# 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

## 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.

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]).

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.

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.

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.

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

## References

- Alomari M, Darus M: On the Hadamard's inequality for log-convex functions on the coordinates.
*Journal of Inequalities and Applications*2009, 2009:-13.Google Scholar - Zhang X-M, Chu Y-M, Zhang X-H: The Hermite-Hadamard type inequality of GA-convex functions and its applications.
*Journal of Inequalities and Applications*2010, 2010:-11.Google Scholar - Dinu C: Hermite-Hadamard inequality on time scales.
*Journal of Inequalities and Applications*2008, 2008:-24.Google Scholar - Dragomir SS, Pearce CEM: Selected Topics on Hermite-Hadamard Inequalities and Applications.
*RGMIA Monographs, Victoria University*, 2000, http://www.staff.vu.edu.au/rgmia/monographs/hermite_hadamard.html RGMIA Monographs, Victoria University, 2000, - Mitrinović DS, Pečarić JE, Fink AM:
*Classical and New Inequalities in Analysis, Mathematics and Its Applications*.*Volume 61*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1993:xviii+740.View ArticleGoogle Scholar - Set E, Özdemir ME, Dragomir SS: On the Hermite-Hadamard inequality and other integral inequalities involving two functions.
*Journal of Inequalities and Applications*2010, -9.Google Scholar - Toader G: Some generalizations of the convexity. In
*Proceedings of the Colloquium on Approximation and Optimization, Cluj-Napoca, Romania*. University of Cluj-Napoca; 1985:329–338.Google Scholar - Miheşan VG: A generalization of the convexity.
*Proceedings of the Seminar on Functional Equations, Approximation and Convexity, 1993, Cluj-Napoca, Romania*Google Scholar - Bakula MK, Özdemir ME, Pečarić J: Hadamard type inequalities for
*m*-convex and -convex functions.*Journal of Inequalities in Pure and Applied Mathematics*2008., 9, article no. 96:Google Scholar - Bakula MK, Pečarić J, Ribičić M: Companion inequalities to Jensen's inequality for
*m*-convex and (*α*,*m*)-convex functions.*Journal of Inequalities in Pure and Applied Mathematics*2006., 7(5, article no. 194): - Pycia M: A direct proof of the
*s*-Hölder continuity of Breckner*s*-convex functions.*Aequationes Mathematicae*2001, 61(1–2):128–130. 10.1007/s000100050165MathSciNetView ArticleMATHGoogle Scholar - Özdemir ME, Avci M, Set E: On some inequalities of Hermite-Hadamard type via m-convexity.
*Applied Mathematics Letters*2010, 23(9):1065–1070. 10.1016/j.aml.2010.04.037MathSciNetView ArticleMATHGoogle Scholar - Gill PM, Pearce CEM, Pečarić J: Hadamard's inequality for
*r*-convex functions.*Journal of Mathematical Analysis and Applications*1997, 215(2):461–470. 10.1006/jmaa.1997.5645MathSciNetView ArticleMATHGoogle Scholar - Pachpatte BG: A note on integral inequalities involving two log-convex functions.
*Mathematical Inequalities & Applications*2004, 7(4):511–515.MathSciNetView ArticleMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.