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

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 [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)

Let , be nonnegative concave functions on . Then, for we have

where

In the special case we have

with

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 [19–21].

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 .

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).

Proof.

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

Then

where

Proof.

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:

Proof.

If , are concave on , then from (1.11) we get

which imply that

On the other hand, if from (1.6) we get

or

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

Then

where

and with

Proof.

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.

1. Pachpatte, BG: On some inequalities for convex functions. RGMIA Research Report Collection E. 6, (2003)

2. Pachpatte, BG: A note on integral inequalities involving two log-convex functions. Mathematical Inequalities & Applications. 7(4), 511–515 (2004). PubMed Abstract | Publisher Full Text

3. Bakula, MK, Özdemir, ME, Pečarić, J: Hadamard type inequalities for -convex and -convex functions. Journal of Inequalities in Pure and Applied Mathematics. 9(4, article 96), (2008)

4. Kirmaci, US, Bakula, MK, Özdemir, ME, Pečarić, J: Hadamard-type inequalities for -convex functions. Applied Mathematics and Computation. 193(1), 26–35 (2007). Publisher Full Text

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. 7(5, article 194), (2006)

6. Bakula, MK, Pečarić, J: Note on some Hadamard-type inequalities. Journal of Inequalities in Pure and Applied Mathematics. 5(3, article 74), (2004)

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. 5(2), 103–165 (2000). Publisher Full Text

8. Hardy, GH, Littlewood, JE, Pólya, G: Inequalities,p. xii+324. Cambridge Mathematical Library, Cambridge , UK (1998)

9. Kirmaci, US, Özdemir, ME: Some inequalities for mappings whose derivatives are bounded and applications to special means of real numbers. Applied Mathematics Letters. 17(6), 641–645 (2004). Publisher Full Text

10. Mitrinović, DS, Pečarić, JE, Fink, AM: Classical and New Inequalities in Analysis, Mathematics and Its Applications (East European Series),p. xviii+740. Kluwer Academic Publishers, Dordrecht, The Netherlands (1993)

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. 143(2-3), 269–274 (2003). Publisher Full Text

12. Pachpatte, BG: Inequalities for Differentiable and Integral Equations, Academic Press, Boston, Mass, USA (1997)

13. Pečarić, J, Pejković, T: On an integral inequality. Journal of Inequalities in Pure and Applied Mathematics. 5(2, article 47), (2004)

14. Pečarić, JE, Proschan, F, Tong, YL: Convex Functions, Partial Orderings, and Statistical Applications, Mathematics in Science and Engineering,p. xiv+467. Academic Press, Boston, Mass, USA (1992)

15. Pogány, TK: On an open problem of F. Qi. Journal of Inequalities in Pure and Applied Mathematics. 3(4, article 54), (2002)

16. Bullen, PS, Mitrinović, DS, Vasić, PM: Means and Their Inequalities, Mathematics and Its Applications (East European Series),p. xx+459. D. Reidel, Dordrecht, The Netherlands (1988)

17. Kirmaci, US, Klaričić, M, Özdemir, ME, Pečarić, J: On some inequalities for -norms. Journal of Inequalities in Pure and Applied Mathematics. 9(1, article 27), (2008)

18. Bougoffa, L: On Minkowski and Hardy integral inequalities. Journal of Inequalities in Pure and Applied Mathematics. 7(2, article 60), (2006)

19. Alomari, M, Darus, M: On the Hadamard's inequality for log-convex functions on the coordinates. Journal of Inequalities and Applications. 2009, (2009)

20. Dinu, C: Hermite-Hadamard inequality on time scales. Journal of Inequalities and Applications. 2008, (2008)

21. Dragomir, SS, Pearce, CEM: Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University, Melbourne, Australia (2000)