# On the Hermite-Hadamard Inequality and Other Integral Inequalities Involving Two Functions

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

**2010**:148102

**DOI: **10.1155/2010/148102

© Erhan Set et al. 2010

**Received: **25 September 2009

**Accepted: **31 March 2010

**Published: **24 May 2010

## Abstract

We establish some new Hermite-Hadamard-type inequalities involving product of two functions. Other integral inequalities for two functions are obtained as well. The analysis used in the proofs is fairly elementary and based on the use of the Minkowski, Hölder, and Young inequalities.

## 1. Introduction

Integral inequalities have played an important role in the development of all branches of Mathematics.

In [1, 2], Pachpatte established some Hermite-Hadamard-type inequalities involving two convex and log-convex functions, respectively. In [3], Bakula et al. improved Hermite-Hadamard type inequalities for products of two -convex and -convex functions. In [4], analogous results for -convex functions were proved by Kirmaci et al.. General companion inequalities related to Jensen's inequality for the classes of -convex and -convex functions were presented by Bakula et al. (see [5]).

For several recent results concerning these types of inequalities, see [6–12] where further references are listed.

The aim of this paper is to establish several new integral inequalities for nonnegative and integrable functions that are related to the Hermite-Hadamard result. Other integral inequalities for two functions are also established.

In order to prove some inequalities related to the products of two functions we need the following inequalities. One of inequalities of this type is the following one.

Barnes-Gudunova-Levin Inequality (see [13–15] and references therein)

To prove our main results we recall some concepts and definitions.

(see, e.g., [10, page 15]).

and is the set of all functions such that .

For several recent results concerning -norms we refer the interested reader to [17].

Also, we need some important inequalities.

Minkowski Integral Inequality (see page 1 in [18])

Hermite-Hadamard's Inequality (see page 10 in [10])

For recent results, refinements, counterparts, generalizations, and new Hermite-Hadamard-type inequalities, see [19–21].

A Reversed Minkowski Integral Inequality (see page 2 in [18])

One of the most important inequalities of analysis is Hölder's integral inequality which is stated as follows (for its variant see [10, page 106]).

Hölder Integral Inequality

with equality holding if and only if almost everywhere, where and are constants.

Remark 1.1.

For , similarly if is concave on the nonnegative function is concave on .

## 2. The Results

Theorem 2.1.

Here is the Barnes-Gudunova-Levin constant given by (1.1).

Proof.

Thus, by applying Barnes-Gudunova-Levin inequality to the right-hand side of (2.4) with (2.6), we get (2.1).

Applying the Hölder inequality to the left-hand side of (2.5) with , we get (2.2).

Theorem 2.2.

Proof.

Since and by applying the Minkowski integral inequality to the right hand side of (2.10), we obtain inequality (2.8).

Theorem 2.3.

Proof.

Theorem 2.4.

Proof.

This completes the proof of the inequality in (2.21).

## Declarations

### Acknowledgment

The authors thank the careful referees for some good advices which have improved the final version of this paper.

## Authors’ Affiliations

## References

- Pachpatte BG: On some inequalities for convex functions.
*RGMIA Research Report Collection E*2003., 6:Google Scholar - Pachpatte BG: A note on integral inequalities involving two log-convex functions.
*Mathematical Inequalities & Applications*2004, 7(4):511–515.MathSciNetView ArticleMATHGoogle Scholar - Bakula MK, Özdemir ME, Pečarić J: Hadamard type inequalities for -convex and -convex functions.
*Journal of Inequalities in Pure and Applied Mathematics*2008., 9(4, article 96):Google Scholar - Kirmaci US, Bakula MK, Özdemir ME, Pečarić J: Hadamard-type inequalities for -convex functions.
*Applied Mathematics and Computation*2007, 193(1):26–35. 10.1016/j.amc.2007.03.030MathSciNetView ArticleMATHGoogle Scholar - Bakula MK, Pečarić J, Ribičić M: Companion inequalities to Jensen's inequality for -convex and -convex functions.
*Journal of Inequalities in Pure and Applied Mathematics*2006., 7(5, article 194):Google Scholar - Bakula MK, Pečarić J: Note on some Hadamard-type inequalities.
*Journal of Inequalities in Pure and Applied Mathematics*2004., 5(3, article 74):Google Scholar - Dragomir SS, Agarwal RP, Barnett NS: Inequalities for beta and gamma functions via some classical and new integral inequalities.
*Journal of Inequalities and Applications*2000, 5(2):103–165. 10.1155/S1025583400000084MathSciNetMATHGoogle Scholar - Hardy GH, Littlewood JE, Pólya G:
*Inequalities*. Cambridge Mathematical Library, Cambridge , UK; 1998:xii+324.MATHGoogle Scholar - Kirmaci US, Özdemir ME: Some inequalities for mappings whose derivatives are bounded and applications to special means of real numbers.
*Applied Mathematics Letters*2004, 17(6):641–645. 10.1016/S0893-9659(04)90098-5MathSciNetView ArticleMATHGoogle Scholar - Mitrinović DS, Pečarić JE, Fink AM:
*Classical and New Inequalities in Analysis, Mathematics and Its Applications (East European Series)*.*Volume 61*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1993:xviii+740.View ArticleMATHGoogle Scholar - Özdemir ME, Kırmacı US: Two new theorem on mappings uniformly continuous and convex with applications to quadrature rules and means.
*Applied Mathematics and Computation*2003, 143(2–3):269–274. 10.1016/S0096-3003(02)00359-4MathSciNetView ArticleMATHGoogle Scholar - Pachpatte BG:
*Inequalities for Differentiable and Integral Equations*. Academic Press, Boston, Mass, USA; 1997.Google Scholar - Pečarić J, Pejković T: On an integral inequality.
*Journal of Inequalities in Pure and Applied Mathematics*2004., 5(2, article 47):Google Scholar - Pečarić JE, Proschan F, Tong YL:
*Convex Functions, Partial Orderings, and Statistical Applications, Mathematics in Science and Engineering*.*Volume 187*. Academic Press, Boston, Mass, USA; 1992:xiv+467.MATHGoogle Scholar - Pogány TK: On an open problem of F. Qi.
*Journal of Inequalities in Pure and Applied Mathematics*2002., 3(4, article 54):Google Scholar - Bullen PS, Mitrinović DS, Vasić PM:
*Means and Their Inequalities, Mathematics and Its Applications (East European Series)*.*Volume 31*. D. Reidel, Dordrecht, The Netherlands; 1988:xx+459.Google Scholar - Kirmaci US, Klaričić M, Özdemir ME, Pečarić J: On some inequalities for -norms.
*Journal of Inequalities in Pure and Applied Mathematics*2008., 9(1, article 27):Google Scholar - Bougoffa L: On Minkowski and Hardy integral inequalities.
*Journal of Inequalities in Pure and Applied Mathematics*2006., 7(2, article 60):Google Scholar - 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 - 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, Melbourne, Australia; 2000.Google Scholar

## Copyright

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