International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2014 / Article

Research Article | Open Access

Volume 2014 |Article ID 163901 | 8 pages | https://doi.org/10.1155/2014/163901

On Some New Hermite-Hadamard Type Inequalities for s-Geometrically Convex Functions

Academic Editor: Henryk Hudzik
Received30 Dec 2013
Revised07 Jun 2014
Accepted09 Jun 2014
Published19 Jun 2014

Abstract

Some new integral inequalities of Hermite-Hadamard type related to the s-geometrically convex functions are established and some applications to special means of positive real numbers are also given.

1. Introduction

In this section, we firstly list several definitions and some known results.

Definition 1. Let be an interval in . Then is said to be convex if for all and .

Definition 2 (see [1]). Let . A function is said to be -convex in the second sense if for all and .

Definition 3 (see [2]). A function is said to be a geometrically convex function if for and .

Definition 4 (see [2]). A function is said to be an -geometrically convex function if for some , where and .

Example 5. (i) , is -geometrically convex for all .
(ii) , is -geometrically convex for all , where is a fixed number.

Let . It is easy to show that is -geometrically convex function on with if and only if is -convex function on .

Let be a convex function defined on the interval of real numbers and with . The following double inequality is well known in the literature as Hermite-Hadamard integral inequality: Recently, several integral inequalities connected with inequalities (5) for the -convex functions have been established by many authors; for example, see [1, 36]. In [2], the authors have established some integral inequalities connected with inequalities (5) for the -geometrically convex and monotonically decreasing functions. In [7], Tunç has established inequalities for -geometrically and geometrically convex functions which are connected with the famous Hermite-Hadamard inequality holding for convex functions. In [7], Tunç also has given the following result for geometrically convex and monotonically decreasing functions.

Corollary 6. Let be geometrically convex and monotonically decreasing on , with ; then one has

Note that inequalities (6) are also true without the condition monotonically decreasing and inequalities (6) are sharp. Also, inequality (6) obtained for geometrically convex functions is analogous with Hermite-Hadamard inequality (5).

Since is a geometrically convex function if and only if is a geometrically convex function, inequality (6) can be given as follows:

In this paper, the author gives new identities for differentiable functions. A consequence of the identities is that the author establishes some new inequalities connected with inequalities (6) for the -geometrically convex functions.

2. Main Results

In order to prove our results, we need the following lemma.

Lemma 7. Let be differentiable on and with . If , then

Proof. Integrating by part and changing variables of integration yield By the following equality, we obtain inequality (8): This completes the proof of Lemma 7.

Theorem 8. Let be differentiable on and with and . If is -geometrically convex on , for and , then where , ,

Proof. (1) Let . Since is -geometrically convex on , from Lemma 7 and power mean inequality, we have If , then (i)If , by (18), we obtain that (ii)If , by (18), we obtain that (iii)If , by (18), we obtain that (iv)If , by (18), we obtain that From (17) to (22), (12) holds.
(2) Let . Since is -geometrically convex on , from Lemma 7 and Hölder inequality, we have (i)If , by (18), we obtain that (ii)If , by (18), we obtain that (iii)If , by (18), we obtain that (iv)If , by (18), we obtain that From (23) to (27), (13) holds. This completes the required proof.

If we take in Theorem 8, we can derive the following corollary.

Corollary 9. Let be differentiable on and with and . If is geometrically convex on , for , then where , , and are the same as in Theorem 8.

If we take in Theorem 8, we can derive the following corollary.

Corollary 10. Let be differentiable on and with and . If is -geometrically convex on , for , then where , , , , and are the same as in Theorem 8.

Theorem 11. Let be differentiable on and with and . If is -geometrically convex on , for and , then where , , and , are the same as in (15).

Proof. (1) Let . Since is -geometrically convex on , from Lemma 7 and Hölder inequality, we have (i)If , by (18), we have (ii)If , by (18), we have (iii)If , by (18), we obtain that (iv)If , by (18), we obtain that From (33) to (37), (30) holds.
(2) Let , Since is -geometrically convex on , from Lemma 7 and Hölder inequality, we have From (38) and (34) to (37), (31) holds.

If taking in Theorem 11, we can derive the following corollary.

Corollary 12. Let be differentiable on and with and . If is geometrically convex on , for , then where , , and are the same as in Theorem 11.

3. Application to Special Means

Let us recall the following special means of two nonnegative numbers with .(1)The arithmetic mean (2)The geometric mean (3)The logarithmic mean (4)The -logarithmic mean

Let , and then the function is -geometrically convex on for (see [2]).

Proposition 13. Let , and . Then where and .

Proof. Let . Then , , . Thus, by Theorem 8, Proposition 13 is proved.

Proposition 14. Let , and . Then