Abstract

In this paper, we introduce a new class of extended (m,r)-convex function and we establish the Hermite-Hadamard inequality for (m,r)-convex functions. Some special cases are discussed. Results represent significant refinement and improvement of the previous results. The definition of (m,r)-convex function is given for the first time in the literature and moreover, the results obtained in special cases coincide with the well-known results in the literature.

Keywords: convex function r-convex function m-convex function -convex Hermite-Hadamard inequality

1. Introduction

Definition 1: A function is said to be convex if the inequality

is valid for all and If this inequality reverses, then is said to be concave on interval This definition is well known in the literature. Denote by the set of the convex functions on the interval

Definition 2: be a convex function defined on the interval of real numbers and with The following inequality

(1.1)

holds. This double inequality is known in the literature as Hermite-Hadamard integral inequality for convex functions.

The inequality (1.1) is well known as the Hermite Hadamard integral inequality. Readers can find more information in ¹. Some refinements of the Hermite-Hadamard integral inequality on convex functions have been extensively investigated by a number of authors (see ^{2, 3, 4}).

Definition 3: ⁵ The function is said to be m-convex, where , if for every and we have

Denote by the set of the m-convex functions on for which

In ⁵, the author gave the following theorem about the inequalities of Hermite-Hadamard type for m-convex functions.

Theorem 1: Let be a m-convex functions with If and then one has the inequality:

Definition 4: ⁶ A positive function is called -convex on interval if for each and

If the equality is reversed, then the function is said to be -concave.

The definition of -convexity naturally complements the concept of -concavity, in which the inequality is reversed ⁷ and which plays an important role in statistics. It is obvious 0-convex functions are simply -convex functions, -convex functions are ordinary convex functions and -convex functions are arithmetically harmonically convex. If is r-convex in the interval then is a convex function and If is -concave in the interval then is a concave function We note that if and are convex and is increasing, then is convex; moreover, since it follows that a -convex function is convex.

Some refinements of the Hadamard integral inequality for -convex functions could be found in ^{8, 9}. In ¹⁰, Bessenyei studied Hermite-Hadamard-type inequalities for generalized -convex functions. In ⁸, the authors showed that if is -convex in and then

In ¹¹, the authors show that is -convex in the interval and then

and they prove the following inequality for -convex functions:

2. Main Results for (m,r)-convex Functions

The main aim of this paper is to establish new inequalities of Hermite-Hadamard type for the class of functions whose derivatives in absolutely value at certain powers are (m,r)-convex.

Definition 5: A positive function is called (m,r)-convex on interval if for each and

(2.1)

If the equality is reversed, then the function is said to be -concave.

Theorem 2: Let be a (m,r)-convex and and If and then the following inequalities hold:

(2.2)

Proof: Suppose that

i. Using the definition of (m,r)-convexity and changing variable as we get

For we obtain,

ii. Let For we obtain,

For we have

This completes the proof of theorem.

Corollary 1: Suppose that all the assumptions of Theorem 2 are satisfied. In the inequality 2.2, If we choose we obtain the inequality in ¹¹.

In ¹², İşcan obtained main results using the following lemma. We will use the same lemma to obtain the main results for (m,r)-convex functions.

Lemma 1: Let be differentiable mapping on with If and then following equality holds:

(2.3)

Theorem 3: Let be a differentiable mapping on such that where with If is (m,r)-convex on for some fixed with and then the following inequalities holds for

(2.4)

Proof: Let From Lemma 1 and the definition of (m,r)-convexity of that is,

we have

Let From Lemma 1, the Power-mean integral inequality and the (m,r)-convexity of we can write

(2.5)

Let calculate the integral in (2.5) respectively: Firstly, sample calculation give us

(2.6)

Using the partial integration, we get

(2.7)

Substituting (2.6) and (2.7) inequalities in (2.5), we obtain

Corollary 2: Under the conditions of Theorem 3,

(i) In the inequality (2.4), for we get the following inequality:

(ii) In the inequality (2.4), for and we get the following:

where is the arithmetic mean.

(iii) In the inequality (2.4), for and we get:

where is the arithmetic mean. This inequality coincides with in ¹² for

Theorem 4: Let be a differentiable mapping on such that where with If is (m,r)-convex on the interval for some fixed with and and then the following inequalities holds for

(2.8)

where

Proof: Using Lemma 1, the Hölder inequality and (m,r)-convexity of we get

where we use the fact that

Corollary 3: Under the conditions of Theorem 4,

(i) In the inequality (2.8), for we get the following inequality:

(ii) In the inequality (2.8), for and we get the following inequality:

where is the arithmetic mean.

(iii) In the inequality (2.8), for and we get the following inequality:

where is the arithmetic mean.

Theorem 5: Let be a differentiable mapping on such that where with If is (m,r)-convex on the interval for some fixed with and then the following inequalities holds for

(2.9)

Proof: From Lemma 1 and using the Power-mean integral inequality and (m,r)-convexity of we get

Corollary 4: Under the conditions of the Theorem 5,

(i) In the inequality (2.9), if we choose we obtain the following inequality:

(ii) In the inequality (2.9), if we choose and we obtain the following inequality:

(iii) In the inequality (2.9), if we choose and we obtain the following inequality:

Corollary 5: Suppose that all the assumptions of Theorem 5 are satisfied. In the inequality (2.9), if we choose and we get the following inequality:

where is the arithmetic mean.

References