Skip to main content

Hermite–Hadamard type inequalities for m-convex and \(( \alpha , m)\)-convex functions

Abstract

In this paper, some new inequalities of the Hermite–Hadamard type for the classes of functions whose derivatives’ absolute values are m-convex and \(( \alpha , m) \)-convex are obtained. The results obtained in this work extend and improve the corresponding ones in the literature. Some applications to special means of real numbers are also given.

Introduction

Let a real function f be defined on some nonempty interval I of real numbers. The function \(f:I\rightarrow \mathbb{R} \) is said to be convex if the inequality

$$ f \bigl( tu+ ( 1-t ) v \bigr) \leq tf ( u ) + ( 1-t ) f ( v ) $$

holds for all \(u,v\in I\) and \(t\in [ 0,1 ] \).

Convexity in connection with integral inequalities is an interesting research area since much attention has been given to studying the concept of convexity and its variant forms in recent years. Some of the most useful inequalities related to the integral mean of a convex function are Hermite–Hadamard’s inequality, Jensen’s inequality, and Hardy’s inequality (see [8, 2325, 31]). Hermite–Hadamard’s inequality provides a necessary and sufficient condition for a function to be convex. This well-known result of Hermite and Hadamard is stated as follows:

If f is a convex function on some nonempty interval I of real numbers and \([ u,v ] \in I\) with \(u< v\), then

$$ f \biggl( \frac{u+v}{2} \biggr) \leq \frac{1}{v-u} \int _{u}^{v}f ( x ) \,dx\leq \frac{f ( u ) +f ( v ) }{2}. $$
(1)

This double inequality may be regarded as a refinement of the concept of convexity, and it follows easily from Jensen’s inequality. Recently, a remarkable variety of generalizations and extensions have been considered for the concept of convexity, and related Hermite–Hadamard type integral inequalities have been studied by many researchers (see, for example, [1, 2, 6, 10, 11, 13, 17, 19, 21, 26, 28, 29, 32, 33] and the references cited therein).

Preliminaries

We recall the following well-known results and concepts.

Toader [36] introduced the concept of m-convex functions as follows.

Definition 2.1

([36])

Let \(m\in [ 0,1 ] \). The function \(f: [ 0,v ] \rightarrow \mathbb{R} \) is said to be m-convex if

$$ f \bigl( tx+m ( 1-t ) y \bigr) \leq tf ( x ) +m ( 1-t ) f ( y ) $$

is satisfied for every \(x,y\in [ 0,v ] \) and \(t\in [ 0,1 ] \).

It can be easily seen that for \(m=1\), m-convexity reduces to the classical convexity of functions.

Miheşan [22] defined the concept of \(( \alpha ,m ) \)-convex functions as follows.

Definition 2.2

([22])

Let \(\alpha ,m \in [ 0,1 ]\). The function \(f: [ 0,v ] \rightarrow \mathbb{R} \) is said to be \(( \alpha ,m ) \)-convex if

$$ f \bigl( tx+m ( 1-t ) y \bigr) \leq t^{\alpha }f ( x ) +m \bigl( 1-t^{\alpha } \bigr) f ( y ) $$

is satisfied for every \(x,y\in [ 0,v ] \) and \(t\in [ 0,1 ] \).

Obviously, \(( \alpha ,m ) \)-convexity reduces to m-convexity for \(\alpha =1\) and classical convexity for \(\alpha =m=1\).

For recent results, improvements and generalizations of the concepts of m-convexity and \((\alpha ,m)\)-convexity, please refer to the monographs [35, 9, 14, 15, 18, 20, 27, 30, 34, 35, 37, 38].

In [7], Dragomir and Agarwal proved the following result connected with the right part of (1).

Lemma 2.1

([7])

Let\(f:I^{\circ }\subseteq \mathbb{R} \rightarrow \mathbb{R} \)be a differentiable function on\(I^{\circ }\) (interior ofI) and\(u,v\in I^{\circ }\)with\(u< v\). If\(f^{\prime }\in L [ a,b ] \), then

$$ \frac{f ( u ) +f ( v ) }{2}-\frac{1}{v-u}\int _{u}^{v}f ( x ) \,dx=\frac{v-u}{2} \int _{0}^{1} ( 1-2t ) f^{\prime } \bigl( tu+ ( 1-t ) v \bigr) \,dt. $$

Bakula et al. [4] established the following result by using Lemma 2.1 and Hölder’s integral inequality.

Theorem 2.1

Suppose thatIis an open real interval such that\([ 0,\infty ) \subset I\), and let\(0\leq u< v<\infty \). Consider the differentiable function\(f:I\rightarrow \mathbb{R} \)onIsuch that\(f^{\prime }\in L [ u,v ]\). If\(\vert f^{\prime } \vert ^{q}\)is anm-convex function on\([ u,v ] \)for some\(m\in ( 0,1 ] \)and\(q\geq 1\), then

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

İşcan [12] obtained the following integral inequality which gives better results than the classical Hölder integral inequality.

Theorem 2.2

(Hölder–İşcan integral inequality)

Letfandgbe two real functions defined on\([ u,v ] \). If\(\vert f \vert ^{p}\)and\(\vert g \vert ^{q}\)are integrable functions on\([ u,v ] \)for\(p>1\)and\(1/p+1/q=1\), then

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

İşcan [12] proved the following Hermite–Hadamard type inequality by using Lemma 2.1 and the Hölder–İşcan integral inequality.

Theorem 2.3

Suppose that\(f:I^{\circ }\subseteq \mathbb{R} \rightarrow \mathbb{R} \)is a differentiable function on\(I^{\circ }\)and\(u,v\in I^{\circ }\)with\(u< v\). If\(\vert f^{\prime } \vert ^{q}\)is a convex function on\([ u,v ]\), then

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

In [16], a different representation of the Hölder–İşcan integral inequality was given as follows.

Theorem 2.4

(Improved power-mean integral inequality)

Letfandgbe two real functions defined on\([ u,v ]\). If\(\vert f \vert \), \(\vert f \vert \vert g \vert ^{q}\)are integrable functions on\([ u,v ] \)for\(q\geq 1\), then

$$\begin{aligned}& \int _{u}^{v} \bigl\vert f ( x ) g ( x ) \bigr\vert \,dx \\& \quad\leq \frac{1}{v-u} \biggl\{ \biggl( \int _{u}^{v} ( v-x ) \bigl\vert f ( x ) \bigr\vert \,dx \biggr) ^{1- \frac{1}{q}} \biggl( \int _{u}^{v} ( v-x ) \bigl\vert f ( x ) \bigr\vert \bigl\vert g ( x ) \bigr\vert ^{q}\,dx \biggr) ^{\frac{1}{q}} \\& \qquad {}+ \biggl( \int _{u}^{v} ( x-u ) \bigl\vert f ( x ) \bigr\vert \,dx \biggr) ^{1- \frac{1}{q}} \biggl( \int _{u}^{v} ( x-u ) \bigl\vert f ( x ) \bigr\vert \bigl\vert g ( x ) \bigr\vert ^{q}\,dx \biggr) ^{\frac{1}{q}} \biggr\} \\& \quad \leq \biggl( \int _{u}^{v} \bigl\vert f ( x ) \bigr\vert \,dx \biggr) ^{1-\frac{1}{q}} \biggl( \int _{u}^{v} \bigl\vert f ( x ) \bigr\vert \bigl\vert g ( x ) \bigr\vert ^{q}\,dx \biggr) ^{\frac{1}{q}}. \end{aligned}$$

Main results

Now we are in a position to establish some new Hermite–Hadamard type inequalities for the classes of m-convex and \(( \alpha , m)\)-convex functions.

Theorem 3.1

Suppose thatIis an open real interval such that\([ 0,\infty ) \subset I\), and let\(0\leq u< v<\infty \). Consider the differentiable function\(f:I\rightarrow \mathbb{R} \)onIsuch that\(f^{\prime }\in L [ u,v ]\). If\(\vert f^{\prime } \vert ^{q}\)is anm-convex function on\([ u,v ] \)for some\(m\in ( 0,1 ] \)and\(q>1\), \(q=\frac{p}{p-1}\), then

$$\begin{aligned} \biggl\vert \frac{f ( u ) +f ( v ) }{2}- \frac{1}{v-u}\int _{u}^{v}f ( x ) \,dx \biggr\vert \leq \frac{v-u}{4 ( p+1 ) ^{\frac{1}{p}}} \bigl( \lambda _{1}^{\frac{1}{q}}+ \lambda _{2}^{\frac{1}{q}} \bigr), \end{aligned}$$
(3)

where

$$\begin{aligned}& \lambda _{1}=\min \biggl\{ \frac{2 \vert f^{\prime } ( u ) \vert ^{q}+m \vert f^{\prime } ( \frac{v}{m} ) \vert ^{q}}{3},\frac{2m \vert f^{\prime } ( \frac{u}{m} ) \vert ^{q}+ \vert f^{\prime } ( v ) \vert ^{q}}{3} \biggr\} , \\& \lambda _{2}=\min \biggl\{ \frac{ \vert f^{\prime } ( u ) \vert ^{q}+2m \vert f^{\prime } ( \frac{v}{m} ) \vert ^{q}}{3},\frac{m \vert f^{\prime } ( \frac{u}{m} ) \vert ^{q}+2 \vert f^{\prime } ( v ) \vert ^{q}}{3} \biggr\} . \end{aligned}$$

Proof

From Lemma 2.1 and the Hölder–İşcan integral inequality, we have

$$\begin{aligned}& \biggl\vert \frac{f ( u ) +f ( v ) }{2}- \frac{1}{v-u}\int _{u}^{v}f ( x ) \,dx \biggr\vert \\& \quad \leq \frac{v-u}{2} \int _{0}^{1} \vert 1-2t \vert \bigl\vert f^{\prime } \bigl( tu+ ( 1-t ) v \bigr) \bigr\vert \,dt \\& \quad \leq \frac{v-u}{2} \biggl\{ \biggl( \int _{0}^{1} ( 1-t ) \vert 1-2t \vert ^{p}\,dt \biggr) ^{\frac{1}{p}} \biggl( \int _{0}^{1} ( 1-t ) \bigl\vert f^{\prime } \bigl( tu+ ( 1-t ) v \bigr) \bigr\vert ^{q}\,dt \biggr) ^{ \frac{1}{q}} \\& \qquad {}+ \biggl( \int _{0}^{1}t \vert 1-2t \vert ^{p} \,dt \biggr) ^{ \frac{1}{p}} \biggl( \int _{0}^{1}t \bigl\vert f^{\prime } \bigl( tu+ ( 1-t ) v \bigr) \bigr\vert ^{q}\,dt \biggr) ^{\frac{1}{q}} \biggr\} . \end{aligned}$$

From m-convexity of \(\vert f^{\prime } \vert ^{q}\) on \([ u,v ] \) for all \(t\in [ 0,1 ] \) we have

$$\begin{aligned} \int _{0}^{1}t \bigl\vert f^{\prime } \bigl( tu+ ( 1-t ) v \bigr) \bigr\vert ^{q}\,dt =& \int _{0}^{1}t \biggl\vert f^{\prime } \biggl( tu+m ( 1-t ) \frac{v}{m} \biggr) \biggr\vert ^{q}\,dt \\ \leq &\frac{2 \vert f^{\prime } ( u ) \vert ^{q}+m \vert f^{\prime } ( \frac{v}{m} ) \vert ^{q}}{6}, \end{aligned}$$

and analogously

$$\begin{aligned} \int _{0}^{1}t \bigl\vert f^{\prime } \bigl( tu+ ( 1-t ) v \bigr) \bigr\vert ^{q}\,dt =& \int _{0}^{1}t \biggl\vert f^{\prime } \biggl( mt\frac{u}{m}+ ( 1-t ) v \biggr) \biggr\vert ^{q} \,dt \\ \leq &\frac{2m \vert f^{\prime } ( \frac{u}{m} ) \vert ^{q}+ \vert f^{\prime } ( v ) \vert ^{q}}{6}. \end{aligned}$$

So we can write

$$\begin{aligned}& \int _{0}^{1}t \bigl\vert f^{\prime } \bigl( tu+ ( 1-t ) v \bigr) \bigr\vert ^{q}\,dt \\& \quad \leq \min \biggl\{ \frac{2 \vert f^{\prime } ( u ) \vert ^{q}+m \vert f^{\prime } ( \frac{v}{m} ) \vert ^{q}}{6}, \frac{2m \vert f^{\prime } ( \frac{u}{m} ) \vert ^{q}+ \vert f^{\prime } ( v ) \vert ^{q}}{6} \biggr\} . \end{aligned}$$
(4)

Similarly, we have

$$\begin{aligned}& \int _{0}^{1} ( 1-t ) \bigl\vert f^{\prime } \bigl( tu+ (1-t ) v \bigr) \bigr\vert ^{q}\,dt \\& \quad \leq \min \biggl\{ \frac{ \vert f^{\prime } ( u ) \vert ^{q}+2m \vert f^{\prime } ( \frac{v}{m} ) \vert ^{q}}{3},\frac{m \vert f^{\prime } ( \frac{u}{m} ) \vert ^{q}+2 \vert f^{\prime } ( v ) \vert ^{q}}{3} \biggr\} . \end{aligned}$$
(5)

Taking into account that

$$\begin{aligned} \int _{0}^{1}t \vert 1-2t \vert ^{p} \,dt =& \int _{0}^{1} ( 1-t ) \vert 1-2t \vert ^{p}\,dt \\ =&\frac{1}{2 ( p+1 ) }, \end{aligned}$$
(6)

we deduce from (4), (5), and (6) inequality (3). □

Remark 3.1

Choosing \(m=1\) in inequality (3), we get inequality (2).

Theorem 3.2

Suppose thatIis an open real interval such that\([ 0,\infty ) \subset I\), and let\(0\leq u< v<\infty \). Consider the differentiable function\(f:I\rightarrow \mathbb{R} \)onIsuch that\(f^{\prime }\in L [ u,v ]\). If\(\vert f^{\prime } \vert ^{q}\)is anm-convex function on\([ u,v ] \)for some\(m\in ( 0,1 ] \)and\(q\geq 1\), then

$$\begin{aligned} \biggl\vert \frac{f ( u ) +f ( v ) }{2}- \frac{1}{v-u}\int _{u}^{v}f ( x ) \,dx \biggr\vert \leq \frac{v-u}{8} \bigl( \mu _{1}^{\frac{1}{q}}+\mu _{2}^{\frac{1}{q}} \bigr), \end{aligned}$$
(7)

where

$$\begin{aligned}& \mu _{1}=\min \biggl\{ \frac{3 \vert f^{\prime } ( u ) \vert ^{q}+m \vert f^{\prime } ( \frac{v}{m} ) \vert ^{q}}{4}, \frac{3m \vert f^{\prime } ( \frac{u}{m} ) \vert ^{q}+ \vert f^{\prime } ( v ) \vert ^{q}}{4} \biggr\} , \\& \mu _{2}=\min \biggl\{ \frac{ \vert f^{\prime } ( u ) \vert ^{q}+3m \vert f^{\prime } ( \frac{v}{m} ) \vert ^{q}}{4}, \frac{m \vert f^{\prime } ( \frac{u}{m} ) \vert ^{q}+3 \vert f^{\prime } ( v ) \vert ^{q}}{4} \biggr\} . \end{aligned}$$

Proof

Using Lemma 2.1 and an improved power-mean integral inequality, we have

$$\begin{aligned}& \biggl\vert \frac{f ( u ) +f ( v ) }{2}- \frac{1}{v-u}\int _{u}^{v}f ( x ) \,dx \biggr\vert \\& \quad \leq \frac{v-u}{2.4^{\frac{1}{p}}} \biggl\{ \biggl( \int _{0}^{1} ( 1-t ) \vert 1-2t \vert \bigl\vert f^{ \prime } \bigl( tu+ ( 1-t ) v \bigr) \bigr\vert ^{q} \,dt \biggr) ^{\frac{1}{q}} \\& \qquad {}+ \biggl( \int _{0}^{1} ( 1-t ) \vert 1-2t \vert \bigl\vert f^{\prime } \bigl( tu+ ( 1-t ) v \bigr) \bigr\vert ^{q} \,dt \biggr) ^{\frac{1}{q}} \biggr\} . \end{aligned}$$

By m-convexity of \(\vert f^{\prime } \vert ^{q}\) on \([ u,v ] \) for all \(t\in [ 0,1 ] \) we have

$$\begin{aligned} \int _{0}^{1}t \vert 1-2t \vert \bigl\vert f^{\prime } \bigl( tu+ ( 1-t ) v \bigr) \bigr\vert ^{q}\,dt \leq \frac{3 \vert f^{\prime } ( u ) \vert ^{q}+m \vert f^{\prime } ( \frac{v}{m} ) \vert ^{q}}{16}, \end{aligned}$$

and analogously

$$ \int _{0}^{1}t \vert 1-2t \vert \bigl\vert f^{\prime } \bigl( tu+ ( 1-t ) v \bigr) \bigr\vert ^{q}\,dt\leq \frac{3m \vert f^{\prime } ( \frac{u}{m} ) \vert ^{q}+ \vert f^{\prime } ( v ) \vert ^{q}}{16}. $$

So we obtain

$$\begin{aligned}& \int _{0}^{1}t \vert 1-2t \vert \bigl\vert f^{\prime } \bigl( tu+ ( 1-t ) v \bigr) \bigr\vert ^{q}\,dt \\& \quad \leq \min \biggl\{ \frac{3 \vert f^{\prime } ( u ) \vert ^{q}+m \vert f^{\prime } ( \frac{v}{m} ) \vert ^{q}}{16}, \frac{3m \vert f^{\prime } ( \frac{u}{m} ) \vert ^{q}+ \vert f^{\prime } ( v ) \vert ^{q}}{16} \biggr\} . \end{aligned}$$
(8)

Similarly, we have

$$\begin{aligned}& \int _{0}^{1} ( 1-t ) \vert 1-2t \vert \bigl\vert f^{\prime } \bigl( tu+ ( 1-t ) v \bigr) \bigr\vert ^{q} \,dt \\& \quad \leq \min \biggl\{ \frac{ \vert f^{\prime } ( u ) \vert ^{q}+3m \vert f^{\prime } ( \frac{v}{m} ) \vert ^{q}}{16},\frac{m \vert f^{\prime } ( \frac{u}{m} ) \vert ^{q}+3 \vert f^{\prime } ( v ) \vert ^{q}}{16} \biggr\} . \end{aligned}$$
(9)

By using inequalities (8), (9) and the fact that \(\int _{0}^{1}t \vert 1-2t \vert \,dt =\frac{1}{4}\), we get inequality (7). □

Corollary 3.1

Let the assumptions of Theorem 3.2be satisfied. If we take\(m=1\), then inequality (7) becomes the following inequality:

$$\begin{aligned}& \biggl\vert \frac{f ( u ) +f ( v ) }{2}- \frac{1}{v-u}\int _{u}^{v}f ( x ) \,dx \biggr\vert \\& \quad \leq \frac{v-u}{8} \biggl\{ \biggl( \frac{3 \vert f^{\prime } ( u ) \vert ^{q}+ \vert f^{\prime } ( v ) \vert ^{q}}{4} \biggr)^{\frac{1}{q}}+ \biggl( \frac{ \vert f^{\prime } ( u ) \vert ^{q}+3 \vert f^{\prime } ( v ) \vert ^{q}}{4} \biggr)^{\frac{1}{q}} \biggr\} . \end{aligned}$$
(10)

Theorem 3.3

Suppose thatIis an open real interval such that\([ 0,\infty ) \subset I\), and let\(0\leq u< v<\infty \). Consider the differentiable function\(f:I\rightarrow \mathbb{R} \)onIsuch that\(f^{\prime }\in L [ u,v ]\). If\(\vert f^{\prime } \vert ^{q}\)is an\((\alpha , m)\)-convex function on\([ u,v ] \)for some\(\alpha ,m\in ( 0,1 ] \)and\(q>1\), \(q=\frac{p}{p-1}\), then

$$\begin{aligned}& \biggl\vert \frac{f ( u ) +f ( v ) }{2}- \frac{1}{v-u}\int _{u}^{v}f ( x ) \,dx \biggr\vert \\& \quad \leq \frac{v-u}{4 ( p+1 ) ^{\frac{1}{p}}} \bigl( \varphi _{1}^{\frac{1}{q}}+ \varphi _{2}^{\frac{1}{q}} \bigr) \leq \frac{v-u}{4} \bigl( \varphi _{1}^{ \frac{1}{q}}+\varphi _{2}^{\frac{1}{q}} \bigr), \end{aligned}$$
(11)

where

$$\begin{aligned}& \varphi _{1}=\min \biggl\{ \frac{\alpha \vert f^{\prime } ( v ) \vert ^{q}+2m \vert f^{\prime } ( \frac{u}{m} ) \vert ^{q}}{\alpha +2}, \frac{2 \vert f^{\prime } ( u ) \vert ^{q}+m\alpha \vert f^{\prime } ( \frac{v}{m} ) \vert ^{q}}{\alpha +2} \biggr\} , \\& \varphi _{2}=\min \biggl\{ \frac{2 \vert f^{\prime } ( u ) \vert ^{q}+m\alpha ( \alpha +3 ) \vert f^{\prime } ( \frac{v}{m} ) \vert ^{q}}{ ( \alpha +1 ) ( \alpha +2 ) }, \frac{2m \vert f^{\prime } ( \frac{u}{m} ) \vert ^{q}+\alpha ( \alpha +3 ) \vert f^{\prime } ( v ) \vert ^{q}}{ ( \alpha +1 ) ( \alpha +2 ) } \biggr\} . \end{aligned}$$

Proof

Using Lemma 2.1 and the Hölder–İşcan integral inequality, we have

$$\begin{aligned}& \biggl\vert \frac{f ( u ) +f ( v ) }{2}- \frac{1}{v-u}\int _{u}^{v}f ( x ) \,dx \biggr\vert \\& \quad \leq \frac{v-u}{2} \biggl\{ \biggl( \int _{0}^{1} ( 1-t ) \vert 1-2t \vert ^{p}\,dt \biggr) ^{\frac{1}{p}} \biggl( \int _{0}^{1} ( 1-t ) \bigl\vert f^{\prime } \bigl( tu+ ( 1-t ) v \bigr) \bigr\vert ^{q}\,dt \biggr) ^{ \frac{1}{q}} \\& \qquad {}+ \biggl( \int _{0}^{1}t \vert 1-2t \vert ^{p} \,dt \biggr) ^{ \frac{1}{p}} \biggl( \int _{0}^{1}t \bigl\vert f^{\prime } \bigl( tu+ ( 1-t ) v \bigr) \bigr\vert ^{q}\,dt \biggr) ^{\frac{1}{q}} \biggr\} . \end{aligned}$$

By \(( \alpha ,m ) \)-convexity of \(\vert f^{\prime } \vert ^{q}\) on \([ u,v ] \) for all \(t\in [ 0,1 ] \), we get

$$\begin{aligned}& \int _{0}^{1}t \bigl\vert f^{\prime } \bigl( tu+ ( 1-t ) v \bigr) \bigr\vert ^{q}\,dt \\& \quad \leq \min \biggl\{ \frac{2 \vert f^{\prime } ( u ) \vert ^{q}+m\alpha \vert f^{\prime } ( \frac{v}{m} ) \vert ^{q}}{2 ( \alpha +2 ) } , \frac{2m \vert f^{\prime } ( \frac{u}{m} ) \vert ^{q}+\alpha \vert f^{\prime } ( v ) \vert ^{q}}{2 ( \alpha +2 ) } \biggr\} \end{aligned}$$
(12)

and

$$\begin{aligned}& \int _{0}^{1} ( 1-t ) \bigl\vert f^{\prime } \bigl( tu+ ( 1-t ) v \bigr) \bigr\vert ^{q}\,dt \\& \quad \leq \min \biggl\{ \frac{2 \vert f^{\prime } ( u ) \vert ^{q}+m\alpha ( \alpha +3 ) \vert f^{\prime } ( \frac{v}{m} ) \vert ^{q}}{2 ( \alpha +1 ) ( \alpha +2 ) }, \frac{2m \vert f^{\prime } ( \frac{u}{m} ) \vert ^{q}+\alpha ( \alpha +3 ) \vert f^{\prime } ( v ) \vert ^{q}}{2 ( \alpha +1 ) ( \alpha +2 ) } \biggr\} .\quad \end{aligned}$$
(13)

The proof of the first inequality in (11) is completed by the combination of inequalities (12) and (13). The proof of the second inequality in (11) is completed using the fact

$$ \frac{1}{2}< \biggl( \frac{1}{p+1} \biggr) ^{\frac{1}{p}}< 1 $$

for \(p>1\). □

Corollary 3.2

Let the assumptions of Theorem 3.3be satisfied. If we take\(m=1\), then inequality (11) becomes the following inequality:

$$\begin{aligned}& \biggl\vert \frac{f ( u ) +f ( v ) }{2}- \frac{1}{v-u} \int f ( x ) \,dx \biggr\vert \\& \quad \leq \frac{v-u}{4 ( p+1 ) ^{\frac{1}{p}}} \biggl[ \biggl( \frac{2 \vert f^{\prime } ( u ) \vert ^{q}+\alpha \vert f^{\prime } ( v ) \vert ^{q}}{\alpha +2} \biggr) ^{\frac{1}{q}}+ \biggl( \frac{2 \vert f^{\prime } ( u ) \vert ^{q}+\alpha ( \alpha +3 ) \vert f^{\prime } ( v ) \vert ^{q}}{2 ( \alpha +1 ) ( \alpha +2 ) } \biggr) ^{\frac{1}{q}} \biggr]. \end{aligned}$$

Remark 3.2

Inequality (11) yields the right-hand side of Hermite–Hadamard inequality (3) for \(\alpha =1\).

Remark 3.3

Choosing \((\alpha ,m)=(1,1)\) in the first part of (11), we get inequality (2).

Theorem 3.4

Suppose thatIis an open real interval such that\([ 0,\infty ) \subset I\), and let\(0\leq u< v<\infty \). Consider the differentiable function\(f:I\rightarrow \mathbb{R} \)onIsuch that\(f^{\prime }\in L [ u,v ]\). If\(\vert f^{\prime } \vert ^{q}\)is an\((\alpha , m)\)-convex function on\([ u,v ] \)for some\(\alpha ,m\in ( 0,1 ] \)and\(q\geq 1\), then

$$\begin{aligned} \biggl\vert \frac{f ( u ) +f ( v ) }{2}- \frac{1}{v-u}\int _{u}^{v}f ( x ) \,dx \biggr\vert \leq \frac{v-u}{2.4^{\frac{1}{p}}} \bigl( \tau _{1}^{\frac{1}{q}}+\tau _{2}^{ \frac{1}{q}} \bigr), \end{aligned}$$
(14)

where

$$\begin{aligned}& \tau _{1}=\min \biggl\{ \kappa _{1} \bigl\vert f^{\prime } ( u ) \bigr\vert ^{q}+m\kappa _{2} \biggl\vert f^{\prime } \biggl( \frac{v}{m} \biggr) \biggr\vert ^{q},m\kappa _{1} \biggl\vert f^{ \prime } \biggl( \frac{u}{m} \biggr) \biggr\vert ^{q}+\kappa _{2} \bigl\vert f^{\prime } ( v ) \bigr\vert ^{q} \biggr\} , \\& \tau _{2}=\min \biggl\{ \kappa _{1}^{\ast } \bigl\vert f^{\prime } ( u ) \bigr\vert ^{q}+m\kappa _{2}^{\ast } \biggl\vert f^{ \prime } \biggl( \frac{v}{m} \biggr) \biggr\vert ^{q},m\kappa _{1}^{ \ast } \biggl\vert f^{\prime } \biggl( \frac{u}{m} \biggr) \biggr\vert ^{q}+\kappa _{2}^{\ast } \bigl\vert f^{\prime } ( v ) \bigr\vert ^{q} \biggr\} \end{aligned}$$

such that

$$\begin{aligned}& \kappa _{1}= \frac{1}{ ( \alpha +2 ) ( \alpha +3 ) } \biggl[ \alpha +1+ \biggl( \frac{1}{2} \biggr) ^{\alpha +1} \biggr] ,\qquad \kappa _{2}= \frac{1}{4}-\kappa _{1} , \\& \kappa _{1}^{\ast }= \frac{1}{ ( \alpha +1 ) ( \alpha +2 ) ( \alpha +3 ) } \biggl[ \alpha -1+ ( \alpha +3 ) \biggl( \frac{1}{2} \biggr) ^{\alpha }- ( \alpha +1 ) \biggl( \frac{1}{2} \biggr) ^{\alpha +1} \biggr] , \qquad \kappa _{2}^{\ast }=\frac{1}{4}- \kappa _{1}^{\ast } . \end{aligned}$$

Proof

Similar to Theorem 3.2 and using \(( \alpha ,m ) \)-convexity of \(\vert f^{\prime } \vert ^{q}\), we get the desired result. □

Remark 3.4

If we take \(\alpha =1\) in Theorem 3.4, inequality (14) reduces to inequality (7) in Theorem 3.2.

Remark 3.5

Choosing \(\alpha =1\) and \(m=1\) in Theorem 3.4, we get inequality (10).

Applications to special means

We now consider the applications of our results to the following special means for positive real numbers u and v\(( u\neq v )\).

(1) The arithmetic mean:

$$\begin{aligned} A:=A ( u,v ) =\frac{u+v}{2}; \end{aligned}$$

(2) The logarithmic mean:

$$\begin{aligned} L:=L ( u,v ) =\frac{v-u}{\ln v-\ln u}; \end{aligned}$$

(3) The generalized logarithmic mean:

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

Proposition 4.1

Let\(u,v\in \mathbb{R} ^{+}\), \(u< v\), \(m\in ( 0,1 ] \), and\(n\in \mathbb{Z} \backslash \{-1,0\}\). Then, for all\(q\geq 1\), we have

$$\begin{aligned}& \bigl\vert A \bigl( u^{n},v^{n} \bigr) -L_{n}^{n} ( u,v ) \bigr\vert \\& \quad \leq n.\frac{v-u}{8} \biggl\{ \biggl( \min \biggl\{ \frac{3 \vert u \vert ^{q ( n-1 ) }+m \vert \frac{v}{m} \vert ^{q ( n-1 ) }}{4}, \frac{3m \vert \frac{u}{m} \vert ^{q ( n-1 ) }+m \vert v \vert ^{q ( n-1 ) }}{4} \biggr\} \biggr) ^{\frac{1}{q}} \\& \qquad {}+ \biggl( \min \biggl\{ \frac{ \vert u \vert ^{q ( n-1 ) }+3m \vert \frac{v}{m} \vert ^{q ( n-1 ) }}{4}, \frac{m \vert \frac{u}{m} \vert ^{q ( n-1 ) }+3 \vert v \vert ^{q ( n-1 ) }}{4} \biggr\} \biggr) ^{\frac{1}{q}} \biggr\} . \end{aligned}$$

If we choose\(m=1\), we obtain

$$\begin{aligned}& \bigl\vert A \bigl( u^{n},v^{n} \bigr) -L_{n}^{n} ( u,v ) \bigr\vert \\& \quad \leq n.\frac{v-u}{8} \biggl[ \biggl( \frac{3 \vert u \vert ^{q ( n-1 ) }+ \vert v \vert ^{q ( n-1 ) }}{4} \biggr) ^{\frac{1}{q}}+ \biggl( \frac{ \vert u \vert ^{q ( n-1 ) }+3 \vert v \vert ^{q ( n-1 ) }}{4} \biggr) ^{ \frac{1}{q}} \biggr]. \end{aligned}$$

Proof

The assertions follow from Theorem 3.2 and Corollary 3.1 applied respectively to the m-convex mapping \(f ( x ) =x^{n}\), \(x\in \mathbb{R} \), \(n\in \mathbb{Z} \). □

Proposition 4.2

Let\(u,v\in \mathbb{R} ^{+}\), \(u< v\), \(\alpha , m\in ( 0,1 ] \), and\(n\in \mathbb{Z} \backslash \{-1,0\}\). Then, for all\(q\geq 1\), we have

$$\begin{aligned}& \bigl\vert A \bigl( u^{n},v^{n} \bigr) -L_{n}^{n} ( u,v ) \bigr\vert \\& \quad \leq n.\frac{v-u}{4 ( p+1 )^{\frac{1}{p}}} \biggl\{ \biggl( \min \biggl\{ \frac{2 \vert u \vert ^{q ( n-1 ) }+m\alpha \vert \frac{v}{m} \vert ^{q ( n-1 ) }}{\alpha +2}, \frac{2m \vert \frac{u}{m} \vert ^{q ( n-1 ) }+\alpha \vert v \vert ^{q ( n-1 ) }}{\alpha +2} \biggr\} \biggr) ^{\frac{1}{q}} \\& \qquad {}+ \biggl( \min \biggl\{ \frac{2 \vert u \vert ^{q ( n-1 ) }+m\alpha ( \alpha +3 ) \vert \frac{v}{m} \vert ^{q ( n-1 ) }}{ ( \alpha +1 ) ( \alpha +2 ) }, \frac{2m \vert \frac{u}{m} \vert ^{q ( n-1 ) }+\alpha ( \alpha +3 ) \vert v \vert ^{q ( n-1 ) }}{ ( \alpha +1 ) ( \alpha +2 ) } \biggr\} \biggr) ^{\frac{1}{q}} \biggr\} . \end{aligned}$$

If we choose\(m=1\), we obtain

$$\begin{aligned}& \bigl\vert A \bigl( u^{n},v^{n} \bigr) -L_{n}^{n} ( u,v ) \bigr\vert \\& \quad \leq n.\frac{v-u}{4 ( p+1 )^{\frac{1}{p}}} \biggl[ \biggl( \frac{2 \vert u \vert ^{q ( n-1 ) }+\alpha \vert v \vert ^{q ( n-1 ) }}{\alpha +2} \biggr) ^{\frac{1}{q}}+ \biggl( \frac{2 \vert u \vert ^{q ( n-1 ) }+\alpha ( \alpha +3 ) \vert v \vert ^{q ( n-1 ) }}{ ( \alpha +1 ) ( \alpha +2 ) } \biggr) ^{\frac{1}{q}} \biggr]. \end{aligned}$$

Proof

The assertions follow from Theorem 3.3 and Corollary 3.2 applied respectively to the \(( \alpha , m) \)-convex mapping \(f ( x ) =x^{n}\), \(x\in \mathbb{R} \), \(n\in \mathbb{Z} \). □

References

  1. 1.

    Ali, M.A., Abbas, M., Zafer, A.A.: On some Hermite–Hadamard integral inequalities in multiplicative calculus. J. Inequal. Spec. Funct. 10(1), 111–122 (2019)

    MathSciNet  Google Scholar 

  2. 2.

    Almuatiri, O., Kılıçman, A.: New fractional inequalities of midpoint type via s-convexity and their application. J. Inequal. Appl. 2019, 267 (2019)

    MathSciNet  Google Scholar 

  3. 3.

    Bai, R.-F., Qi, F., Xi, B.-Y.: Hermite–Hadamard type inequalities for the m- and \(( \alpha , m)\)-logarithmically convex functions. Filomat 27(1), 1–7 (2013)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Bakula, M.K., Özdemir, M.E., Pec̆arić, J.: Hadamard type inequalities for m-convex and \(( \alpha , m)\)-convex functions. J. Inequal. Pure Appl. Math. 9(4), Article ID 96 (2008)

    MathSciNet  Google Scholar 

  5. 5.

    Bakula, M.K., Pec̆arić, J., Ribićić, M.: Companion inequalities to Jensen’s inequality for m-convex and \(( \alpha , m)\)-convex functions. J. Inequal. Pure Appl. Math. 7(5), Article ID 194 (2006)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Budak, H., Sarıkaya, M.Z.: Some new generalized Hermite–Hadamard inequalities for generalized convex functions and applications. J. Math. Ext. 12(4), 51–66 (2018)

    Google Scholar 

  7. 7.

    Dragomir, S.S., Agarwal, R.P.: Two inequalities for differentiable mappings and applications to special means of real numbers and trapezoidal formula. Appl. Math. Lett. 11(5), 91–95 (1998)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Dragomir, S.S., Pearce, C.E.M.: Selected Topics on Hermite–Hadamard Inequalities and Applications. RGMIA Monographs. Victoria University (2000)

    Google Scholar 

  9. 9.

    Dragomir, S.S., Toader, G.: Some inequalities for m-convex functions. Stud. Univ. Babeş–Bolyai, Math. 38(1), 21–28 (1993)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Guessab, A., Schmeisser, G.: Sharp integral inequalities of the Hermite–Hadamard type. J. Approx. Theory 115(2), 260–288 (2002)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Guessab, A., Schmeisser, G.: Convexity results and sharp error estimates in approximate multivariate integration. Math. Comput. 73(247), 1365–1384 (2004)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    İşcan, İ.: New refinements for integral and sum forms of Hölder inequality. J. Inequal. Appl. 2019, 304 (2019)

    Google Scholar 

  13. 13.

    İşcan, İ., Turhan, S., Maden, S.: Hermite–Hadamard and Simpson-like type inequalities for differentiable p-quasi-convex functions. Filomat 31(19), 5945–5953 (2017)

    MathSciNet  Google Scholar 

  14. 14.

    Kadakal, H.: \(( \alpha , m_{1}, m_{2})\)-Convexity and some inequalities of Hermite–Hadamard type. Commun. Fac. Sci. Univ. Ank. Sér. A1 Math. Stat. 68(2), 2128–2142 (2019)

    MathSciNet  Google Scholar 

  15. 15.

    Kadakal, M.: Some Hermite–Hadamard type inequalities for \((P;m)\)-function and quasi m-convex functions. Int. J. Optim. Control Theor. Appl. 10(1), 78–84 (2020)

    Google Scholar 

  16. 16.

    Kadakal, M., İşcan, İ., Kadakal, H., Bekar, K.: On improvements of some integral inequalities (2019) https://doi.org/10.13140/RG.2.2.15052.46724. Researchgate preprint

  17. 17.

    Kunt, M., İşcan, İ.: Hermite–Hadamard–Fejer type inequalities for p-convex functions. Arab J. Math. Sci. 23, 215–230 (2017)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Kunt, M., İşcan, İ.: Hermite–Hadamard type inequalities for harmonically \(( \alpha , m)\)-convex functions by using fractional integrals. Konuralp J. Math. 5(1), 201–213 (2017)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Latif, M.A.: Inequalities of Hermite–Hadamard type for functions whose derivatives in absolute value are convex with applications. Arab J. Math. Sci. 21, 84–97 (2015)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Latif, M.A., Dragomir, S.S., Momoniat, E.: On Hermite–Hadamard type integral inequalities for n-times differentiable m- and \((\alpha , m)\)-logarithmically convex functions. Filomat 30(11), 3101–3114 (2016)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Mehrez, K., Agarwal, P.: New Hermite–Hadamard type integral inequalities for convex functions and their applications. J. Comput. Appl. Math. 350, 274–285 (2019)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Miheşan, V.G.: A generalization of the convexity. In: Seminar on Functional Equations, Approximation and Convexity, Cluj-Napoca, Romania (1993)

    Google Scholar 

  23. 23.

    Milovanovic, G.V., Rassias, M.T. (eds.): Analytic Number Theory, Approximation Theory and Special Functions Springer, Berlin (2014)

    Google Scholar 

  24. 24.

    Mitrinovic, D.S., Pecaric, J.E., Fink, A.M.: Inequalities Involving Functions and Their Integrals and Derivatives. Kluwer Academic, Norwell (1991)

    Google Scholar 

  25. 25.

    Mitrinovic, D.S., Pecaric, J.E., Fink, A.M.: Classical and New Inequalities in Analysis. Kluwer Academic, Norwell (1993)

    Google Scholar 

  26. 26.

    Noor, M.A., Qi, F., Awan, M.U.: Some Hermite–Hadamard type inequalities for \(log-h\)-convex functions. Analysis 33, 1–9 (2013)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Önalan, H.K., Akdemir, A.O., Set, E., Sarıkaya, M.Z.: Simpson’s type inequalities for m- and \((\alpha , m)\)-geometrically convex functions. Konuralp J. Math. 2(1), 90–101 (2014)

    MATH  Google Scholar 

  28. 28.

    Özcan, S., İşcan, İ.: Some new Hermite–Hadamard type inequalities for s-convex functions and their applications. J. Inequal. Appl. 2019, 201 (2019)

    MathSciNet  Google Scholar 

  29. 29.

    Özdemir, M.E., Önalan, H.K., Ardıç, M.A.: Hermite–Hadamard type inequalities for \((h(\alpha , m))\)-convex functions. J. Concr. Appl. Math. 13(1), 96–107 (2015)

    MathSciNet  MATH  Google Scholar 

  30. 30.

    Özdemir, M.E., Set, E., Sarıkaya, M.Z.: Some new Hadamard type inequalities for co-ordinated m-convex and \(( \alpha , m)\)-convex functions. Hacet. J. Math. Stat. 40(2), 219–229 (2011)

    MathSciNet  MATH  Google Scholar 

  31. 31.

    Pecaric, J.E., Proschan, F., Tong, Y.L.: Convex Functions, Partial Orderings and Statistical Applications. Academic Press, Boston (1992)

    Google Scholar 

  32. 32.

    Sarikaya, M.Z., Budak, H.: On generalized Hermite–Hadamard inequality for generalized convex function. Int. J. Nonlinear Anal. Appl. 8(2), 209–222 (2017)

    MATH  Google Scholar 

  33. 33.

    Set, E., İşcan, İ., Sarıkaya, M.Z., Özdemir, M.E.: On new inequalities of Hermite–Hadamard–Fejer type for convex functions via fractional integrals. Appl. Math. Comput. 259, 875–881 (2015)

    MathSciNet  MATH  Google Scholar 

  34. 34.

    Set, E., Özdemir, M.E., Sarıkaya, M.Z., Karakoç, F.: Hermite–Hadamard type inequalities for \(( \alpha , m)\)-convex functions via fractional integrals. Moroccan J. Pure Appl. Anal. 3(1), 15–21 (2017)

    MathSciNet  Google Scholar 

  35. 35.

    Set, E., Sardari, M., Özdemir, M.E., Rooin, J.: On generalizations of the Hadamard inequality for \(( \alpha , m)\)-convex functions. Kyungpook Math. J. 52, 307–317 (2012)

    MathSciNet  MATH  Google Scholar 

  36. 36.

    Toader, G.: Some generalizations of the convexity. In: Proc. Colloq. Approx. Optim. Cluj-Napoca, Romania, Univ. Cluj-Napoca, pp. 329–338 (1985)

    Google Scholar 

  37. 37.

    Tunç, M.: Hermite–Hadamard type inequalities via m and \(( \alpha , m)\)-convexity. Demonstr. Math. XLVI(3), 475–483 (2013)

    MathSciNet  MATH  Google Scholar 

  38. 38.

    Wang, S.-H., Wu, S.-H.: Some new Hermite–Hadamard type inequalities for operator m-convex and \(( \alpha , m)\)-convex functions on the co-ordinates. J. Comput. Anal. Appl. 25(3), 474–487 (2018)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

The author is thankful to the editor and anonymous referees for their valuable comments and suggestions.

Availability of data and materials

Not applicable.

Funding

There is no funding for this research article.

Author information

Affiliations

Authors

Contributions

All authors read and approved the final manuscript.

Corresponding author

Correspondence to Serap Özcan.

Ethics declarations

Competing interests

The author declares that no competing interests exist.

Rights and permissions

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

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Özcan, S. Hermite–Hadamard type inequalities for m-convex and \(( \alpha , m)\)-convex functions. J Inequal Appl 2020, 175 (2020). https://doi.org/10.1186/s13660-020-02442-5

Download citation

MSC

  • 26D15
  • 26A51

Keywords

  • m-convex function
  • \(( \alpha , m) \)-convex function
  • Hermite–Hadamard type inequalities
  • Integral inequalities