International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 493531 | 13 pages | https://doi.org/10.1155/2011/493531

Generalization of Some Simpson-Like Type Inequalities via Differentiable s-Convex Mappings in the Second Sense

Academic Editor: Jewgeni Dshalalow
Received31 Mar 2011
Accepted24 Jun 2011
Published17 Aug 2011

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 [15].

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 [13, 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.

References

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Copyright © 2011 Jaekeun Park. 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.


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