#### 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:

for and .

Now, using the results of Section 2, some new inequalities are derived for the following means:

(1.1) Let .

(a) In Theorem 2.2,

(i) if and , then we get

and,

(ii) if and , then we have

(b) In Theorem 2.4,

(i) if , and then we get:

and

(ii) if , and , then we have

(c) In Theorem 2.6,

(i) if , and then we get

and

(ii) if , and , then we have In Theorem 2.8,

(i) if , and then we get

where

and

(ii) if , and , then we have

where

(2.2) Let .

(a) In Theorem 2.2,

(i) if and , then we get

and

(ii) if and , then we have

(b) In Theorem 2.4,

(i) if and , then we get

and

(ii) if and , then we have

(c) In Theorem 2.6,

(i) if and , then we get

and

(ii) if and , then we have

(d) In Theorem 2.8,

(i) if , and then we get

where

and

(ii) if , and , then we have

where

#### Acknowledgments

The author is thankful to Professor Merve Avci and Uur S. Kirmaci, Atatürk University, K. K. Education Faculty, Deptartment of Mathematics, for their valuable suggestions and for the improvement of this paper. This work is financially supported by Hanseo University research fund no. 111-S-101.