International Journal of Analysis

International Journal of Analysis / 2016 / Article

Research Article | Open Access

Volume 2016 |Article ID 6749213 | https://doi.org/10.1155/2016/6749213

Badreddine Meftah, "Some New Ostrwoski’s Inequalities for Functions Whose th Derivatives Are -Convex", International Journal of Analysis, vol. 2016, Article ID 6749213, 7 pages, 2016. https://doi.org/10.1155/2016/6749213

Some New Ostrwoski’s Inequalities for Functions Whose th Derivatives Are -Convex

Academic Editor: Ahmed Zayed
Received20 Jul 2016
Accepted14 Nov 2016
Published28 Nov 2016

Abstract

We prove some new Ostrowski’s inequalities for functions whose th derivatives are -convex.

1. Introduction

In 1938, A. M. Ostrowski proved an interesting integral inequality, given by the following theorem.

Theorem 1 ([1]). Let be a differentiable mapping on (interior of ), and let with . If for all , then

Inequality (1) has attracted much interest due to its diversity of applications in numerical analysis, probability theory, and other areas. We note that a numerous variants, extensions, and generalizations of inequality (1) have been discovered.

In [2], Cerone et al. proved the following identity.

Lemma 2 ([2, Lemma ]). Let be a mapping such that is absolutely continuous on . Then for all one has the identitywhere the kernel is given byand is natural number, .

We also recall that a positive function is said to be -convex on , if the following inequalityholds for all and ; see [3].

In this paper we establish some new Ostrwoski’s inequalities for functions whose th derivatives are -convex.

2. Main Results

In order to establish our results, we need these lemmas.

Lemma 3 ([4]). For and , ,where .

Lemma 4 ([5]). For and , the following algebraic inequalities are true:

Theorem 5. For , let be -time differentiable on such that . If is -convex, then the following inequality holds for all , where , , and

Proof. From Lemma 2 and properties of modulus, we have Since is -convex function, for , the use of Lemma 4 gives where is defined as in (9).
In the case where , (10) becomesand we distinguish 4 cases.
If , then (12) givesIf , then (12) becomeswhere we have used (5).
If , then (12) becomes where we have used (6) with .
In the case where and , using Lemma 3, (12) givesThe desired result follows from (11) and (13)–(16).

Theorem 6. For , let be -time differentiable on such that and let . If is -convex, then the following inequalityholds for all , where , , , and is defined as in (9).

Proof. From Lemma 2, properties of modulus, and Hölder’s inequality, we haveIn the case where , the use of -convexity of and Lemma 4 givesFor , using the -convexity of , (18) gives Analogously to Theorem 5, we will treat the 4 cases.
If , then (20) givesIf , then (20) becomes If , then (20) becomesIn the case where and , (20) givesThe desired result follows from (19) and (21)–(24).

Theorem 7. For , let be -time differentiable on such that and let . If is convex, then the following inequality holds for all , where , , and is defined as in (9).

Proof. From Lemma 2, properties of modulus, and power mean inequality, we have If , using -convexity of and Lemma 4, we getNow, suppose that , and from the convexity of , (25) becomesIf , then (27) givesIf , then (27) becomesIf , then (27) becomesIn the case where and , (27) gives