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  • Research Article
  • Open Access

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

Journal of Inequalities and Applications20102010:148102

  • Received: 25 September 2009
  • Accepted: 31 March 2010
  • Published:


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.


  • Convex Function
  • Integrable Function
  • Positive Function
  • Type Inequality
  • Concave Function

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 [612] 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 [1315] and references therein)

Let , be nonnegative concave functions on . Then, for we have
In the special case we have

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

Let and be two positive -tuples, and let . Then, on putting the th power mean of with weights is defined [16] by
Note that if , then

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

Let and . The -norm of the function on is defined by

and is the set of all functions such that .

One can rewrite the inequality (1.1) as follows:

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])

Let , and . Then

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

Let be a convex function on interval of real numbers and with . Then the following Hermite-Hadamard inequality for convex functions holds:
If the function is concave, the inequality (1.10) can be written as follows:

For recent results, refinements, counterparts, generalizations, and new Hermite-Hadamard-type inequalities, see [1921].

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

Let and be positive functions satisfying
Then, putting , we have

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

Let and If and are real functions defined on and if and are integrable functions on , then

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

Remark 1.1.

Observe that whenever, is concave on the nonnegative function is also concave on . Namely,
that is,
and ; using the power-mean inequality (1.6), we obtain

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

2. The Results

Theorem 2.1.

Let and let , , be nonnegative functions such that and are concave on . Then
and if , then one has

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


Since are concave functions on , then from (1.11) and Remark 1.1 we get
By multiplying the above inequalities, we obtain (2.4) and (2.5)
If , then it easy to show that

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.

Let , and , and let be positive functions with



Since are positive, as in the proof of the inequality (1.13) (see [18, page 2]), we have that
By multiplying the above inequalities, we get

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

Theorem 2.3.

Let and be as in Theorem 2.1. Then the following inequality holds:


If , are concave on , then from (1.11) we get
which imply that
On the other hand, if from (1.6) we get
which imply that
Combining (2.13) and (2.16), we obtain the desired inequality as
that is,
To prove the following theorem we need the following Young-type inequality (see [7, page 117]):

Theorem 2.4.

Let be functions such that , and are in , and

and with


From we have
From (2.19) with (2.23) we obtain
Using the elementary inequality , ( and ) in (2.24), we get

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.

Authors’ Affiliations

Department of Mathematics, K. K. Education Faculty, Atatürk University, 25240 Erzurum, Turkey
Graduate School of Natural and Applied Sciences, Ağrı İbrahim Çeçen University, Ağrı, 04100, Turkey
Research Group in Mathematical Inequalities & Applications, School of Engineering & Science, Victoria University, P.O. Box 14428, Melbourne, 8001, Australia


  1. Pachpatte BG: On some inequalities for convex functions. RGMIA Research Report Collection E 2003., 6:Google Scholar
  2. Pachpatte BG: A note on integral inequalities involving two log-convex functions. Mathematical Inequalities & Applications 2004, 7(4):511–515.MathSciNetView ArticleMATHGoogle Scholar
  3. 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
  4. 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
  5. 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
  6. 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
  7. 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
  8. Hardy GH, Littlewood JE, Pólya G: Inequalities. Cambridge Mathematical Library, Cambridge , UK; 1998:xii+324.MATHGoogle Scholar
  9. 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
  10. 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
  11. 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
  12. Pachpatte BG: Inequalities for Differentiable and Integral Equations. Academic Press, Boston, Mass, USA; 1997.Google Scholar
  13. Pečarić J, Pejković T: On an integral inequality. Journal of Inequalities in Pure and Applied Mathematics 2004., 5(2, article 47):Google Scholar
  14. 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
  15. Pogány TK: On an open problem of F. Qi. Journal of Inequalities in Pure and Applied Mathematics 2002., 3(4, article 54):Google Scholar
  16. 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
  17. 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
  18. Bougoffa L: On Minkowski and Hardy integral inequalities. Journal of Inequalities in Pure and Applied Mathematics 2006., 7(2, article 60):Google Scholar
  19. 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
  20. Dinu C: Hermite-Hadamard inequality on time scales. Journal of Inequalities and Applications 2008, 2008:-24.Google Scholar
  21. Dragomir SS, Pearce CEM: Selected Topics on Hermite-Hadamard Inequalities and Applications. RGMIA Monographs, Victoria University, Melbourne, Australia; 2000.Google Scholar


© Erhan Set et al. 2010

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