# On Hadamard-Type Inequalities Involving Several Kinds of Convexity

- Erhan Set
^{1}Email author, - MEmin Özdemir
^{1}and - SeverS Dragomir
^{2, 3}

**2010**:286845

**DOI: **10.1155/2010/286845

© Erhan Set et al. 2010

**Received: **14 May 2010

**Accepted: **23 August 2010

**Published: **26 August 2010

## Abstract

We do not only give the extensions of the results given by Gill et al. (1997) for log-convex functions but also obtain some new Hadamard-type inequalities for log-convex -convex, and -convex functions.

## 1. Introduction

where is a convex function on the interval of real numbers and with This inequality is one of the most useful inequalities in mathematical analysis. For new proofs, note worthy extension, generalizations, and numerous applications on this inequality; see ([1–6]) where further references are given.

(see [5], Page 1). Geometrically, this means that if , and are three distinct points on the graph of with between and , then is on or below chord

It is said to be log-concave if the inequality in (1.3) is reversed.

In [7], Toader defined -convexity as follows.

Definition 1.1.

for all and We say that is -concave if is -convex.

Denote by the class of all -convex functions on such that (if ). Obviously, if we choose Definition 1.1 recaptures the concept of standard convex functions on

In [8], Miheşan defined -convexity as in the following:

Definition 1.2.

for all and .

Denote by the class of all -convex functions on for which . It can be easily seen that for -convexity reduces to -convexity and for , -convexity reduces to the concept of usual convexity defined on , .

For recent results and generalizations concerning -convex and -convex functions, see ([9–12]).

(for , we put ).

In [13], Gill et al. established the following results.

Theorem 1.3.

where is a logarithmic mean of the positive real numbers as in (1.6).

For a positive -concave function, the inequality is reversed.

Corollary 1.4.

For some recent results related to the Hadamard's inequalities involving two -convex functions, see [14] and the references cited therein. The main purpose of this paper is to establish the general version of inequalities (1.7) and new Hadamard-type inequalities involving two -convex functions, two -convex functions, or two -convex functions using elementary analysis.

## 2. Main Results

We start with the following theorem.

Theorem 2.1.

where is a logarithmic mean of positive real numbers.

For a positive -concave function, the inequality is reversed.

Proof.

for all and

the theorem is proved.

Remark 2.2.

By taking and in Theorem 2.1 we obtain (1.7).

Corollary 2.3.

Proof.

for all , whence (2.7). Similarly we can prove (2.8).

Remark 2.4.

By taking and in (2.7) and (2.8) we obtain the inequalities of Corollary 1.4.

We will now point out some new results of the Hadamard type for log-convex, -convex, and -convex functions, respectively.

Theorem 2.5.

Proof.

for all and .

Combining (2.16), we get the desired inequalities (2.10). The proof is complete.

Theorem 2.6.

where is a logarithmic mean of positive real numbers.

Proof.

for all and

Combining (2.21), we get the required inequalities (2.17). The proof is complete.

Theorem 2.7.

Proof.

Rewriting (2.27) and (2.28), we get the required inequality in (2.22). The proof is complete.

Theorem 2.8.

Proof.

Rewriting (2.34) and (2.35), we get the required inequality in (2.29). The proof is complete.

Remark 2.9.

In Theorem 2.8, if we choose , we obtain the inequality of Theorem 2.7.

## Authors’ Affiliations

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