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Open Access Research Article

On Hadamard-Type Inequalities Involving Several Kinds of Convexity

Erhan Set1*, MEmin Özdemir1 and SeverS Dragomir23

Author Affiliations

1 Department of Mathematics, K.K. Education Faculty, Atatürk University, Campus, 25240 Erzurum, Turkey

2 Research Group in Mathematical Inequalities & Applications, School of Engineering & Science, Victoria University, P.O. Box 14428, Melbourne City, VIC 8001, Australia

3 School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa

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Journal of Inequalities and Applications 2010, 2010:286845  doi:10.1155/2010/286845

The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2010/1/286845


Received:14 May 2010
Accepted:23 August 2010
Published:26 August 2010

© 2010 Erhan Set et al.

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.

We do not only give the extensions of the results given by Gill et al. (1997) for log-convex functions but also obtain some new Hadamard-type inequalities for log-convex -convex, and -convex functions.

1. Introduction

The following inequality is well known in the literature as Hadamard's inequality:

(11)

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 ([16]) where further references are given.

Let be on interval in . Then is said to be convex if, for all and ,

(12)

(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

Recall that a function is said to be log-convex function if, for all and , one has the inequality (see [5], Page 3)

(13)

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.

The function , is said to be -convex, where , if one has

(14)

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.

The function , , is said to be -convex, where , if one has

(15)

for all and .

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 ([912]).

In the literature, the logarithmic mean of the positive real numbers is defined as the following:

(16)

(for , we put ).

In [13], Gill et al. established the following results.

Theorem 1.3.

Let be a positive, -convex function on . Then

(17)

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.

Let be positive -convex functions on . Then

(18)

If is a positive -concave function, then

(19)

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.

Let    be -convex functions on and with . Then the following inequality holds:

(21)

where is a logarithmic mean of positive real numbers.

For a positive -concave function, the inequality is reversed.

Proof.

Since are -convex functions on , we have

(22)

for all and Writing (2.2) for and multiplying the resulting inequalities, it is easy to observe that

(23)

for all and

Integrating inequality (2.3) on over , we get

(24)

As

(25)

(26)

the theorem is proved.

Remark 2.2.

By taking and in Theorem 2.1 we obtain (1.7).

Corollary 2.3.

Let be -convex functions on and with . Then

(27)

If are positive -concave functions, then

(28)

Proof.

Let be positive -convex functions. Then by Theorem 2.1 we have that

(29)

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.

Let be -convex functions on and with Then the following inequalities hold:

(210)

Proof.

We can write

(211)

Using the elementary inequality ( reals) and equality (2.11), we have

(212)

Since are -convex functions, we obtain

(213)

for all and .

Rewriting (2.12) and (2.13), we have

(214)

(215)

Integrating both sides of (2.14) and (2.15) on over , respectively, we obtain

(216)

Combining (2.16), we get the desired inequalities (2.10). The proof is complete.

Theorem 2.6.

Let be -convex functions on and with Then the following inequalities hold:

(217)

where is a logarithmic mean of positive real numbers.

Proof.

From inequality (2.14), we have

(218)

for all and

Using the elementary inequality ( reals) on the right side of the above inequality, we have

(219)

Since are -convex functions, then we get

(220)

Integrating both sides of (2.19) and (2.20) on over , respectively, we obtain

(221)

Combining (2.21), we get the required inequalities (2.17). The proof is complete.

Theorem 2.7.

Let be such that is in , where . If is nonincreasing -convex function and is nonincreasing -convex function on for some fixed then the following inequality holds:

(222)

where

(223)

(224)

Proof.

Since is -convex function and is -convex function, we have

(225)

for all . It is easy to observe that

(226)

Using the elementary inequality ( reals), (2.25) on the right side of (2.26) and making the charge of variable and since is nonincreasing, we have

(227)

Analogously we obtain

(228)

Rewriting (2.27) and (2.28), we get the required inequality in (2.22). The proof is complete.

Theorem 2.8.

Let be such that is in , where . If is nonincreasing -convex function and is nonincreasing -convex function on for some fixed Then the following inequality holds:

(229)

where

(230)

(231)

Proof.

Since is -convex function and is -convex function, then we have

(232)

for all . It is easy to observe that

(233)

Using the elementary inequality ( reals), (2.32) on the right side of (2.33) and making the charge of variable and since is nonincreasing, we have

(234)

Analogously we obtain

(235)

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.

References

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

  2. 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)

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

  4. 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 webcite

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

  6. Set, E, Özdemir, ME, Dragomir, SS: On the Hermite-Hadamard inequality and other integral inequalities involving two functions. Journal of Inequalities and Applications (2010)

  7. Toader, G: Some generalizations of the convexity. Proceedings of the Colloquium on Approximation and Optimization, Cluj-Napoca, Romania, pp. 329–338. University of Cluj-Napoca (1985)

  8. Miheşan, VG: A generalization of the convexity. In: Proceedings of the Seminar on Functional Equations, Approximation and Convexity, 1993, Cluj-Napoca, Romania

  9. Bakula, MK, Özdemir, ME, Pečarić, J: Hadamard type inequalities for m-convex and -convex functions. Journal of Inequalities in Pure and Applied Mathematics. 9, article no. 96, (2008)

  10. 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. 7(5, article no. 194), (2006)

  11. Pycia, M: A direct proof of the s-Hölder continuity of Breckner s-convex functions. Aequationes Mathematicae. 61(1-2), 128–130 (2001). Publisher Full Text OpenURL

  12. Özdemir, ME, Avci, M, Set, E: On some inequalities of Hermite-Hadamard type via m-convexity. Applied Mathematics Letters. 23(9), 1065–1070 (2010). Publisher Full Text OpenURL

  13. Gill, PM, Pearce, CEM, Pečarić, J: Hadamard's inequality for r-convex functions. Journal of Mathematical Analysis and Applications. 215(2), 461–470 (1997). Publisher Full Text OpenURL

  14. 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 OpenURL