On the Hermite-Hadamard Inequality and Other Integral Inequalities Involving Two Functions
© Erhan Set et al. 2010
Received: 25 September 2009
Accepted: 31 March 2010
Published: 24 May 2010
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.
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 , Bakula et al. improved Hermite-Hadamard type inequalities for products of two -convex and -convex functions. In , 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 ).
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.
To prove our main results we recall some concepts and definitions.
(see, e.g., [10, page 15]).
For several recent results concerning -norms we refer the interested reader to .
Also, we need some important inequalities.
Minkowski Integral Inequality (see page 1 in )
Hermite-Hadamard's Inequality (see page 10 in )
A Reversed Minkowski Integral Inequality (see page 2 in )
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
2. The Results
Thus, by applying Barnes-Gudunova-Levin inequality to the right-hand side of (2.4) with (2.6), we get (2.1).
This completes the proof of the inequality in (2.21).
The authors thank the careful referees for some good advices which have improved the final version of this paper.
- 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
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.