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

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) giveswhere we have used Lemma 3. The desired result follows from (26) and (28)–(31).

Competing Interests

The author declares that there are no competing interests regarding the publication of this paper.