International Journal of AnalysisVolume 2014 (2014), Article ID 125439, 6 pageshttp://dx.doi.org/10.1155/2014/125439

Research Article

Hermite-Hadamard and Simpson Type Inequalities for Differentiable P-GA-Functions

İmdat İşcan

Department of Mathematics, Faculty of Arts and Sciences, Giresun University, Giresun, Turkey

Received 21 February 2014; Accepted 14 May 2014; Published 22 May 2014

Academic Editor: Seenith Sivasundaram

Copyright © 2014 İmdat İşcan. 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.

Abstract.

The author introduces the concept of the -GA-functions, gives Hermite-Hadamard's inequalities for -GA-functions, and defines a new identity. By using this identity, the author obtains new estimates on generalization of Hadamard and Simpson type inequalities for -GA-functions. Some applications to special means of real numbers are also given.

1. Introduction

Let real function be defined on some nonempty interval of real line . The function is said to be convex on if inequality holds for all and .

We recall that a function is said to be -function on or belong to the class if it is nonnegative and for all and . Note that contain all nonnegative convex and quasiconvex functions [1].

The following inequalities are well known in the literature as Hermite-Hadamard inequality and Simpson inequality, respectively.

Theorem 1. Let be a convex function defined on the interval of real numbers and with . The following double inequality holds:

Theorem 2. Let be a four times continuously differentiable mapping on and . Then the following inequality holds:

Definition 3 (see [2, 3]). A function is said to be GA-convex (geometric-arithmetically convex) if for all and .

In recent years, many authors have studied errors estimations for Hermite-Hadamard and Simpson inequalities; for refinements, counterparts, and generalization concerning -functions and GA-convex, see [4–11].

In this paper, the concept of the -GA-function is introduced, Hermite-Hadamard’s inequalities for -GA-functions are established, and a new identity for differentiable functions is defined. By using this identity, the author obtains a generalization of Hadamard and Simpson type inequalities for -GA-functions.

2. Main Results

Let be a differentiable function on , the interior of ; throughout this section we will take where with and .

Definition 4. A function is said to be -GA-function (-geometric-arithmetic function) on if for any and .

Proposition 5. Let . If is -function and nondecreasing, then is -GA-function on .

Proof. This follows from for all and .

Proposition 6. Let . If is -GA-function and nonincreasing, then is -function on .

Proof. The conclusion follows from for all and , respectively.

Hermite-Hadamard’s inequalities can be represented for -GA-functions as follows.

Theorem 7. Let be a function such that ( is integrable on ), where with . If is a -GA-function on , then the following inequalities hold: with .

Proof. Since is a -GA-function on , we have for all (with in inequality (7)) Choosing ,  , we get Integrating the resulting inequality with respect to over , we obtain and the first inequality is proved.For the proof of the second inequality in (10) we first note that if is a -GA-function, then, for , it yields By adding side to side these inequalities and taking square root we have and, integrating the resulting inequality with respect to over , we obtain The proof is completed.

In order to prove our main results we need the following identity.

Lemma 8. Let be a differentiable function on such that , where with . Then for all , , and one has

Proof. By integration by parts and changing the variable, we can state and similarly we get Adding the resulting identities we obtain the desired result.

Theorem 9. Let be a differentiable function on such that , where with . If is -GA-function on for some fixed , , then the following inequality holds: where

Proof. Since is -GA-function on , for all , Hence, using Lemma 8 and power mean inequality, we get which completes the proof.

Corollary 10. Under the assumptions of Theorem 9 with , inequality (20) reduced to the following inequality:

Corollary 11. Under the assumptions of Theorem 9 with , inequality (20) reduced to the following inequality: In particular, for , we get For , we get and, for ,

Theorem 12. Let be a differentiable function on such that , where with . If is -GA-function on for some fixed , , then the following inequality holds: where and is n-logarithmic mean defined with , .

Proof. Since is -GA-function on and using Lemma 8 and Hölder inequality, we get Here it is seen by simple computation that Hence, the proof is completed.

Corollary 13. Under the assumptions of Theorem 12 with , inequality (29) reduced to the following inequality: In particular, for , we get For , we get and, for , we get

3. Application to Special Means

Let us recall the following special means of two nonnegative numbers , with :(1)the arithmetic mean (2)the weighted arithmetic mean (3)the geometric mean (4)the weighted geometric mean (5)the logarithmic mean (6)the -logarithmic mean

Proposition 14. For , , , and , one has where is defined as in (21).

Proof. Let , , , and.

Proposition 15. For , , , and , one has

Proof. Let , , , and .

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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