartheta )} \int _{x}^{v} (\ell -x)^{\vartheta -1}g(t)\,d\ell , \quad x< v. \end{aligned}$$

It is clear that inequality (3) is a generalization of Hermite–Hadamard inequality (2). If we take $\vartheta =1$ in (3) we obtain (2). Many inequalities have been established in view of inequality (3); for more details see [5, 10, 11, 14, 15, 21, 23].

Recently, in [12], Mehrez and Agarwal obtained a new modification of the Hermite–Hadamard inequality (2); this is given by

$$\begin{aligned} g \biggl(\frac{u+v}{2} \biggr) &\leq \frac{1}{v-u} \int _{u}^{v} g(x)\,dx \leq \frac{g (\frac{3v-u}{2} )+2g (\frac{u+v}{2} )+g (\frac{3u-v}{2} )}{4}. \end{aligned}$$

(4)

Furthermore, Mehrez and Agarwal obtained many inequalities in view of inequalities (4); for which we refer the reader to their interesting paper [12].

The aim of this paper is to establish new inequalities of Hermite–Hadamard type for convex functions via Riemann–Liouville fractional integrals.

2 Preliminary lemmas

In order to obtain our main results, we need some qualities which are stated in the following lemmas.

Lemma 1

([23])

Let$g:[u,v]\subseteq \mathbb{R}\to \mathbb{R}$be a differentiable mapping on$(u,v)$with$u< v$. If$g'\in L_{1}[u,v]$, then we have

$$\begin{aligned} &\frac{\varGamma (\vartheta +1)}{2(v-u)^{\vartheta }} \bigl[\mathfrak{J} _{u^{+}}^{\vartheta }g(v)+ \mathfrak{J}_{v^{-}}^{\vartheta }g(u) \bigr]-g \biggl( \frac{u+v}{2} \biggr) \\ &\quad =\frac{v-u}{2} \biggl[ \int _{0}^{1} \kappa g' \bigl( \ell u+(1- \ell )v \bigr)\,d\ell - \int _{0}^{1} \bigl[(1-\ell )^{\vartheta }- \ell ^{\vartheta } \bigr]g' \bigl(\ell u+(1-\ell )v \bigr)\,d\ell \biggr], \end{aligned}$$

(5)

where

$$\begin{aligned} \kappa &= \textstyle\begin{cases} 1 & 0\leq \ell < \frac{1}{2}, \\ -1 & \frac{1}{2}\leq \ell < 1. \end{cases}\displaystyle \end{aligned}$$

Lemma 2

([5])

Let$g:\mathcal{I}\subseteq \mathbb{R}\to \mathbb{R}$be a twice differentiable function on$\mathcal{I}^{o}$ (the interior of$\mathcal{I}$). Assume that$u,v\in \mathcal{I}^{o}$with$u< v$. If$g''\in L_{1}[u,v]$, then for$\vartheta >0$we have

$$\begin{aligned} &\frac{g(u)+g(v)}{2}- \frac{\varGamma (\vartheta +1)}{2(v-u)^{\vartheta }} \bigl[ \mathfrak{J}_{u^{+}}^{\vartheta }g(v)+\mathfrak{J}_{v^{-}} ^{\vartheta }g(u) \bigr] \\ &\quad =\frac{(v-u)^{2}}{2(\vartheta +1)} \int _{0}^{1}\ell \bigl(1- \ell ^{\vartheta } \bigr) \bigl[g'' \bigl(\ell u+(1-\ell )v \bigr)+g'' \bigl((1-\ell )u+\ell v \bigr) \bigr]\,d\ell . \end{aligned}$$

(6)

Lemma 3

Let$g:\mathcal{I}\subseteq \mathbb{R}\to \mathbb{R}$be a differentiable function on$\mathcal{I}^{o}$and$g\in L_{1}[u,v]$. Ifgis a convex function on$[u,v]$. then for$\vartheta >0$we have

$$\begin{aligned} &g \biggl(\frac{u+v}{2} \biggr) \leq \frac{\varGamma (\vartheta +1)}{2(v-u)^{ \vartheta }} \bigl[\mathfrak{J}_{u^{+}}^{\vartheta }g(v)+ \mathfrak{J}_{v^{-}}^{\vartheta }g(u) \bigr] \leq \frac{g (\frac{3v-u}{2} )+2g (\frac{u+v}{2} )+g (\frac{3u-v}{2} )}{4} \end{aligned}$$

(7)

and

$$\begin{aligned} & \biggl\vert \frac{\varGamma (\vartheta +1)}{2(v-u)^{\vartheta }} \bigl[\mathfrak{J} _{u^{+}}^{\vartheta }g(v)+\mathfrak{J}_{v^{-}}^{\vartheta }g(u) \bigr] -\frac{1}{2}g \biggl(\frac{u+v}{2} \biggr) \biggr\vert \leq \frac{g (\frac{3v-u}{2} )+g (\frac{3u-v}{2} )}{4}. \end{aligned}$$

(8)

Proof

From the definition of Riemann–Liouville fractional integral, we have

$$\begin{aligned} &\frac{\varGamma (\vartheta +1)}{2(v-u)^{\vartheta }} \bigl[\mathfrak{J} _{u^{+}}^{\vartheta }g(v)+ \mathfrak{J}_{v^{-}}^{\vartheta }g(u) \bigr] \\ &\quad =\frac{\vartheta }{2(v-u)^{\vartheta }} \biggl[ \int _{u}^{v}(v-x)^{ \vartheta -1}g(x)\,dx+ \int _{u}^{v}(x-u)^{\vartheta -1} g(x)\,dx \biggr]. \end{aligned}$$

By using the change of the variable $x=\frac{3}{4}\ell +\frac{u+v}{4}$ for $\ell \in [\frac{3u-v}{3},\frac{3v-u}{3} ]$, we obtain

$$\begin{aligned} &\frac{\varGamma (\vartheta +1)}{2(v-u)^{\vartheta }} \bigl[\mathfrak{J} _{u^{+}}^{\vartheta }g(v) +\mathfrak{J}_{v^{-}}^{\vartheta }g(u) \bigr] \\ &\quad =\frac{3\vartheta }{8(v-u)^{\vartheta }} \biggl[ \int _{\frac{3u-v}{3}} ^{\frac{3v-u}{3}} \biggl(\frac{3v-u}{4}- \frac{3}{4}\ell \biggr)^{ \vartheta -1}g \biggl(\frac{3}{4} \ell +\frac{u+v}{4} \biggr) \,d\ell \\ &\qquad {}+ \int _{\frac{3u-v}{3}}^{\frac{3v-u}{3}} \biggl(\frac{3}{4}\ell - \frac{3u-v}{4} \biggr)^{\vartheta -1}g \biggl(\frac{3}{4}\ell + \frac{u+v}{4} \biggr) \,d\ell \biggr]. \end{aligned}$$

(9)

Since g is convex on $[u,v]$, we have

$$\begin{aligned} g \biggl(\frac{3}{4}\ell +\frac{u+v}{4} \biggr)=g \biggl( \frac{ \frac{3}{2}\ell +\frac{u+v}{2}}{2} \biggr) \leq \frac{1}{2}g \biggl( \frac{3}{2} \ell \biggr) +\frac{1}{2}g \biggl( \frac{u+v}{2} \biggr). \end{aligned}$$

It follows from this and (9) that

$$\begin{aligned} &\frac{\varGamma (\vartheta +1)}{2(v-u)^{\vartheta }} \bigl[\mathfrak{J} _{u^{+}}^{\vartheta }g(v)+ \mathfrak{J}_{v^{-}}^{\vartheta }g(u) \bigr] \\ &\quad \leq \frac{3\vartheta }{16(v-u)^{\vartheta }} \biggl[ \int _{ \frac{3u-v}{3}}^{\frac{3v-u}{3}} \biggl(\frac{3v-u}{4}- \frac{3}{4} \ell \biggr)^{\vartheta -1} \biggl\{ g \biggl( \frac{3}{2}\ell \biggr) +g \biggl(\frac{u+v}{2} \biggr) \biggr\} \,d\ell \\ &\qquad {}+ \int _{\frac{3u-v}{3}}^{\frac{3v-u}{3}} \biggl(\frac{3}{4}\ell - \frac{3u-v}{4} \biggr)^{\vartheta -1} \biggl\{ g \biggl( \frac{3}{2} \ell \biggr) +g \biggl(\frac{u+v}{2} \biggr) \biggr\} \,d\ell \biggr] \\ &\quad =\frac{g (\frac{u+v}{2} )}{2}+\frac{3\vartheta }{16(v-u)^{ \vartheta }} \biggl[ \int _{\frac{3u-v}{3}}^{\frac{3v-u}{3}} \biggl(\frac{3v-u}{4}- \frac{3}{4}\ell \biggr)^{\vartheta -1}g \biggl(\frac{3}{2} \ell \biggr) \,d\ell \\ &\qquad {}+ \int _{\frac{3u-v}{3}}^{\frac{3v-u}{3}} \biggl(\frac{3}{4}\ell - \frac{3u-v}{4} \biggr)^{\vartheta -1}g \biggl(\frac{3}{2}\ell \biggr) \,d\ell \biggr]. \end{aligned}$$

Again, by using the change of the variable $z=\frac{3}{2}\ell$ for $\ell \in [\frac{3u-v}{2},\frac{3v-u}{2} ]$, we obtain

$$\begin{aligned} &\frac{\varGamma (\vartheta +1)}{2(v-u)^{\vartheta }} \bigl[\mathfrak{J} _{u^{+}}^{\vartheta }g(v) +\mathfrak{J}_{v^{-}}^{\vartheta }g(u) \bigr] \\ &\quad \leq \frac{g (\frac{u+v}{2} )}{2}+\frac{\vartheta }{2^{ \vartheta +2}(v-u)^{\vartheta }} \\ &\qquad {}\times \biggl[ \int _{\frac{3u-v}{2}}^{\frac{3v-u}{2}} \biggl(\frac{3u-v}{2}-z \biggr) ^{\vartheta -1}g (z ) \,dz + \int _{\frac{3u-v}{2}}^{ \frac{3v-u}{2}} \biggl(z-\frac{3v-u}{2} \biggr)^{\vartheta -1}g (z ) \,dz \biggr] \\ &\quad =\frac{g (\frac{u+v}{2} )}{2}+\frac{\varGamma (\vartheta +1)}{2^{ \vartheta +2}(v-u)^{\vartheta }} \biggl[\mathfrak{J}_{ (\frac{3u-v}{2} ) ^{+}}^{\vartheta }g \biggl(\frac{3v-u}{2} \biggr) +\mathfrak{J}_{ (\frac{3v-u}{2} ) ^{-}}^{\vartheta }g \biggl(\frac{3u-v}{2} \biggr) \biggr]. \end{aligned}$$

(10)

Replace u by $\frac{3u-v}{2}$ and v by $\frac{3v-u}{2}$ in the right-hand side of inequality (3) and multiply both sides by $\frac{1}{2}$, we get

$$\begin{aligned} \frac{\varGamma (\vartheta +1)}{2^{\vartheta +2}(v-u)^{\vartheta }} \biggl[\mathfrak{J}_{ (\frac{3u-v}{2} )^{+}}^{\vartheta }g \biggl(\frac{3v-u}{2} \biggr) +\mathfrak{J}_{ (\frac{3v-u}{2} ) ^{-}}^{\vartheta }g \biggl(\frac{3u-v}{2} \biggr) \biggr] \leq \frac{g (\frac{3v-u}{2} )+g (\frac{3u-v}{2} )}{4}. \end{aligned}$$

(11)

From (10) and (11), we obtain the desired inequality (7) and from (7) we can easily obtain the inequality (8). These complete the proof of Lemma 3. □

Remark 1

If we use $\vartheta =1$ in Lemma 3, then Lemma 3 reduces to Lemma 3 in [12]. In particular, inequalities (7) reduces to the inequalities (4).

3 Hermite–Hadamard type inequalities

Our main results start from the following theorem.

Theorem 1

Let$g:\mathcal{I}^{o}\subseteq \mathbb{R}\to \mathbb{R}$be a differentiable mapping on$\mathcal{I}^{o}$, $u,v\in \mathcal{I}^{o}$with$u< v$. Let$g'\in L_{1} [\frac{3u-v}{2},\frac{3v-u}{2} ]$and$g': [\frac{3u-v}{2},\frac{3v-u}{2} ]\to \mathbb{R}$be a continuous function on$[\frac{3u-v}{2},\frac{3v-u}{2} ]$. If$|g'|^{q}, q\geq 1$is a convex function on$[\frac{3u-v}{2}, \frac{3v-u}{2} ]$, then

$$\begin{aligned} \begin{aligned}[b] & \biggl\vert \frac{\varGamma (\vartheta +1)}{2(v-u)^{\vartheta }} \bigl[\mathfrak{J} _{u^{+}}^{\vartheta }g(v)+ \mathfrak{J}_{v^{-}}^{\vartheta }g(u) \bigr] -g \biggl( \frac{u+v}{2} \biggr) \biggr\vert \\ &\quad \leq \frac{v-u}{2} \biggl(\frac{2^{ \vartheta }-1}{2^{\vartheta }(\vartheta +1)}+\frac{1}{2} \biggr) \biggl( \biggl\vert g' \biggl(\frac{3u-v}{2} \biggr) \biggr\vert ^{q} + \biggl\vert g' \biggl( \frac{3v-u}{2} \biggr) \biggr\vert ^{q} \biggr)^{\frac{1}{q}}. \end{aligned} \end{aligned}$$

(12)

Proof

First we prove the theorem for $q=1$. By changing the variables $u\to \frac{3u-v}{2}$ and $v\to \frac{3v-u}{2}$ in Lemma 1, we get

$$\begin{aligned} &\frac{\varGamma (\vartheta +1)}{2^{\vartheta +1}(v-u)^{\vartheta }} \biggl[\mathfrak{J}_{ (\frac{3u-v}{2} )^{+}}^{\vartheta } g \biggl(\frac{3v-u}{2} \biggr)+\mathfrak{J}_{ (\frac{3v-u}{2} ) ^{-}}^{\vartheta } g \biggl(\frac{3u-v}{2} \biggr) \biggr] \\ &\quad =g \biggl(\frac{u+v}{2} \biggr) +(v-u) \biggl[ \int _{0}^{1} \kappa g' \biggl( \frac{3v-u}{2}+2(v-u) \ell \biggr)\,d\ell \\ &\qquad {}- \int _{0}^{1} \bigl[(1-\ell )^{\vartheta }- \ell ^{\vartheta } \bigr] g' \biggl(\frac{3v-u}{2}+2(v-u) \ell \biggr)\,d\ell \biggr] \\ &\quad =g \biggl(\frac{u+v}{2} \biggr)+(v-u) \biggl[ \int _{0}^{\frac{1}{2}} g' \biggl( \frac{3v-u}{2}+2(v-u)\ell \biggr)\,d\ell \\ &\qquad {} - \int _{\frac{1}{2}} ^{1} g' \biggl( \frac{3v-u}{2}+2(v-u)\ell \biggr)\,d\ell \\ &\qquad {}+ \int _{0}^{1} \bigl[\ell ^{\vartheta }-(1- \ell )^{\vartheta } \bigr] g' \biggl(\frac{3v-u}{2}+2(v-u) \ell \biggr)\,d\ell \biggr]. \end{aligned}$$

(13)

From (10) and (13), we find

$$\begin{aligned} & \biggl\vert \frac{\varGamma (\vartheta +1)}{2(v-u)^{\vartheta }} \bigl[\mathfrak{J} _{u^{+}}^{\vartheta }g(v)+\mathfrak{J}_{v^{-}}^{\vartheta }g(u) \bigr] -g \biggl(\frac{u+v}{2} \biggr) \biggr\vert \\ &\quad \leq \frac{v-u}{2} \biggl[ \int _{0}^{\frac{1}{2}} \bigl[(1-\ell )^{ \vartheta }- \ell ^{\vartheta }+1 \bigr] \biggl\vert g' \biggl( \frac{3v-u}{2}+2(v-u) \ell \biggr) \biggr\vert \,d\ell \\ &\qquad {}+ \int _{\frac{1}{2}}^{1} \bigl[\ell ^{\vartheta }-(1- \ell )^{\vartheta }+1 \bigr] \biggl\vert g' \biggl( \frac{3v-u}{2}+2(v-u)\ell \biggr) \biggr\vert \,d\ell \biggr]. \end{aligned}$$

(14)

Using the convexity of $|g'|$ on $[\frac{3u-v}{2}, \frac{3v-u}{2} ]$, we obtain

$$\begin{aligned} & \int _{0}^{\frac{1}{2}} \bigl[(1-\ell )^{\vartheta }- \ell ^{\vartheta }+1 \bigr] \biggl\vert g' \biggl( \frac{3v-u}{2}+2(v-u)\ell \biggr) \biggr\vert \,d\ell \\ &\quad = \int _{0}^{\frac{1}{2}} \bigl[(1-\ell )^{\vartheta }- \ell ^{\vartheta }+1 \bigr] \biggl\vert g' \biggl( \frac{3u-v}{2}\ell +\frac{3v-u}{2}(1- \ell ) \biggr) \biggr\vert \,d\ell \\ &\quad \leq \int _{0}^{\frac{1}{2}} \bigl[(1-\ell )^{\vartheta }- \ell ^{\vartheta }+1 \bigr] \biggl\{ \ell \biggl\vert g' \biggl(\frac{3u-v}{2} \biggr) \biggr\vert +(1- \ell ) \biggl\vert g' \biggl(\frac{3v-u}{2} \biggr) \biggr\vert \biggr\} \,d\ell \\ &\quad = \biggl(\frac{1}{8}+\frac{1}{(\vartheta +1)(\vartheta +2)}-\frac{1}{( \vartheta +1)2^{\vartheta +1}} \biggr) \biggl\vert g' \biggl(\frac{3u-v}{2} \biggr) \biggr\vert \\ &\qquad {}+ \biggl(\frac{3}{8}+\frac{1}{\vartheta +2}-\frac{1}{(\vartheta +1)2^{ \vartheta +1}} \biggr) \biggl\vert g' \biggl(\frac{3v-u}{2} \biggr) \biggr\vert . \end{aligned}$$

(15)

Analogously, we obtain

$$\begin{aligned} & \int _{\frac{1}{2}}^{1} \bigl[\ell ^{\vartheta }-(1- \ell )^{\vartheta }+1 \bigr] \biggl\vert g' \biggl( \frac{3v-u}{2}+2(v-u)\ell \biggr) \biggr\vert \,d\ell \\ &\quad \leq \biggl(\frac{3}{8}+\frac{1}{\vartheta +2}-\frac{1}{(\vartheta +1)2^{\vartheta +1}} \biggr) \biggl\vert g' \biggl(\frac{3u-v}{2} \biggr) \biggr\vert \\ &\qquad {}+ \biggl(\frac{1}{8}+\frac{1}{(\vartheta +1)(\vartheta +2)}-\frac{1}{( \vartheta +1)2^{\vartheta +1}} \biggr) \biggl\vert g' \biggl(\frac{3v-u}{2} \biggr) \biggr\vert . \end{aligned}$$

(16)

Using (15) and (16) in (14), we get inequality (12) for $q=1$.

For $q>1$ we use the Hölder inequality and the convexity of $|g'|^{q}$ on $[\frac{3u-v}{2},\frac{3v-u}{2} ]$ to obtain

$$\begin{aligned} \begin{aligned}[b] & \int _{0}^{\frac{1}{2}} \bigl[(1-\ell )^{\vartheta }- \ell ^{\vartheta }+1 \bigr] \biggl\vert g' \biggl( \frac{3v-u}{2}+2(v-u)\ell \biggr) \biggr\vert \,d\ell \\ &\quad \leq \biggl( \int _{0}^{\frac{1}{2}} \bigl[(1-\ell )^{\vartheta }- \ell ^{\vartheta }+1 \bigr]\,d\ell \biggr)^{1-\frac{1}{q}} \\ &\qquad {}\times \biggl( \int _{0}^{\frac{1}{2}} \bigl[(1-\ell )^{\vartheta }- \ell ^{\vartheta }+1 \bigr] \biggl\vert g' \biggl( \frac{3u-v}{2}\ell + \frac{3v-u}{2}(1-\ell ) \biggr) \biggr\vert ^{q}\,d\ell \biggr)^{ \frac{1}{q}} \\ &\quad \leq \biggl( \int _{0}^{\frac{1}{2}} \bigl[(1-\ell )^{\vartheta }- \ell ^{\vartheta }+1 \bigr]\,d\ell \biggr)^{1-\frac{1}{q}} \\ &\qquad {}\times \biggl( \int _{0}^{\frac{1}{2}} \bigl[(1-\ell )^{\vartheta }- \ell ^{\vartheta }+1 \bigr] \biggl\{ \ell \biggl\vert g' \biggl(\frac{3u-v}{2} \biggr) \biggr\vert ^{q}+(1-\ell ) \biggl\vert g' \biggl(\frac{3v-u}{2} \biggr) \biggr\vert ^{q} \biggr\} \,d\ell \biggr)^{\frac{1}{q}} \\ &\quad = \biggl(\frac{1}{2}+\frac{1}{\vartheta +1} \biggl(1- \frac{1}{2^{ \vartheta }} \biggr) \biggr)^{1-\frac{1}{q}} \biggl( \biggl[ \frac{1}{8}+\frac{1}{( \vartheta +1)(\vartheta +2)}-\frac{1}{(\vartheta +1)2^{\vartheta +1}} \biggr] \biggl\vert g' \biggl(\frac{3u-v}{2} \biggr) \biggr\vert ^{q} \\ &\qquad {}+ \biggl[\frac{3}{8}+\frac{1}{\vartheta +2}-\frac{1}{(\vartheta +1)2^{ \vartheta +1}} \biggr] \biggl\vert g' \biggl(\frac{3v-u}{2} \biggr) \biggr\vert ^{q} \biggr)^{\frac{1}{q}}. \end{aligned} \end{aligned}$$

(17)

Analogously,

$$\begin{aligned} & \int _{\frac{1}{2}}^{1} \bigl[\ell ^{\vartheta }-(1- \ell )^{\vartheta }+1 \bigr] \biggl\vert g' \biggl( \frac{3v-u}{2}+2(v-u)\ell \biggr) \biggr\vert \,d\ell \\ &\quad \leq \biggl(\frac{1}{2}+\frac{1}{\vartheta +1} \biggl(1- \frac{1}{2^{ \vartheta }} \biggr) \biggr)^{1-\frac{1}{q}} \biggl( \biggl[ \frac{3}{8}+\frac{1}{ \vartheta +2}-\frac{1}{(\vartheta +1)2^{\vartheta +1}} \biggr] \biggl\vert g' \biggl(\frac{3u-v}{2} \biggr) \biggr\vert ^{q} \\ &\qquad {}+ \biggl[\frac{1}{8}+\frac{1}{(\vartheta +1)(\vartheta +2)}-\frac{1}{( \vartheta +1)2^{\vartheta +1}} \biggr] \biggl\vert g' \biggl(\frac{3v-u}{2} \biggr) \biggr\vert ^{q} \biggr)^{\frac{1}{q}}. \end{aligned}$$

(18)

Using (17) and (18) in (14), we get

$$\begin{aligned} & \biggl\vert \frac{\varGamma (\vartheta +1)}{2(v-u)^{\vartheta }} \bigl[\mathfrak{J} _{u^{+}}^{\vartheta }g(v)+\mathfrak{J}_{v^{-}}^{\vartheta }g(u) \bigr] -g \biggl(\frac{u+v}{2} \biggr) \biggr\vert \\ &\quad \leq \frac{v-u}{2} \biggl(\frac{1}{2}+\frac{1}{\vartheta +1} \biggl(1-\frac{1}{2^{ \vartheta }} \biggr) \biggr)^{1-\frac{1}{q}} \bigl\{ (c_{1}+d_{1})^{ \frac{1}{q}}+(c_{2}+d_{2})^{\frac{1}{q}} \bigr\} \end{aligned}$$

(19)

where

$$\begin{aligned} &c_{1}= \biggl[\frac{1}{8}+\frac{1}{(\vartheta +1)(\vartheta +2)}- \frac{1}{( \vartheta +1)2^{\vartheta +1}} \biggr] \biggl\vert g' \biggl( \frac{3u-v}{2} \biggr) \biggr\vert ^{q}, \\ &c_{2}= \biggl[\frac{3}{8}+\frac{1}{\vartheta +2}- \frac{1}{(\vartheta +1)2^{\vartheta +1}} \biggr] \biggl\vert g' \biggl( \frac{3u-v}{2} \biggr) \biggr\vert ^{q}, \\ &d_{1}= \biggl[\frac{3}{8}+\frac{1}{\vartheta +2}- \frac{1}{(\vartheta +1)2^{\vartheta +1}} \biggr] \biggl\vert g' \biggl( \frac{3v-u}{2} \biggr) \biggr\vert ^{q}, \\ &d_{2}= \biggl[\frac{1}{8}+\frac{1}{(\vartheta +1)(\vartheta +2)}- \frac{1}{( \vartheta +1)2^{\vartheta +1}} \biggr] \biggl\vert g' \biggl( \frac{3v-u}{2} \biggr) \biggr\vert ^{q}. \end{aligned}$$

Applying the formula

$$\begin{aligned} \sum_{k=1}^{n} (c_{i}+d_{i} )^{m}\leq \sum_{k=1}^{n}c_{i} ^{m}+\sum_{k=1}^{n}d_{i}^{m}, \quad 0\leq m< 1, \end{aligned}$$

for (19) and then using the fact that $\vert x_{1}^{r}+x _{2}^{r} \vert \leq \vert x_{1}+x_{2} \vert ^{r}$, $x_{1}, x_{2}, r \in [0,1]$, we obtain the inequality (12). This completes the proof of Theorem 1. □

Corollary 1

With similar assumptions to Theorem1, if$\vartheta =1$, then

$$\begin{aligned} & \biggl\vert \frac{1}{v-u} \int _{u}^{v} g(x)\,dx-g \biggl( \frac{u+v}{2} \biggr) \biggr\vert \leq \frac{3(v-u)}{8} \biggl( \biggl\vert g' \biggl(\frac{3u-v}{2} \biggr) \biggr\vert ^{q} + \biggl\vert g' \biggl(\frac{3v-u}{2} \biggr) \biggr\vert ^{q} \biggr)^{\frac{1}{q}}, \end{aligned}$$

(20)

which is obtained by Mehrez and Agarwal in [12, Theorem 1].

Remark 2

In [23], the following inequality has been established:

$$\begin{aligned} & \biggl\vert \frac{1}{v-u} \int _{u}^{v} g(x)\,dx-g \biggl( \frac{u+v}{2} \biggr) \biggr\vert \leq \frac{3(v-u)}{8} \bigl( \bigl\vert g'(u) \bigr\vert + \bigl\vert g'(v) \bigr\vert \bigr). \end{aligned}$$

(21)

We show an analytical and numerical comparison between the left-hand side of inequalities (20) and (21).

• 1. Let $q=1$ and $u,v\in \mathbb{R}$ with $u< v$. Then:

  • (a) If the function $|g'|$ is increasing on $[\frac{3u-v}{2}, \frac{3v-u}{2} ]$. Since $\frac{3u-v}{2}< u< v<\frac{3v-u}{2}$, we obtain

    $$\begin{aligned} g' \biggl(\frac{3u-v}{2} \biggr)< g'(u)\quad \text{and}\quad g'(v)< g' \biggl( \frac{3v-u}{2} \biggr); \end{aligned}$$

    or if the function $|g'|$ is decreasing on $[\frac{3u-v}{2}, \frac{3v-u}{2} ]$, we obtain

    $$\begin{aligned} g'(u)< g' \biggl(\frac{3u-v}{2} \biggr)\quad \text{and}\quad g' \biggl(\frac{3v-u}{2} \biggr)< g'(v). \end{aligned}$$

    In those cases, comparison does not occur analytically between inequalities (20) and (21).

  • (b) If the function $|g'|$ is increasing on $[\frac{3u-v}{2},u ]$, and decreasing on $[v, \frac{3v-u}{2} ]$, then we have

    $$\begin{aligned} g' \biggl(\frac{3u-v}{2} \biggr)< g'(u)\quad \text{and}\quad g' \biggl(\frac{3v-u}{2} \biggr)< g'(v). \end{aligned}$$

    This tells us the right-hand side of inequality (20) is better than the right-hand side of inequality (21).

  • (c) If the function $|g'|$ is decreasing on $[\frac{3u-v}{2},u ]$, and increasing on $[v, \frac{3v-u}{2} ]$, then we conclude that the right-hand side of inequality (21) is better than the right-hand side of inequality (20).

• 2. Suppose that m and n represent the right-hand side of inequalities (20) and (21), respectively. Let $[u,v]= [-1,-\frac{1}{2} ]$ and $g(x)=e^{x}$, then we obtain $\mathbf{m}=0.199744$ and $\mathbf{n}=0.167444$ when $q=1$ and $\mathbf{m}=0.155592$ when $q=2$. Then we conclude that the right-hand side of inequality (20) is worse than the right-hand side of inequalities (21) when $q=1$, but better when $q=2$.

Theorem 2

Let$g:\mathcal{I}^{o}\subseteq \mathbb{R}\to \mathbb{R}$be a differentiable mapping on$\mathcal{I}^{o}$, $u,v\in \mathcal{I}^{o}$with$u< v$. Let$g'\in L_{1} [\frac{3u-v}{2},\frac{3v-u}{2} ]$and$g': [\frac{3u-v}{2},\frac{3v-u}{2} ]\to \mathbb{R}$be a continuous function on$[\frac{3u-v}{2},\frac{3v-u}{2} ]$. If$|g'|^{q}, q>1$is a convex function on$[\frac{3u-v}{2}, \frac{3v-u}{2} ]$, then

$$\begin{aligned} \begin{aligned} [b]& \biggl\vert \frac{\varGamma (\vartheta +1)}{2(v-u)^{\vartheta }} \bigl[\mathfrak{J} _{u^{+}}^{\vartheta }g(v) + \mathfrak{J}_{v^{-}}^{\vartheta }g(u) \bigr]-g \biggl( \frac{u+v}{2} \biggr) \biggr\vert \\ &\quad \leq \frac{v-u}{2} \biggl( \frac{2^{ \vartheta p}-1}{2^{\vartheta p}(\vartheta p+1)}+\frac{1}{2} \biggr) ^{\frac{1}{p}} \\ &\qquad {}\times \biggl\{ \biggl(\frac{ \vert g' (\frac{3u-v}{2} ) \vert ^{q} +3 \vert g' (\frac{3v-u}{2} ) \vert ^{q}}{8} \biggr) ^{\frac{1}{q}} + \biggl(\frac{3 \vert g' (\frac{3u-v}{2} ) \vert ^{q} + \vert g' (\frac{3v-u}{2} ) \vert ^{q}}{8} \biggr) ^{\frac{1}{q}} \biggr\} \\ &\quad \leq \frac{v-u}{2} \biggl(\frac{2^{\vartheta p}-1}{2^{\vartheta p}(\vartheta p+1)}+\frac{1}{2} \biggr)^{ \frac{1}{p}} \biggl(\frac{ \vert g' (\frac{3u-v}{2} ) \vert ^{q} + \vert g' (\frac{3v-u}{2} ) \vert ^{q}}{2} \biggr) ^{\frac{1}{q}}, \end{aligned} \end{aligned}$$

(22)

where$\frac{1}{p}+\frac{1}{q}=1$.

Proof

Applying Hölder’s inequality and the convexity of $|g'|^{q}, q>1$ on $[\frac{3u-v}{2},\frac{3v-u}{2} ]$, we obtain

$$\begin{aligned} & \int _{0}^{\frac{1}{2}} \bigl[(1-\ell )^{\vartheta }- \ell ^{\vartheta }+1 \bigr] \biggl\vert g' \biggl( \frac{3u-v}{2}\ell +\frac{3v-u}{2}(1- \ell ) \biggr) \biggr\vert \,d\ell \\ &\quad \leq \biggl( \int _{0}^{\frac{1}{2}} \bigl[(1-\ell )^{\vartheta }- \ell ^{\vartheta }+1 \bigr]^{p}\,d\ell \biggr)^{\frac{1}{p}} \\ &\qquad {}\times \biggl( \int _{0}^{\frac{1}{2}} \biggl\vert g' \biggl(\frac{3u-v}{2}\ell +\frac{3v-u}{2}(1- \ell ) \biggr) \biggr\vert ^{q}\,d\ell \biggr)^{\frac{1}{q}} \\ &\quad \leq \biggl( \int _{0}^{\frac{1}{2}} \bigl[(1-\ell )^{\vartheta }- \ell ^{\vartheta }+1 \bigr]^{p}\,d\ell \biggr)^{\frac{1}{p}} \\ &\qquad {}\times \biggl( \int _{0}^{\frac{1}{2}} \biggl\{ \ell \biggl\vert g' \biggl(\frac{3u-v}{2} \biggr) \biggr\vert ^{q}+(1-\ell ) \biggl\vert g' \biggl( \frac{3v-u}{2} \biggr) \biggr\vert ^{q} \biggr\} \,d\ell \biggr) ^{\frac{1}{q}}. \end{aligned}$$

(23)

Since $(H_{1}-H_{2})^{q}\leq H_{1}^{q}-H_{2}^{q}$ for each $H_{1}, H _{2}>0$ and $q>1$, (23) becomes

$$\begin{aligned} & \int _{0}^{\frac{1}{2}} \bigl[(1-\ell )^{\vartheta }- \ell ^{\vartheta }+1 \bigr] \biggl\vert g' \biggl( \frac{3v-u}{2}+2(v-u)\ell \biggr) \biggr\vert \,d\ell \\ &\quad \leq \biggl( \int _{0}^{\frac{1}{2}} \bigl[1+(1-\ell )^{\vartheta p}-\ell ^{\vartheta p} \bigr]\,d\ell \biggr)^{\frac{1}{p}} \\ &\qquad {}\times \biggl( \biggl\vert g' \biggl(\frac{3u-v}{2} \biggr) \biggr\vert ^{q} \int _{0}^{\frac{1}{2}}\ell \,d\ell + \biggl\vert g' \biggl(\frac{3v-u}{2} \biggr) \biggr\vert ^{q} \int _{0}^{\frac{1}{2}}(1-\ell )\,d\ell \biggr)^{\frac{1}{q}} \\ &\quad = \biggl(\frac{2^{\vartheta p}-1}{2^{\vartheta p}(\vartheta p+1)}+ \frac{1}{2} \biggr)^{\frac{1}{p}} \biggl(\frac{ \vert g' (\frac{3u-v}{2} ) \vert ^{q} +3 \vert g' (\frac{3v-u}{2} ) \vert ^{q}}{8} \biggr) ^{\frac{1}{q}}. \end{aligned}$$

(24)

In a similar manner, we get

$$\begin{aligned} \begin{aligned}[b] & \int _{\frac{1}{2}}^{1} \bigl[\ell ^{\vartheta }-(1- \ell )^{\vartheta }+1 \bigr] \biggl\vert g' \biggl( \frac{3u-v}{2}\ell +\frac{3v-u}{2}(1- \ell ) \biggr) \biggr\vert \,d\ell \\ &\quad \leq \biggl(\frac{2^{\vartheta p}-1}{2^{\vartheta p}(\vartheta p+1)}+\frac{1}{2} \biggr)^{\frac{1}{p}} \biggl(\frac{3 \vert g' (\frac{3u-v}{2} ) \vert ^{q} + \vert g' (\frac{3v-u}{2} ) \vert ^{q}}{8} \biggr)^{\frac{1}{q}}. \end{aligned} \end{aligned}$$

(25)

Using (24) and (25) in (14), we get

$$\begin{aligned} & \biggl\vert \frac{\varGamma (\vartheta +1)}{2(v-u)^{\vartheta }} \bigl[\mathfrak{J} _{u^{+}}^{\vartheta }g(v) +\mathfrak{J}_{v^{-}}^{\vartheta }g(u) \bigr]-g \biggl( \frac{u+v}{2} \biggr) \biggr\vert \\ &\quad \leq \frac{v-u}{2} \biggl( \frac{2^{ \vartheta p}-1}{2^{\vartheta p}(\vartheta p+1)}+\frac{1}{2} \biggr) ^{\frac{1}{p}} \\ &\qquad {}\times \biggl\{ \biggl(\frac{ \vert g' (\frac{3u-v}{2} ) \vert ^{q} +3 \vert g' (\frac{3v-u}{2} ) \vert ^{q}}{8} \biggr) ^{\frac{1}{q}} + \biggl(\frac{3 \vert g' (\frac{3u-v}{2} ) \vert ^{q} + \vert g' (\frac{3v-u}{2} ) \vert ^{q}}{8} \biggr) ^{\frac{1}{q}} \biggr\} , \end{aligned}$$

which proves the first inequality of (22). Applying the fact

$$\begin{aligned} \bigl\vert c_{1}^{m}+c_{2}^{m} \bigr\vert \leq \vert c_{1}+c_{2} \vert ^{m} , \quad 0\leq m< 1,\; c_{1}, c_{2}\in [0,1] \end{aligned}$$

to the last inequality, we get the second inequality of (22). This completes the proof. □

Collecting both of Theorems 1 and 2 we obtain the following corollary.

Corollary 2

Let$\frac{1}{p}+\frac{1}{q}=1$, then from Theorems1and2, we have

$$\begin{aligned} & \biggl\vert \frac{\varGamma (\vartheta +1)}{2(v-u)^{\vartheta }} \bigl[\mathfrak{J} _{u^{+}}^{\vartheta }g(v) +\mathfrak{J}_{v^{-}}^{\vartheta }g(u) \bigr]-g \biggl( \frac{u+v}{2} \biggr) \biggr\vert \\ &\quad \leq \frac{v-u}{2} \biggl( \biggl\vert g' \biggl( \frac{3u-v}{2} \biggr) \biggr\vert ^{q} + \biggl\vert g' \biggl(\frac{3v-u}{2} \biggr) \biggr\vert ^{q} \biggr)^{ \frac{1}{q}}\min \{\gamma _{1},\gamma _{2}\}, \end{aligned}$$

where$\gamma _{1}= (\frac{2^{\vartheta }-1}{2^{\vartheta }( \vartheta +1)}+\frac{1}{2} )$and$\gamma _{2}= (\frac{2^{ \vartheta p}-1}{2^{\vartheta p+\frac{p}{q}}(\vartheta p+1)} +\frac{1}{2^{ \frac{p}{q}+1}} )^{\frac{1}{p}}$.

Corollary 3

With similar assumptions to Theorem2if$\vartheta =1$, we have

$$\begin{aligned} & \biggl\vert \frac{1}{v-u} \int _{u}^{v} g(x)\,dx-g \biggl( \frac{u+v}{2} \biggr) \biggr\vert \\ &\quad \leq (v-u) \biggl( \frac{1}{2^{p+1}(p+1)} \biggr) ^{\frac{1}{p}} \biggl(\frac{ \vert g' (\frac{3u-v}{2} ) \vert ^{q} + \vert g' (\frac{3v-u}{2} ) \vert ^{q}}{2} \biggr) ^{\frac{1}{q}}, \end{aligned}$$

which is obtained by Mehrez and Agarwal in [12, Theorem 2].

Theorem 3

Let$g:\mathcal{I}^{o}\subseteq \mathbb{R}\to \mathbb{R}$be a differentiable mapping on$\mathcal{I}^{o}$, $u,v\in \mathcal{I}^{o}$with$u< v$. Let$g'': [\frac{3u-v}{2},\frac{3v-u}{2} ] \to \mathbb{R}$be a continuous function on$[\frac{3u-v}{2}, \frac{3v-u}{2} ]$. If$|g''|^{q}, q\geq 1$is a convex function on$[\frac{3u-v}{2},\frac{3v-u}{2} ]$, then

$$\begin{aligned} \begin{aligned}[b] & \biggl\vert \frac{\varGamma (\vartheta +1)}{2(v-u)^{\vartheta }} \bigl[\mathfrak{J} _{u^{+}}^{\vartheta }g(v) + \mathfrak{J}_{v^{-}}^{\vartheta }g(u) \bigr]-\frac{g (\frac{3v-u}{2} )+2g (\frac{u+v}{2} ) +g (\frac{3u-v}{2} )}{4} \biggr\vert \\ &\quad \leq \frac{\vartheta (v-u)^{2}}{2(\vartheta +1)(\vartheta +2)} \biggl\{ \biggl(\frac{2 \vartheta +4}{3\vartheta +9} \biggl\vert g'' \biggl(\frac{3u-v}{2} \biggr) \biggr\vert ^{q}+\frac{\vartheta +5}{3\vartheta +9} \biggl\vert g'' \biggl(\frac{3v-u}{2} \biggr) \biggr\vert ^{q} \biggr)^{\frac{1}{q}} \\ &\qquad {}+ \biggl(\frac{\vartheta +5}{3\vartheta +9} \biggl\vert g'' \biggl(\frac{3u-v}{2} \biggr) \biggr\vert ^{q} + \frac{2\vartheta +4}{3\vartheta +9} \biggl\vert g'' \biggl( \frac{3v-u}{2} \biggr) \biggr\vert ^{q} \biggr) ^{\frac{1}{q}} \biggr\} . \end{aligned} \end{aligned}$$

(26)

Proof

From Lemma 2 we have

$$\begin{aligned} &\frac{\varGamma (\vartheta +1)}{2^{\vartheta +1}(v-u)^{\vartheta }} \biggl[\mathfrak{J}_{ (\frac{3u-v}{2} )^{+}}^{\vartheta }g \biggl(\frac{3v-u}{2} \biggr) +\mathfrak{J}_{ (\frac{3v-u}{2} ) ^{-}}^{\vartheta }g \biggl(\frac{3u-v}{2} \biggr) \biggr] \\ &\quad =\frac{g (\frac{3u-v}{2} )+g (\frac{3v-u}{2} )}{2} \\ &\qquad {}-\frac{2(v-u)^{2}}{\vartheta +1} \int _{0}^{1}\ell \bigl(1- \ell ^{\vartheta } \bigr) \biggl[g'' \biggl(\ell \biggl( \frac{3u-v}{2} \biggr)+(1- \ell ) \biggl(\frac{3v-u}{2} \biggr) \biggr) \\ &\qquad {}+g'' \biggl((1-\ell ) \biggl( \frac{3u-v}{2} \biggr)+\ell \biggl(\frac{3v-u}{2} \biggr) \biggr) \biggr]\,d\ell . \end{aligned}$$

(27)

From (27) and (10), we have

$$\begin{aligned} & \biggl\vert \frac{\varGamma (\vartheta +1)}{2(v-u)^{\vartheta }} \bigl[\mathfrak{J} _{u^{+}}^{\vartheta }g(v) +\mathfrak{J}_{v^{-}}^{\vartheta }g(u) \bigr]-\frac{g (\frac{3v-u}{2} )+2g (\frac{u+v}{2} ) +g (\frac{3u-v}{2} )}{4} \biggr\vert \\ &\quad \leq \frac{(v-u)^{2}}{ \vartheta +1} \int _{0}^{1}\ell \bigl(1-\ell ^{\vartheta } \bigr) \\ &\qquad {}\times \biggl[ \biggl\vert g'' \biggl(\ell \biggl(\frac{3u-v}{2} \biggr)+(1- \ell ) \biggl(\frac{3v-u}{2} \biggr) \biggr) \biggr\vert \\ &\qquad {} + \biggl\vert g'' \biggl((1- \ell ) \biggl(\frac{3u-v}{2} \biggr)+\ell \biggl( \frac{3v-u}{2} \biggr) \biggr) \biggr\vert \biggr]\,d\ell . \end{aligned}$$

(28)

Using the convexity of $|g''|^{q}, q>1$ on $[\frac{3u-v}{2}, \frac{3v-u}{2} ]$ and Hölder’s inequality, we have

$$\begin{aligned} & \int _{0}^{1}\ell \bigl(1-\ell ^{\vartheta } \bigr) \biggl[ \biggl\vert g'' \biggl(\ell \biggl( \frac{3u-v}{2} \biggr)+(1-\ell ) \biggl(\frac{3v-u}{2} \biggr) \biggr) \biggr\vert \\ &\qquad {}+ \biggl\vert g'' \biggl((1-\ell ) \biggl( \frac{3u-v}{2} \biggr)+\ell \biggl(\frac{3v-u}{2} \biggr) \biggr) \biggr\vert \biggr]\,d\ell \\ &\quad \leq \biggl( \int _{0}^{1}\ell \bigl(1-\ell ^{\vartheta } \bigr)\,d\ell \biggr)^{1-\frac{1}{q}} \\ &\qquad {}\times \biggl[ \biggl( \int _{0}^{1}\ell \bigl(1-\ell ^{\vartheta } \bigr) \biggl\{ \ell \biggl\vert g'' \biggl( \biggl(\frac{3u-v}{2} \biggr) \biggr\vert ^{q} +(1-\ell ) \biggl\vert g'' \biggl(\frac{3v-u}{2} \biggr) \biggr) \biggr\vert ^{q} \biggr\} \,d\ell \biggr)^{\frac{1}{q}} \\ &\qquad {}+ \biggl( \int _{0}^{1}\ell \bigl(1-\ell ^{\vartheta } \bigr) \biggl\{ (1-\ell ) \biggl\vert g'' \biggl( \biggl(\frac{3u-v}{2} \biggr) \biggr\vert ^{q} + \ell \biggl\vert g'' \biggl(\frac{3v-u}{2} \biggr) \biggr) \biggr\vert ^{q} \biggr\} \,d\ell \biggr)^{\frac{1}{q}} \biggr] \\ &\quad =\frac{\vartheta }{2(\vartheta +2)} \biggl\{ \biggl(\frac{2\vartheta +4}{3\vartheta +9} \biggl\vert g'' \biggl(\frac{3u-v}{2} \biggr) \biggr\vert ^{q}+\frac{\vartheta +5}{3\vartheta +9} \biggl\vert g'' \biggl(\frac{3v-u}{2} \biggr) \biggr\vert ^{q} \biggr)^{\frac{1}{q}} \\ &\qquad {}+ \biggl(\frac{\vartheta +5}{3\vartheta +9} \biggl\vert g'' \biggl(\frac{3u-v}{2} \biggr) \biggr\vert ^{q} + \frac{2\vartheta +4}{3\vartheta +9} \biggl\vert g'' \biggl( \frac{3v-u}{2} \biggr) \biggr\vert ^{q} \biggr)^{\frac{1}{q}} \biggr\} . \end{aligned}$$

(29)

Using (29) in (28) we get (26) for $q>1$.

Now by using the convexity of $|g''|$, we find

$$\begin{aligned} & \int _{0}^{1}\ell \bigl(1-\ell ^{\vartheta } \bigr) \biggl[ \biggl\vert g'' \biggl(\ell \biggl( \frac{3u-v}{2} \biggr)+(1-\ell ) \biggl(\frac{3v-u}{2} \biggr) \biggr) \biggr\vert \\ &\qquad {}+ \biggl\vert g'' \biggl((1-\ell ) \biggl( \frac{3u-v}{2} \biggr)+\ell \biggl(\frac{3v-u}{2} \biggr) \biggr) \biggr\vert \biggr]\,d\ell \\ &\quad \leq \biggl( \int _{0}^{1}\ell \bigl(1-\ell ^{\vartheta } \bigr) \biggl\{ \ell \biggl\vert g'' \biggl( \biggl(\frac{3u-v}{2} \biggr) \biggr\vert +(1-\ell ) \biggl\vert g'' \biggl(\frac{3v-u}{2} \biggr) \biggr) \biggr\vert \biggr\} \,d\ell \biggr) \\ &\qquad {}+ \biggl( \int _{0}^{1}\ell \bigl(1-\ell ^{\vartheta } \bigr) \biggl\{ (1-\ell ) \biggl\vert g'' \biggl( \biggl(\frac{3u-v}{2} \biggr) \biggr\vert + \ell \biggl\vert g'' \biggl(\frac{3v-u}{2} \biggr) \biggr) \biggr\vert \biggr\} \,d\ell \biggr) \\ &\quad = \biggl[\frac{\vartheta }{3(\vartheta +3)} \biggl\vert g'' \biggl(\frac{3u-v}{2} \biggr) \biggr\vert +\frac{ \vartheta (\vartheta +5)}{6(\vartheta +2)(\vartheta +3)} \biggl\vert g'' \biggl(\frac{3v-u}{2} \biggr) \biggr\vert \\ &\qquad {}+\frac{\vartheta }{3(\vartheta +3)} \biggl\vert g'' \biggl( \frac{3v-u}{2} \biggr) \biggr\vert +\frac{ \vartheta (\vartheta +5)}{6(\vartheta +2)(\vartheta +3)} \biggl\vert g'' \biggl(\frac{3u-v}{2} \biggr) \biggr\vert \biggr] \\ &\quad =\frac{\vartheta }{2(\vartheta +2)} \biggl( \biggl\vert g'' \biggl(\frac{3u-v}{2} \biggr) \biggr\vert + \biggl\vert g'' \biggl(\frac{3v-u}{2} \biggr) \biggr\vert \biggr). \end{aligned}$$

(30)

Substituting (30) into (28) we deduce that the inequality (26) holds true for $q=1$. Hence the proof of Theorem 3 is completed. □

Corollary 4

With similar assumptions to Theorem3if$\vartheta =1$, we have

$$\begin{aligned} & \biggl\vert \frac{1}{v-u} \int _{u}^{v} g(x)\,dx-\frac{g (\frac{3v-u}{2} )+2g (\frac{u+v}{2} ) +g (\frac{3u-v}{2} )}{4} \biggr\vert \\ &\quad \leq \frac{(v-u)^{2}}{6} \biggl(\frac{ \vert g'' (\frac{3u-v}{2} ) \vert ^{q} + \vert g'' (\frac{3v-u}{2} ) \vert ^{q}}{2} \biggr) ^{\frac{1}{q}}. \end{aligned}$$

(31)

Remark 3

In [12, Theorem 3], Mehrez and Agarwal obtained the following inequality:

$$\begin{aligned} & \biggl\vert \frac{1}{v-u} \int _{u}^{v} g(x)\,dx-\frac{g (\frac{3v-u}{2} )+2g (\frac{u+v}{2} ) +g (\frac{3u-v}{2} )}{4} \biggr\vert \\ &\quad \leq \frac{(v-u)^{2}}{3} \biggl(\frac{ \vert g'' (\frac{3u-v}{2} ) \vert ^{q} + \vert g'' (\frac{3v-u}{2} ) \vert ^{q}}{2} \biggr) ^{\frac{1}{q}}. \end{aligned}$$

(32)

The right-hand side of (31) confirms the modification of our work compared with (32).

Remark 4

If $g''(x)$ is bounded on the interval $[\frac{3u-v}{2}, \frac{3v-u}{2} ]$, then Theorem 3 reduces to

$$\begin{aligned} & \biggl\vert \frac{\varGamma (\vartheta +1)}{2(v-u)^{\vartheta }} \bigl[\mathfrak{J} _{u^{+}}^{\vartheta }g(v) +\mathfrak{J}_{v^{-}}^{\vartheta }g(u) \bigr] -\frac{g (\frac{3v-u}{2} )+2g (\frac{u+v}{2} )+g (\frac{3u-v}{2} )}{4} \biggr\vert \leq \frac{M \vartheta (v-u)^{2}}{( \vartheta +1)(\vartheta +2)} \end{aligned}$$

for some $M\in \mathbb{R}$.

Theorem 4

Let$g:\mathcal{I}^{o}\subseteq \mathbb{R}\to \mathbb{R}$be a differentiable mapping on$\mathcal{I}^{o}$, $u,v\in \mathcal{I}^{o}$with$u< v$. Let$g'': [\frac{3u-v}{2},\frac{3v-u}{2} ] \to \mathbb{R}$be a continuous function on$[\frac{3u-v}{2}, \frac{3v-u}{2} ]$. If$|g''|^{q}, q>1$is a convex function on$[\frac{3u-v}{2},\frac{3v-u}{2} ]$, then

$$\begin{aligned} \begin{aligned} & \biggl\vert \frac{\varGamma (\vartheta +1)}{2(v-u)^{\vartheta }} \bigl[\mathfrak{J} _{u^{+}}^{\vartheta }g(v) + \mathfrak{J}_{v^{-}}^{\vartheta }g(u) \bigr]-\frac{g (\frac{3v-u}{2} ) +2g (\frac{u+v}{2} )+g (\frac{3u-v}{2} )}{4} \biggr\vert \\ &\quad \leq \frac{2(v-u)^{2}}{\vartheta +1}\beta ^{\frac{1}{p}}(p+1, \vartheta p+1) \biggl( \frac{ \vert g'' (\frac{3u-v}{2} ) \vert ^{q}+ \vert g'' (\frac{3v-u}{2} ) \vert ^{q}}{2} \biggr) ^{\frac{1}{q}}, \end{aligned} \end{aligned}$$

(33)

where$\frac{1}{p}+\frac{1}{q}=1$.

Proof

From inequality (28) and the Hölder inequality, we have

$$\begin{aligned} & \biggl\vert \frac{\varGamma (\vartheta +1)}{2(v-u)^{\vartheta }} \bigl[\mathfrak{J} _{u^{+}}^{\vartheta }g(v) +\mathfrak{J}_{v^{-}}^{\vartheta }g(u) \bigr] -\frac{g (\frac{3v-u}{2} )+2g (\frac{u+v}{2} )+g (\frac{3u-v}{2} )}{4} \biggr\vert \\ &\quad \leq \frac{(v-u)^{2}}{\vartheta +1} \biggl( \int _{0}^{1}\ell ^{p} \bigl(1-\ell ^{\vartheta } \bigr)^{p}\,d\ell \biggr)^{\frac{1}{p}} \\ &\qquad {}\times\biggl[ \biggl( \int _{0}^{1} \biggl\vert g'' \biggl(\ell \biggl(\frac{3u-v}{2} \biggr) +(1-\ell ) \biggl( \frac{3v-u}{2} \biggr) \biggr) \biggr\vert ^{q} \,d\ell \biggr) ^{\frac{1}{q}} \\ &\qquad {}+ \biggl( \int _{0}^{1} \biggl\vert g'' \biggl(\ell \biggl(\frac{3v-u}{2} \biggr) +(1- \ell ) \biggl( \frac{3u-v}{2} \biggr) \biggr) \biggr\vert ^{q} \,d\ell \biggr) ^{\frac{1}{q}} \biggr]. \end{aligned}$$

Using the fact that $|g''|^{q}, q>1$ is convex on $[\frac{3u-v}{2}, \frac{3v-u}{2} ]$, we have

$$\begin{aligned} & \biggl\vert \frac{\varGamma (\vartheta +1)}{2(v-u)^{\vartheta }} \bigl[\mathfrak{J} _{u^{+}}^{\vartheta }g(v) +\mathfrak{J}_{v^{-}}^{\vartheta }g(u) \bigr] -\frac{g (\frac{3v-u}{2} )+2g (\frac{u+v}{2} )+g (\frac{3u-v}{2} )}{4} \biggr\vert \\ &\quad \leq \frac{(v-u)^{2}}{\vartheta +1} \biggl( \int _{0}^{1}\ell ^{p} \bigl(1- \ell ^{\vartheta } \bigr)^{p}\,d\ell \biggr)^{\frac{1}{p}} \\ &\qquad {}\times \biggl[ \biggl( \int _{0}^{1} \biggl\{ \ell \biggl\vert g'' \biggl( \biggl(\frac{3u-v}{2} \biggr) \biggr\vert +(1- \ell ) \biggl\vert g'' \biggl( \frac{3v-u}{2} \biggr) \biggr) \biggr\vert ^{q} \biggr\} \,d\ell \biggr)^{\frac{1}{q}} \\ &\qquad {}+ \biggl( \int _{0}^{1} \biggl\{ (1-\ell ) \biggl\vert g'' \biggl( \biggl(\frac{3u-v}{2} \biggr) \biggr\vert ^{q} +\ell \biggl\vert g'' \biggl(\frac{3v-u}{2} \biggr) \biggr) \biggr\vert ^{q} \biggr\} \,d\ell \biggr)^{\frac{1}{q}} \biggr] \\ &\quad =\frac{2(v-u)^{2}}{\vartheta +1}\beta ^{\frac{1}{p}}(p+1,\vartheta p+1) \biggl( \frac{ \vert g'' (\frac{3u-v}{2} ) \vert ^{q}+ \vert g'' (\frac{3v-u}{2} ) \vert ^{q}}{2} \biggr) ^{\frac{1}{q}}. \end{aligned}$$

(34)

Observe that $\ell ^{p} (1-\ell ^{\vartheta } )^{p}\leq \ell ^{p} ([1-\ell ]^{\vartheta } )^{p} =\ell ^{p} (1- \ell )^{\vartheta p}$. So, (34) completes the proof of Theorem 4. □

Corollary 5

With similar assumptions to Theorem4if$\vartheta =1$, we have

$$\begin{aligned} & \biggl\vert \frac{1}{v-u} \int _{u}^{v} g(x)\,dx-\frac{g (\frac{3v-u}{2} )+2g (\frac{u+v}{2} ) +g (\frac{3u-v}{2} )}{4} \biggr\vert \\ &\quad \leq \frac{(v-u)^{2}}{4} \biggl(\frac{\sqrt{\pi }\varGamma (p+1)}{2 \varGamma (p+\frac{3}{2} )} \biggr)^{\frac{1}{p}} \biggl(\frac{ \vert g'' (\frac{3u-v}{2} ) \vert ^{q} + \vert g'' (\frac{3v-u}{2} ) \vert ^{q}}{2} \biggr)^{\frac{1}{q}}. \end{aligned}$$

(35)

Proof

The proof of this corollary follows from the facts that

$$\begin{aligned} \beta (p+1,+1)=\frac{1}{2^{2p+1}}\frac{\sqrt{\pi } \varGamma (p+1)}{ \varGamma (p+\frac{3}{2} )}. \end{aligned}$$

 □

Remark 5

The right-hand side of inequality (35) confirms the modification of our work compared with the right-hand side of inequality (3.24) in [12, Theorem 4].

Remark 6

If $g''(x)$ is bounded on the interval $[\frac{3u-v}{2}, \frac{3v-u}{2} ]$, then Theorem 4 reduces to

$$\begin{aligned} & \biggl\vert \frac{\varGamma (\vartheta +1)}{2(v-u)^{\vartheta }} \bigl[\mathfrak{J} _{u^{+}}^{\vartheta }g(v) +\mathfrak{J}_{v^{-}}^{\vartheta }g(u) \bigr]-\frac{g (\frac{3v-u}{2} ) +2g (\frac{u+v}{2} )+g (\frac{3u-v}{2} )}{4} \biggr\vert \\ &\quad \leq \beta ^{\frac{1}{p}}(p+1,\vartheta p+1) \frac{2M(v-u)^{2}}{ \vartheta +1} \end{aligned}$$

for $\frac{1}{p}=1-\frac{1}{q}$ and for some $M\in \mathbb{R}$.

Theorem 5

With similar assumptions to Theorem4, we have

$$\begin{aligned} \begin{aligned}[b] & \biggl\vert \frac{\varGamma (\vartheta +1)}{2(v-u)^{\vartheta }} \bigl[\mathfrak{J} _{u^{+}}^{\vartheta }g(v) + \mathfrak{J}_{v^{-}}^{\vartheta }g(u) \bigr]-\frac{g (\frac{3v-u}{2} ) +2g (\frac{u+v}{2} )+g (\frac{3u-v}{2} )}{4} \biggr\vert \\ &\quad \leq \frac{(v-u)^{2}}{\vartheta +1} \biggl(\frac{1}{p+1} \biggr) ^{\frac{1}{p}} \biggl(\frac{1}{(\vartheta q+1)(\vartheta q+2)} \biggr) ^{\frac{1}{q}} \\ &\qquad {}\times \biggl[ \biggl( \biggl\vert g'' \biggl( \frac{3u-v}{2} \biggr) \biggr\vert ^{q}+ (\vartheta q+1) \biggl\vert g'' \biggl(\frac{3v-u}{2} \biggr) \biggr\vert ^{q} \biggr)^{\frac{1}{q}} \\ &\qquad+ \biggl((\vartheta q+1) \biggl\vert g'' \biggl(\frac{3u-v}{2} \biggr) \biggr\vert ^{q}+ \biggl\vert g'' \biggl(\frac{3v-u}{2} \biggr) \biggr\vert ^{q} \biggr)^{\frac{1}{q}} \biggr]. \end{aligned} \end{aligned}$$

(36)

Proof

By using the Hölder inequality and the convexity of $|g''|^{q}, q>1$ on $[\frac{3u-v}{2},\frac{3v-u}{2} ]$, we have

$$\begin{aligned} & \int _{0}^{1}\ell \bigl(1-\ell ^{\vartheta } \bigr) \biggl[ \biggl\vert g'' \biggl(\ell \biggl( \frac{3u-v}{2} \biggr)+(1-\ell ) \biggl(\frac{3v-u}{2} \biggr) \biggr) \biggr\vert \\ &\qquad {}+ \biggl\vert g'' \biggl((1-\ell ) \biggl( \frac{3u-v}{2} \biggr)+\ell \biggl(\frac{3v-u}{2} \biggr) \biggr) \biggr\vert \biggr]\,d\ell \\ &\quad \leq \biggl( \int _{0}^{1}\ell ^{p}\,d\ell \biggr)^{ \frac{1}{p}} \\ &\qquad {}\times \biggl[ \biggl( \int _{0}^{1} \bigl(1-\ell ^{\vartheta } \bigr) ^{q} \biggl\{ \ell \biggl\vert g'' \biggl( \biggl(\frac{3u-v}{2} \biggr) \biggr\vert ^{q} +(1- \ell ) \biggl\vert g'' \biggl( \frac{3v-u}{2} \biggr) \biggr) \biggr\vert ^{q} \biggr\} \,d\ell \biggr)^{\frac{1}{q}} \\ &\qquad {}+ \biggl( \int _{0}^{1} \bigl(1-\ell ^{\vartheta } \bigr)^{q} \biggl\{ (1-\ell ) \biggl\vert g'' \biggl( \biggl(\frac{3u-v}{2} \biggr) \biggr\vert ^{q} + \ell \biggl\vert g'' \biggl(\frac{3v-u}{2} \biggr) \biggr) \biggr\vert ^{q} \biggr\} \,d\ell \biggr)^{\frac{1}{q}} \biggr] \\ &\quad = \biggl(\frac{1}{p+1} \biggr)^{\frac{1}{p}} \biggl[ \biggl( \frac{1}{( \vartheta q+1)(\vartheta q+2)} \biggl\vert g'' \biggl( \frac{3u-v}{2} \biggr) \biggr\vert ^{q}+ \frac{1}{\vartheta q+2} \biggl\vert g'' \biggl( \frac{3v-u}{2} \biggr) \biggr\vert ^{q} \biggr)^{\frac{1}{q}} \\ &\qquad {}+ \biggl(\frac{1}{\vartheta q+2} \biggl\vert g'' \biggl(\frac{3u-v}{2} \biggr) \biggr\vert ^{q}+ \frac{1}{(\vartheta q+1)(\vartheta q+2)} \biggl\vert g'' \biggl( \frac{3v-u}{2} \biggr) \biggr\vert ^{q} \biggr)^{\frac{1}{q}} \biggr]. \end{aligned}$$

(37)

Using (37) in (28), we obtain the required inequality (36). □

Corollary 6

With similar assumptions to Theorem5if$\vartheta =1$, we have

$$\begin{aligned} & \biggl\vert \frac{1}{v-u} \int _{u}^{v} g(x)\,dx-\frac{g (\frac{3v-u}{2} )+2g (\frac{u+v}{2} ) +g (\frac{3u-v}{2} )}{4} \biggr\vert \\ &\quad \leq (v-u)^{2} \biggl(\frac{1}{p+1} \biggr)^{\frac{1}{p}} \biggl(\frac{1}{(q+1)(q+2)} \biggr) ^{\frac{1}{q}} \\ &\qquad {}\times \biggl( \biggl\vert g'' \biggl(\frac{3u-v}{2} \biggr) \biggr\vert ^{q} +(q+1) \biggl\vert g'' \biggl(\frac{3v-u}{2} \biggr) \biggr\vert ^{q} \biggr) ^{\frac{1}{q}}. \end{aligned}$$

(38)

Remark 7

The right-hand side of inequality (38) confirms the modification of our work compared with the right-hand side of inequality (3.27) in [12, Theorem 5].

Theorem 6

With similar assumptions to Theorem3, we have

$$\begin{aligned} \begin{aligned}[b] & \biggl\vert \frac{\varGamma (\vartheta +1)}{2(v-u)^{\vartheta }} \bigl[\mathfrak{J} _{u^{+}}^{\vartheta }g(v)+ \mathfrak{J}_{v^{-}}^{\vartheta }g(u) \bigr] -\frac{g (\frac{3v-u}{2} )+2g (\frac{u+v}{2} )+g (\frac{3u-v}{2} )}{4} \biggr\vert \\ &\quad \leq \frac{(v-u)^{2}}{2( \vartheta +1)} \biggl(\frac{2}{(\vartheta q+1)(\vartheta q+2)( \vartheta q+3)} \biggr)^{\frac{1}{q}} \\ &\qquad {}\times \biggl[ \biggl(2 \biggl\vert g'' \biggl( \frac{3u-v}{2} \biggr) \biggr\vert ^{q}+ (\vartheta q+1) \biggl\vert g'' \biggl(\frac{3v-u}{2} \biggr) \biggr\vert ^{q} \biggr)^{ \frac{1}{q}} \\ &\qquad {}+ \biggl((\vartheta q+1) \biggl\vert g'' \biggl(\frac{3u-v}{2} \biggr) \biggr\vert ^{q}+ 2 \biggl\vert g'' \biggl(\frac{3v-u}{2} \biggr) \biggr\vert ^{q} \biggr) ^{\frac{1}{q}} \biggr]. \end{aligned} \end{aligned}$$

(39)

Proof

Let $q>1$, then, by using the Hölder inequality and the convexity of $|g''|^{q}$ on $[\frac{3u-v}{2},\frac{3v-u}{2} ]$, we have

$$\begin{aligned} & \int _{0}^{1}\ell \bigl(1-\ell ^{\vartheta } \bigr) \biggl[ \biggl\vert g'' \biggl(\ell \biggl( \frac{3u-v}{2} \biggr)+(1-\ell ) \biggl(\frac{3v-u}{2} \biggr) \biggr) \biggr\vert \\ &\qquad {}+ \biggl\vert g'' \biggl((1-\ell ) \biggl( \frac{3u-v}{2} \biggr)+\ell \biggl(\frac{3v-u}{2} \biggr) \biggr) \biggr\vert \biggr]\,d\ell \leq \biggl( \int _{0}^{1}\ell \,d\ell \biggr)^{1- \frac{1}{q}} \\ &\qquad {}\times \biggl[ \biggl( \int _{0}^{1}\ell \bigl(1-\ell ^{\vartheta } \bigr) ^{q} \biggl\{ \ell \biggl\vert g'' \biggl( \biggl(\frac{3u-v}{2} \biggr) \biggr\vert ^{q} +(1- \ell ) \biggl\vert g'' \biggl( \frac{3v-u}{2} \biggr) \biggr) \biggr\vert ^{q} \biggr\} \,d\ell \biggr)^{\frac{1}{q}} \\ &\qquad {}+ \biggl( \int _{0}^{1}\ell \bigl(1-\ell ^{\vartheta } \bigr)^{q} \biggl\{ (1-\ell ) \biggl\vert g'' \biggl( \biggl(\frac{3u-v}{2} \biggr) \biggr\vert ^{q}+\ell \biggl\vert g'' \biggl(\frac{3v-u}{2} \biggr) \biggr) \biggr\vert ^{q} \biggr\} \,d\ell \biggr)^{\frac{1}{q}} \biggr] \\ &\quad \leq \biggl(\frac{1}{2} \biggr)^{1-\frac{1}{q}} \biggl[ \biggl(\beta (3, \vartheta q+1) \biggl\vert g'' \biggl( \frac{3u-v}{2} \biggr) \biggr\vert ^{q}+ \beta (2,\vartheta q+2) \biggl\vert g'' \biggl( \frac{3v-u}{2} \biggr) \biggr\vert ^{q} \biggr)^{\frac{1}{q}} \\ &\qquad {}+ \biggl(\beta (2,\vartheta q+2) \biggl\vert g'' \biggl(\frac{3u-v}{2} \biggr) \biggr\vert ^{q}+ \beta (3, \vartheta q+1) \biggl\vert g'' \biggl( \frac{3v-u}{2} \biggr) \biggr\vert ^{q} \biggr)^{\frac{1}{q}} \biggr] \\ &\quad =\frac{1}{2} \biggl(\frac{2}{(\vartheta q+1)(\vartheta q+2)( \vartheta q+3)} \biggr)^{\frac{1}{q}} \biggl[ \biggl(2 \biggl\vert g'' \biggl( \frac{3u-v}{2} \biggr) \biggr\vert ^{q}+ (\vartheta q+1) \biggl\vert g'' \biggl(\frac{3v-u}{2} \biggr) \biggr\vert ^{q} \biggr)^{\frac{1}{q}} \\ &\qquad {}+ \biggl((\vartheta q+1) \biggl\vert g'' \biggl(\frac{3u-v}{2} \biggr) \biggr\vert ^{q}+ 2 \biggl\vert g'' \biggl(\frac{3v-u}{2} \biggr) \biggr\vert ^{q} \biggr) ^{\frac{1}{q}} \biggr]. \end{aligned}$$

(40)

Using (40) in (28) we obtain the inequality (39) for $q>1$.

Now, using the convexity of $|g''|$ and the properties of the modulus, we find

$$\begin{aligned} & \int _{0}^{1}\ell \bigl(1-\ell ^{\vartheta } \bigr) \biggl[ \biggl\vert g'' \biggl(\ell \biggl( \frac{3u-v}{2} \biggr)+(1-\ell ) \biggl(\frac{3v-u}{2} \biggr) \biggr) \biggr\vert \\ &\qquad {}+ \biggl\vert g'' \biggl((1-\ell ) \biggl( \frac{3u-v}{2} \biggr)+\ell \biggl(\frac{3v-u}{2} \biggr) \biggr) \biggr\vert \biggr]\,d\ell \\ &\quad = \int _{0}^{1}\ell ^{0} \biggl[\ell \bigl(1-\ell ^{\vartheta } \bigr) \biggl\vert g'' \biggl(\ell \biggl(\frac{3u-v}{2} \biggr)+(1-\ell ) \biggl( \frac{3v-u}{2} \biggr) \biggr) \biggr\vert \\ &\qquad {}+\ell \bigl(1-\ell ^{\vartheta } \bigr) \biggl\vert g'' \biggl((1-\ell ) \biggl(\frac{3u-v}{2} \biggr)+\ell \biggl( \frac{3v-u}{2} \biggr) \biggr) \biggr\vert \biggr]\,d\ell \\ &\quad \leq \biggl( \int _{0}^{1}\,d\ell \biggr) [ \biggl( \int _{0}^{1} \ell (1-\ell )^{\vartheta } \biggl\{ \ell \biggl\vert g'' \biggl( \biggl( \frac{3u-v}{2} \biggr) \biggr\vert +(1-\ell ) \biggl\vert g'' \biggl(\frac{3v-u}{2} \biggr) \biggr) \biggr\vert \biggr\} \,d\ell \biggr) \\ &\qquad {}+ \biggl( \int _{0}^{1}\ell (1-\ell )^{\vartheta } \biggl\{ (1-\ell ) \biggl\vert g'' \biggl( \biggl( \frac{3u-v}{2} \biggr) \biggr\vert + \ell \biggl\vert g'' \biggl(\frac{3v-u}{2} \biggr) \biggr) \biggr\vert \biggr\} \,d\ell \biggr) \\ &\quad = \biggl(\frac{2}{(\vartheta +1)(\vartheta +2)} \biggr) \biggl( \biggl\vert g'' \biggl(\frac{3u-v}{2} \biggr) \biggr\vert + \biggl\vert g'' \biggl( \frac{3v-u}{2} \biggr) \biggr\vert \biggr). \end{aligned}$$

(41)

Substituting (41) into (28) we deduce that the inequality (39) holds true for $q=1$. Thus (40), (41) and (28) complete the proof of Theorem 6. □

Corollary 7

With similar assumptions to Theorem5if$\vartheta =1$, we have

$$\begin{aligned} & \biggl\vert \frac{1}{v-u} \int _{u}^{v} g(x)\,dx-\frac{g (\frac{3v-u}{2} )+2g (\frac{u+v}{2} ) +g (\frac{3u-v}{2} )}{4} \biggr\vert \\ &\quad \leq \frac{(v-u)^{2}}{2} \biggl(\frac{2}{(q+1)(q+2)(q+3)} \biggr) ^{\frac{1}{q}} \\ &\qquad {}\times \biggl(2 \biggl\vert g'' \biggl(\frac{3u-v}{2} \biggr) \biggr\vert ^{q} +(q+1) \biggl\vert g'' \biggl(\frac{3v-u}{2} \biggr) \biggr\vert ^{q} \biggr) ^{\frac{1}{q}}. \end{aligned}$$

(42)

Remark 8

The right-hand side of inequality (42) confirms the modification of our work compared with the right-hand side of inequality (3.29) in [12, Theorem 6].

Collecting Theorems 3–6 we obtain the following corollary.

Corollary 8

From Theorems3–6we deduce that

$$\begin{aligned} & \biggl\vert \frac{\varGamma (\vartheta +1)}{2(v-u)^{\vartheta }} \bigl[\mathfrak{J} _{u^{+}}^{\vartheta }g(v) +\mathfrak{J}_{v^{-}}^{\vartheta }g(u) \bigr]-\frac{g (\frac{3v-u}{2} )+2g (\frac{u+v}{2} ) +g (\frac{3u-v}{2} )}{4} \biggr\vert \\ &\quad \leq \frac{(v-u)^{2}}{2(\vartheta +1)}\min \{\gamma _{3},\gamma _{4}, \gamma _{5},\gamma _{6}\}, \end{aligned}$$

where

$$\begin{aligned} & \begin{aligned} \gamma _{3}={}&\frac{\vartheta }{(\vartheta +2)} \biggl\{ \biggl( \frac{2 \vartheta +4}{3\vartheta +9} \biggl\vert g'' \biggl( \frac{3u-v}{2} \biggr) \biggr\vert ^{q}+ \frac{\vartheta +5}{3\vartheta +9} \biggl\vert g'' \biggl( \frac{3v-u}{2} \biggr) \biggr\vert ^{q} \biggr)^{\frac{1}{q}} \\ &{}+ \biggl(\frac{\vartheta +5}{3\vartheta +9} \biggl\vert g'' \biggl(\frac{3u-v}{2} \biggr) \biggr\vert ^{q} + \frac{2\vartheta +4}{3\vartheta +9} \biggl\vert g'' \biggl( \frac{3v-u}{2} \biggr) \biggr\vert ^{q} \biggr)^{\frac{1}{q}} \biggr\} , \end{aligned} \\ & \gamma _{4} =\beta ^{\frac{1}{p}}(p+1,\vartheta p+1) \biggl( \frac{ \vert g'' (\frac{3u-v}{2} ) \vert ^{q}+ \vert g'' (\frac{3v-u}{2} ) \vert ^{q}}{2} \biggr)^{\frac{1}{q}}, \\ & \begin{aligned} \gamma _{5} ={}&\frac{1}{2} \biggl(\frac{1}{p+1} \biggr)^{\frac{1}{p}} \biggl(\frac{1}{(\vartheta q+1)(\vartheta q+2)} \biggr)^{ \frac{1}{q}}\\ &{}\times \biggl[ \biggl( \biggl\vert g'' \biggl( \frac{3u-v}{2} \biggr) \biggr\vert ^{q}+ (\vartheta q+1) \biggl\vert g'' \biggl(\frac{3v-u}{2} \biggr) \biggr\vert ^{q} \biggr)^{\frac{1}{q}} \\ &{}+ \biggl((\vartheta q+1) \biggl\vert g'' \biggl(\frac{3u-v}{2} \biggr) \biggr\vert ^{q}+ \biggl\vert g'' \biggl(\frac{3v-u}{2} \biggr) \biggr\vert ^{q} \biggr)^{ \frac{1}{q}} \biggr], \end{aligned} \\ & \begin{aligned} \gamma _{6} ={}& \biggl(\frac{2}{(\vartheta q+1)(\vartheta q+2)( \vartheta q+3)} \biggr)^{\frac{1}{q}}\\ &{}\times \biggl[ \biggl(2 \biggl\vert g'' \biggl( \frac{3u-v}{2} \biggr) \biggr\vert ^{q}+ (\vartheta q+1) \biggl\vert g'' \biggl(\frac{3v-u}{2} \biggr) \biggr\vert ^{q} \biggr)^{\frac{1}{q}} \\ &{}+ \biggl((\vartheta q+1) \biggl\vert g'' \biggl(\frac{3u-v}{2} \biggr) \biggr\vert ^{q}+ 2 \biggl\vert g'' \biggl(\frac{3v-u}{2} \biggr) \biggr\vert ^{q} \biggr) ^{\frac{1}{q}} \biggr] , \end{aligned} \end{aligned}$$

for$q>1$.

A few results for concave functions will be extended in the following theorems.

Theorem 7

Let$g:\mathcal{I}^{o}\subseteq \mathbb{R}\to \mathbb{R}$be a differentiable mapping on$\mathcal{I}^{o}$, $u,v\in \mathcal{I}^{o}$with$u< v$. Let$g'': [\frac{3u-v}{2},\frac{3v-u}{2} ] \to \mathbb{R}$be a continuous function on$[\frac{3u-v}{2}, \frac{3v-u}{2} ]$. If$|g''|^{q}$is concave on$[\frac{3u-v}{2}, \frac{3v-u}{2} ]$, then

$$\begin{aligned} \begin{aligned}[b] & \biggl\vert \frac{\varGamma (\vartheta +1)}{2(v-u)^{\vartheta }} \bigl[\mathfrak{J} _{u^{+}}^{\vartheta }g(v)+ \mathfrak{J}_{v^{-}}^{\vartheta }g(u) \bigr] -\frac{g (\frac{3v-u}{2} )+2g (\frac{u+v}{2} )+g (\frac{3u-v}{2} )}{4} \biggr\vert \\ &\quad \leq \frac{2(v-u)^{2}}{ \vartheta +1}\beta ^{\frac{1}{p}}(p+1,\vartheta p+1) \biggl\vert g'' \biggl(\frac{u+v}{2} \biggr) \biggr\vert , \end{aligned} \end{aligned}$$

(43)

where$q=\frac{p}{p-1}$such that$p\in \mathbb{R}$, $p>1$.

Proof

Applying Hölder’s inequality to (28), we get

$$\begin{aligned} & \biggl\vert \frac{\varGamma (\vartheta +1)}{2(v-u)^{\vartheta }} \bigl[\mathfrak{J} _{u^{+}}^{\vartheta }g(v)+\mathfrak{J}_{v^{-}}^{\vartheta }g(u) \bigr] -\frac{g (\frac{3v-u}{2} )+2g (\frac{u+v}{2} )+g (\frac{3u-v}{2} )}{4} \biggr\vert \\ &\quad \leq \biggl(\frac{(v-u)^{2}}{\vartheta +1} \int _{0}^{1}\ell ^{p} \bigl(1-\ell ^{\vartheta } \bigr)^{p}\,d\ell \biggr)^{\frac{1}{p}} \\ &\qquad {}\times\biggl\{ \biggl[ \int _{0}^{1} \biggl\vert g'' \biggl(\ell \biggl(\frac{3u-v}{2} \biggr) +(1-\ell ) \biggl( \frac{3v-u}{2} \biggr) \biggr) \biggr\vert \,d\ell \biggr] ^{\frac{1}{q}} \\ &\qquad {}+ \biggl[ \int _{0}^{1} \biggl\vert g'' \biggl(\ell \biggl(\frac{3v-u}{2} \biggr) +(1- \ell ) \biggl( \frac{3u-v}{2} \biggr) \biggr) \biggr\vert \,d\ell \biggr]^{ \frac{1}{q}} \biggr\} . \end{aligned}$$

(44)

By using the concavity of $|g''|^{q}$ on $[\frac{3u-v}{2}, \frac{3v-u}{2} ]$ and the integral Jensen’s inequality, we get

$$\begin{aligned} & \int _{0}^{1} \biggl\vert g'' \biggl(\ell \biggl(\frac{3u-v}{2} \biggr) +(1- \ell ) \biggl( \frac{3v-u}{2} \biggr) \biggr) \biggr\vert \,d\ell \\ &\quad \leq \biggl( \int _{0}^{1}\ell ^{0}\,d\ell \biggr) \biggl\vert g'' \biggl(\frac{ \int _{0}^{1}\ell ^{0} (\ell (\frac{3u-v}{2} ) +(1- \ell ) (\frac{3v-u}{2} ) )\,d\ell }{\int _{0}^{1}\ell ^{0}\,d\ell } \biggr) \biggr\vert ^{q} = \biggl\vert g'' \biggl(\frac{u+v}{2} \biggr) \biggr\vert ^{q} \end{aligned}$$

(45)

and analogously

$$\begin{aligned} & \int _{0}^{1} \biggl\vert g'' \biggl(\ell \biggl(\frac{3v-u}{2} \biggr) +(1- \ell ) \biggl( \frac{3u-v}{2} \biggr) \biggr) \biggr\vert \,d\ell \leq \biggl\vert g'' \biggl(\frac{u+v}{2} \biggr) \biggr\vert ^{q}. \end{aligned}$$

(46)

Thus substituting the obtained results of (45) and (46) in (44), we get (43) as desired. □

Theorem 8

Let$g:\mathcal{I}^{o}\subseteq \mathbb{R}\to \mathbb{R}$be a differentiable mapping on$\mathcal{I}^{o}$, $u,v\in \mathcal{I}^{o}$with$u< v$. Let$g'': [\frac{3u-v}{2},\frac{3v-u}{2} ] \to \mathbb{R}$be a continuous function on$[\frac{3u-v}{2}, \frac{3v-u}{2} ]$. Assume that$\frac{1}{p}+\frac{1}{q}=1$with$p\geq 1$such that$|g''|^{q}$is concave on$[\frac{3u-v}{2}, \frac{3v-u}{2} ]$. Then

$$\begin{aligned} \begin{aligned}[b] & \biggl\vert \frac{\varGamma (\vartheta +1)}{2(v-u)^{\vartheta }} \bigl[\mathfrak{J} _{u^{+}}^{\vartheta }g(v) + \mathfrak{J}_{v^{-}}^{\vartheta }g(u) \bigr]-\frac{g (\frac{3v-u}{2} ) +2g (\frac{u+v}{2} )+g (\frac{3u-v}{2} )}{4} \biggr\vert \\ &\quad \leq \frac{\vartheta (v-u)^{2}}{2(\vartheta +1)(\vartheta +2)}\\ &\qquad {}\times \biggl( \biggl\vert g'' \biggl(\frac{5\vartheta +7}{6(\vartheta +3)}u+\frac{ \vartheta +11}{6(\vartheta +3)}v \biggr) \biggr\vert + \biggl\vert g'' \biggl(\frac{ \vartheta +11}{6(\vartheta +3)}u+ \frac{5\vartheta +7}{6(\vartheta +3)}v \biggr) \biggr\vert \biggr). \end{aligned} \end{aligned}$$

(47)

Proof

From the concavity of $|g''|^{q}$ and the power-mean inequality, we have

$$\begin{aligned} \bigl\vert g''\bigl(\ell x+(1-\ell )y\bigr) \bigr\vert ^{q} &>\ell \bigl\vert g''(x) \bigr\vert ^{q}+(1-\ell ) \bigl\vert g''(y) \bigr\vert ^{q} \\ &\geq \bigl(\ell \bigl\vert g''(x) \bigr\vert +(1-\ell ) \bigl\vert g''(y) \bigr\vert \bigr)^{q} \end{aligned}$$

for all $x,y\in [\frac{3u-v}{2},\frac{3v-u}{2} ]$ and $\ell \in [0,1]$. This also gives

$$\begin{aligned} \bigl\vert g''\bigl(\ell x+(1-\ell )y\bigr) \bigr\vert \geq \ell \bigl\vert g''(x) \bigr\vert +(1-\ell ) \bigl\vert g''(y) \bigr\vert \end{aligned}$$

this means that $|g''|$ is also concave. Again by the Jensen integral inequality, we obtain

$$\begin{aligned} & \int _{0}^{1} \biggl\vert g'' \biggl(\ell \biggl(\frac{3u-v}{2} \biggr) +(1- \ell ) \biggl( \frac{3v-u}{2} \biggr) \biggr) \biggr\vert \,d\ell \\ &\quad \leq \biggl( \int _{0}^{1}\ell \bigl(1-\ell ^{\vartheta } \bigr)\,d\ell \biggr) \biggl\vert g'' \biggl( \frac{\int _{0}^{1}\ell (1- \ell ^{\vartheta } ) (\ell (\frac{3u-v}{2} ) +(1- \ell ) (\frac{3v-u}{2} ) )\,d\ell }{\int _{0}^{1}\ell (1-\ell ^{\vartheta } )\,d\ell } \biggr) \biggr\vert ^{q} \\ &\quad =\frac{\vartheta }{2(\vartheta +2)} \biggl\vert g'' \biggl( \frac{5\vartheta +7}{6(\vartheta +3)}u+\frac{\vartheta +11}{6(\vartheta +3)}v \biggr) \biggr\vert ^{q}, \end{aligned}$$

(48)

and analogously

$$\begin{aligned} & \int _{0}^{1} \biggl\vert g'' \biggl(\ell \biggl(\frac{3v-u}{2} \biggr) +(1- \ell ) \biggl( \frac{3u-v}{2} \biggr) \biggr) \biggr\vert \,d\ell \leq \biggl\vert g'' \biggl(\frac{\vartheta +11}{6(\vartheta +3)}u + \frac{5\vartheta +7}{6( \vartheta +3)}v \biggr) \biggr\vert ^{q}. \end{aligned}$$

(49)

Thus substituting the obtained results of (48) and (49) in (44), we get (47) as desired. □

4 Applications

In this section some applications are presented to demonstrate the usefulness of our obtained results in the previous sections.

4.1 Applications to special means

Let u and v are two arbitrary positive real numbers such that $u\neq v$, we consider the following special means [17].

• (i) The arithmetic mean:

  $$A=A(u,v)=\frac{u+v}{2}. $$

• (ii) The inverse arithmetic mean:

  $$H=H(u,v)=\frac{2}{\frac{1}{u}+\frac{1}{v}}, \quad u,v\neq 0. $$

• (iii) The geometric mean:

  $$G=G(u,v)=\sqrt{u v}. $$

• (iv) The logarithmic mean:

  $$L(u,v)=\frac{v-u}{\log (v)-\log (u)}, \quad u\neq v. $$

• (v) The generalized logarithmic mean:

  $$L_{n}(u,v)= \biggl[\frac{v^{n+1}-u^{n+1}}{(v-u)(n+1)} \biggr]^{ \frac{1}{n}}, \quad n\in \mathbb{Z}\setminus \{-1,0\}. $$

Proposition 1

Let$u,v\in \mathbb{R}$with$0< u< v$and$n\in \mathbb{Z}/\{0,-1\}$, then we have

$$\begin{aligned} \begin{aligned}[b] &\bigl\vert L_{n}^{n}(u,v)-A^{n}(u,v) \bigr\vert \\ &\quad \leq \min \{\delta _{1},\delta _{2}\} \frac{ \vert n \vert (v-u)}{2^{\frac{1}{p}}} \biggl[A \biggl( \biggl\vert \biggl(\frac{3u-v}{2} \biggr) \biggr\vert ^{(n-1)q}, \biggl\vert \biggl( \frac{3v-u}{2} \biggr) \biggr\vert ^{(n-1)q} \biggr) \biggr] ^{\frac{1}{q}}, \end{aligned} \end{aligned}$$

(50)

where$p=\frac{q}{q-1}$.

Proof

The proof of this proposition follows from Corollary 2 with $\vartheta =1$ and $g(x)=x^{n}$. □

Proposition 2

Let$u,v\in \mathbb{R}$with$0< u< v$and$n\in \mathbb{Z}/\{0,-1\}$, then we have

$$\begin{aligned} \begin{aligned}[b] &\bigl\vert G^{-1}(u,v)-A^{-1}(u,v) \bigr\vert \\ &\quad \leq \min \{\delta _{1},\delta _{2}\} \frac{v-u}{2^{1-\frac{1}{q}}} \biggl[A \biggl( \biggl\vert \biggl(\frac{3u-v}{2} \biggr) \biggr\vert ^{-2q}, \biggl\vert \biggl( \frac{3v-u}{2} \biggr) \biggr\vert ^{-2q} \biggr) \biggr] ^{\frac{1}{q}} \end{aligned} \end{aligned}$$

(51)

for$q\geq 1$.

Proof

The assertion follows from Corollary 2 with $\vartheta =1$ and $g(x)=\frac{1}{x}$. □

Proposition 3

Let$|n|\geq 3$and$u, v\in \mathbb{R}$with$0< u< v$, then

$$\begin{aligned} & \bigl\vert L_{n}^{n} \bigl(v^{-1},u^{-1}-H^{-n}(v,u) \bigr) \bigr\vert \\ &\quad \leq \min \{\delta _{1},\delta _{2}\} \frac{ \vert n \vert (v^{-1}-u^{-1} )}{2^{ \frac{1}{p}}} \biggl[H \biggl( \biggl\vert \biggl(\frac{3u-v}{2} \biggr) \biggr\vert ^{(n-1)q}, \biggl\vert \biggl( \frac{3v-u}{2} \biggr) \biggr\vert ^{(n-1)q} \biggr) \biggr] ^{\frac{-1}{q}} \end{aligned}$$

(52)

and

$$\begin{aligned} &\bigl\vert L^{-1} \bigl(v^{-1},u^{-1} \bigr)-H(v,u) \bigr\vert \\ &\quad \leq \min \{ \delta _{1},\delta _{2}\} \frac{v^{-1}-u^{-1}}{2^{1-\frac{1}{q}}} \biggl[A \biggl( \biggl\vert \biggl( \frac{3u-v}{2} \biggr) \biggr\vert ^{-2q}, \biggl\vert \biggl(\frac{3v-u}{2} \biggr) \biggr\vert ^{-2q} \biggr) \biggr]^{\frac{1}{q}} \end{aligned}$$

(53)

for$q\geq 1$.

Proof

Observe that $A^{-1} (u^{-1},v^{-1} )=H(u,v)=\frac{2}{ \frac{1}{u}+\frac{1}{v}}$. So, making the change of variables $u\to v^{-1}$ and $v\to u^{-1}$ in the inequalities (50) and (51), we can deduce the desired inequalities (52) and (53), respectively. □

Proposition 4

Let$u,v\in \mathbb{R}$with$0< u< v$and$n\in \mathbb{Z}/\{0,-1\}$, then we have

$$\begin{aligned} \bigl\vert G^{-2}(u,v)-A^{-2}(u,v) \bigr\vert &\leq \min \{\delta _{1},\delta _{2}\} \frac{v-u}{2^{\frac{2}{p}-1}} \biggl[A \biggl( \biggl\vert \biggl(\frac{3u-v}{2} \biggr) \biggr\vert ^{-3q}, \biggl\vert \biggl(\frac{3v-u}{2} \biggr) \biggr\vert ^{-3q} \biggr) \biggr] ^{\frac{1}{q}}, \end{aligned}$$

where$p=\frac{q}{q-1}$.

Proof

The proof of this proposition follows from Corollary 2 with $\vartheta =1$ and $g(x)=\frac{1}{x^{2}}$. □

4.2 The midpoint formula

Let d be a partition of the interval $[u,v]$ such that $u=x_{0}< x _{1}< x_{2}<\cdots <x_{m-1}<x_{m}=v$. Consider the quadrature formula [17]

$$\begin{aligned} \int _{u}^{v}g(x)\,dx=\mathcal{T}(g,d)+ \mathcal{E}(g,d), \end{aligned}$$

(54)

where $\mathcal{E}(g,d)$ represents the associated approximation error and

$$\begin{aligned} \mathcal{T}(g,d) &=\sum_{k=0}^{m-1}g \biggl(\frac{x_{i}+x_{k+1)}}{2} \biggr) (x _{k+1}-x_{k}) \end{aligned}$$

is the midpoint version.

Proposition 5

Let$g:\mathcal{I}^{o}\subseteq \mathbb{R}\to \mathbb{R}$be a differentiable mapping on$\mathcal{I}^{o}$, $u,v\in \mathcal{I}^{o}$with$u< v$. Let$g'\in L_{1} [\frac{3u-v}{2},\frac{3v-u}{2} ]$and$g': [\frac{3u-v}{2},\frac{3v-u}{2} ]\to \mathbb{R}$be a continuous function on$[\frac{3u-v}{2},\frac{3v-u}{2} ]$. If$|g'|^{q}, q\geq 1$is a convex function, then, for every partitiondof$[\frac{3u-v}{2},\frac{3v-u}{2} ]$in (54), we have

$$\begin{aligned} \begin{aligned}[b] &\bigl\vert \mathcal{E}(f,d) \bigr\vert \\ &\quad \leq \frac{1}{2}\min \{\delta _{1}, \delta _{2}\}\sum_{k=1}^{m-1} (x_{k+1}-x_{k})^{2} \biggl( \biggl\vert g' \biggl(\frac{3x _{k}-x_{k+1}}{2} \biggr) \biggr\vert ^{q} + \biggl\vert g' \biggl(\frac{3x_{k+1}-x _{k}}{2} \biggr) \biggr\vert ^{q} \biggr)^{\frac{1}{q}} \\ &\quad \leq \min \{ \delta _{1},\delta _{2}\}\sum _{k=1}^{m-1}(x_{k+1}-x_{k})^{2} \max \biggl( \biggl\vert g' \biggl(\frac{3x_{k}-x_{k+1}}{2} \biggr) \biggr\vert , \biggl\vert g' \biggl(\frac{3x _{k+1}-x_{k}}{2} \biggr) \biggr\vert \biggr). \end{aligned} \end{aligned}$$

(55)

Proof

Applying Corollary 2 with $\vartheta =1$ on the subinterval $[\frac{3x_{k}-x_{k+1}}{2},\frac{3x_{k+1}-x_{k}}{2} ]$ ($k=0,1,\ldots, m-1$) of the partition d, we obtain

$$\begin{aligned} & \biggl\vert \int _{x_{k}}^{x_{k+1}} g(x)\,dx-(x_{k+1}-x_{k})g \biggl(\frac{x _{k}+x_{k+1}}{2} \biggr) \biggr\vert \\ &\quad \leq \frac{1}{2}\min \{\delta _{1},\delta _{2}\}(x_{k+1}-x_{k})^{2} \biggl( \biggl\vert g' \biggl(\frac{3x_{k}-x_{k+1}}{2} \biggr) \biggr\vert ^{q} + \biggl\vert g' \biggl( \frac{3x_{k+1}-x_{k}}{2} \biggr) \biggr\vert ^{q} \biggr) ^{\frac{1}{q}}. \end{aligned}$$

Summing over k from 0 to $m-1$ and taking into account that $|g'|$ is convex, we obtain, by the triangle inequality,

$$\begin{aligned} &\biggl\vert \int _{u}^{v} g(x)\,dx-T(g,d) \biggr\vert \\ &\quad = \Biggl\vert \sum_{k=0}^{m-1} \biggl[ \int _{x_{k}}^{x_{k+1}} g(x)\,dx-(x_{k+1}-x_{k})g \biggl(\frac{x _{k}+x_{k+1}}{2} \biggr) \biggr] \Biggr\vert \\ &\quad \leq \sum_{k=0}^{m-1} \biggl\vert \int _{x_{k}}^{x_{k+1}}g(x)\,dx-(x_{k+1}-x _{k})g \biggl(\frac{x_{k}+x_{k+1}}{2} \biggr) \biggr\vert \\ &\quad \leq \frac{1}{2}\min \{\delta _{1},\delta _{2}\}\sum_{k=1}^{m-1} (x _{k+1}-x_{k})^{2} \biggl( \biggl\vert g' \biggl(\frac{3x_{k}-x_{k+1}}{2} \biggr) \biggr\vert ^{q} + \biggl\vert g' \biggl(\frac{3x_{k+1}-x_{k}}{2} \biggr) \biggr\vert ^{q} \biggr) ^{\frac{1}{q}} \\ &\quad \leq \min \{\delta _{1},\delta _{2}\}\sum _{k=1}^{m-1}(x_{k+1}-x_{k})^{2} \max \biggl( \biggl\vert g' \biggl(\frac{3x_{k}-x_{k+1}}{2} \biggr) \biggr\vert , \biggl\vert g' \biggl(\frac{3x_{k+1}-x_{k}}{2} \biggr) \biggr\vert \biggr). \end{aligned}$$

This completes the proof of (55). □

5 Conclusion

In this paper, we generalized the modified Hermite–Hadamard inequality obtained by Mehrez and Agarwal in [12], it can be found in Lemma 3 and Theorems 1–6. Corollaries 4–7 confirm that our results modified the existing results of [12]. Furthermore, Theorems 7–8 modified the existing Theorems 5–6 of [5].