Abstract

The author obtained new generalizations and refinements of some inequalities based on differentiable -convex mappings in the second sense. Also, some applications to special means of real numbers are given.

1. Introduction

Recall that the function , is said to be -convex in the second sense for if for all and [1–5].

In (1.1), if we let , is said to be a convex mapping on an interval [1].

Let us denote the set of -convex mappings in the second sense on by .

For some further properties of the -convex mappings, see [1–3, 6]. In recent, M. Z. Sarikaya et al. [4], and U. S. Kirmaci et al. [7] established a more general result of the Hermite-Hadamard inequalities.

For recent years many authors have established error estimations for the Simpson’s inequality: for refinements, counterparts, generalizations, and new Simpson's type inequalities, see [1, 4, 6, 8].

S. S. Dragomir et al. [9], and M. Alomari et al. [8] proved the following developments on Simpson's inequality for which the remainder is expressed in terms of lower derivatives than the twice.

In the sequel, denote the interior of an interval by .

Theorem 1.1. Let be an absolutely continuous mapping on such that , where with . Then the following inequality holds:

In this article, the author gives some generalized Simpson’s type inequalities based on -convex mappings in the second sense by using the following lemma.

2. Generalization of Inequalities Based on s-Convex Mappings

In this article, for the simplicity of the notation, let for with for any integer .

In order to generalize the classical Simpson-like type inequalities, we need the following lemma [1, 6].

Lemma 2.1. Let , be a differentiable mapping on such that , where with and . If , then, for with for any the following equality holds: for each , where

Proof. By the integration by parts, we have which completes the proof.

Theorem 2.2. Let , be a differentiable mapping on such that , where with and . If , for some , then for with for any the following inequality holds: where

Proof. From Lemma 2.1 and since is -convex on , by using Hölder integral inequality, we have which implies the theorem.

Corollary 2.3. In Theorem 2.2, one has:
(i)
(ii) which implies that Corollary 2.3 is a generalization of Theorem 1.1.

Theorem 2.4. Let , be a differentiable mapping on such that , where with and . If , for some fixed and with , then for with for any the following inequality holds:

Proof. From Lemma 2.1, using the Hölder inequality we get Since for a fixed , we have(a)(b) By (2.11) and (2.12), the assertion (2.10) holds.

Corollary 2.5. In Theorem 2.4, letting and , one has

Theorem 2.6. Let , be a differentiable mapping on such that , where with and . If , for some fixed and with , then for with for any the following inequality holds:

Proof. Note that By (2.11) and (2.16), the assertion (2.15) of this theorem holds.

Corollary 2.7. In Theorem 2.6, letting , then one has which implies that Theorem 2.6 is a generalization of Theorem 1.1.

Theorem 2.8. Let , be a differentiable mapping on such that , where with and . If , for some fixed and with , then for with for any the following inequality holds: where

Proof. Suppose that . From Lemma 2.1, using the power mean inequality one has
Since is -convex on , we have By the above facts (2.20) and (2.21), the assertion (2.18) in this theorem is proved.

Corollary 2.9. In Theorem 2.8, letting , one has which implies that
Especially, in Theorem 2.8, letting and , one has

3. Applications to Special Means

We now consider the applications of our theorems to the followings special means.

(a) The arithmetic mean: .

(b) The -logarithmic mean: