On Hadamard-Type Inequalities Involving Several Kinds of Convexity

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.

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

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 ([1–6]) where further references are given.

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

(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)

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

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

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 ([9–12]).

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

(for , we put ).

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

Theorem 1.3.

Let be a positive, -convex function on . Then

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

If is a positive -concave function, then

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.

We start with the following theorem.

Theorem 2.1.

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

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

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

for all and

Integrating inequality (2.3) on over , we get

As

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

If are positive -concave functions, then

Proof.

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

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:

Proof.

We can write

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

Since are -convex functions, we obtain

for all and .

Rewriting (2.12) and (2.13), we have

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

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:

where is a logarithmic mean of positive real numbers.

Proof.

From inequality (2.14), we have

for all and

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

Since are -convex functions, then we get

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

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:

where

Proof.

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

for all . It is easy to observe that

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

Analogously we obtain

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:

where

Proof.

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

for all . It is easy to observe that

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

Analogously we obtain

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.

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

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

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

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