`International Journal of Mathematics and Mathematical SciencesVolumeÂ 2012, Article IDÂ 715751, 16 pageshttp://dx.doi.org/10.1155/2012/715751`
Research Article

Generalizations of the Simpson-Like Type Inequalities for Co-Ordinated -Convex Mappings in the Second Sense

Department of Mathematics, Hanseo University, Chungnam-do, Seosan-si 356-706, Republic of Korea

Received 14 October 2011; Revised 6 December 2011; Accepted 21 December 2011

Copyright Â© 2012 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.

Abstract

A generalized identity for some partial differentiable mappings on a bidimensional interval is obtained, and, by using this result, the author establishes generalizations of Simpson-like type inequalities for coordinated s-convex mappings in the second sense.

1. Introduction

In recent years, a number of authors have considered error estimate inequalities for some known and some new quadrature formulas. Sometimes they have considered generalizations of the Simpson-like type inequality which gives an error bound for the well-known Simpson rule.

Theorem 1.1. Let be a four-time continuous differentiable mapping on and . Then, the following inequality holds:

It is well known that the mapping is neither four times differentiable nor is the fourth derivative bounded on , then we cannot apply the classical Simpson quadrature formula.

For recent results on Simpson type inequalities, you may see the papers [1â€“5].

In [2, 6â€“8], Dragomir et al. and Park considered among others the class of mappings which are -convex on the coordinates.

In the sequel, in this paper let be a bidimensional interval in with and .

Definition 1.2. A mapping will be called -convex in the second sense on if the following inequality: holds, for all , and .

Modification for convex and -convex mapping on , which are also known as co-ordinated convex, -convex mapping, and --convex, respectively, were introduced by Dragomir, Sarikaya [5, 9, 10], and Park [4, 8, 11, 12].

Definition 1.3. A mapping will be called coordinated -convex in the second sense on if the partial mappings are -convex in the second sense, for all , , and [5, 9, 10].

A formal definition for coordinated -convex mappings may be stated as follow [8].

Definition 1.4. A mapping will be called coordinated -convex in the second sense on if the following inequality: holds, for all , , and .

In [2], S.S. Dragomir established the following theorem.

Theorem 1.5. Let be convex on the coordinates on . Then, one has the inequalities:

In [13], Hwang et al. gave a refinement of Hadamardâ€™s inequality on the coordinates and they proved some inequalities for coordinated convex mappings.

In [1, 6, 14], Alomari and Darus proved inequalities for coordinated -convex mappings.

In [15], Latif and Alomari defined coordinated -convex mappings, established some inequalities for co-ordinated -convex mappings and proved inequalities involving product of convex mappings on the coordinates.

In [3], Ă–zdemir et al. gave the following theorems:

Theorem 1.6. Let be a partial differentiable mapping on . If is convex on the coordinates on , then the following inequality holds: where

Theorem 1.7. Let be a partial differentiable mapping on . If is bounded, that is, for all , then the following inequality holds: where is defined in Theorem 1.6.

In [3], Ă–zdemir et al. proved a new equality and, by using this equality, established some inequalities on coordinated convex mappings.

In this paper the author give a generalized identity for some partial differentiable mappings on a bidimensional interval and, by using this result, establish a generalizations of Simpson-like type inequalities for coordinated -convex mappings in the second sense.

2. Main Results

To prove our main results, we need the following lemma.

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

Proof. By the definitions of and , we can write where
By integration by parts, we have
By using the equalities (2.5) and (2.6) in (2.3), we have where
Note that
By the equalities (2.9) and (2.10), we have
By the similar way, we get the following:
By the equalities (2.7) and (2.11)â€“(2.14) and using the change of the variables and for , then multiplying both sides with , we have the required result (2.1), which completes the proof.

Remark 2.2. Lemma 2.1 is a generalization of the results which proved by Sarikaya, Set, Ă–zdemir, and Dragomir [3, 5, 9, 10].

Theorem 2.3. Let be a partial differentiable mapping on . If is in and is a coordinated -convex mapping in the second sense on , then, for and with , and the following inequality holds: where for

Proof. From Lemma 2.1 and by the coordinated -convexity in the second sense of , we can write
Note that
By (2.18) and (2.19), we get the inequality (2.15) by the simple calculations.

Remark 2.4. In Theorem 2.3,(i)if we choose , , and in (2.15), then we get (ii)if we choose , , and in (2.15), then we get where which implies that Theorem 2.3 is a generalization of Theorem 1.6.

Theorem 2.5. Let be a partial differentiable mapping on . If is bounded, that is, for all , then, for and with and , the following inequality holds:

Proof. From Lemma 2.1, using the property of modulus and the boundedness of , we get
By the simple calculations, we have
By using the inequality (2.25) and the equalities (2.26)-(2.27), the assertion (2.24) holds.

Remark 2.6. In Theorem 2.5,(i)if we choose and , then we get (ii)if we choose and , then we get which implies that Theorem 2.5 is a generalization of Theorem 1.7.

The following theorem is a generalization of Theorem 1.6.

Theorem 2.7. Let be a partial differentiable mapping on . If is in and is a coordinated -convex mapping in the second sense on , then, for and with and , the following inequality holds: where

Proof. From Lemma 2.1, we can write
Hence, by the inequality (2.32) and the coordinated -convexity in the second sense of , it follows that
Note that
By the inequality (2.33) and the equalities (2.34) and (2.35), the assertion (2.30) holds.

Remark 2.8. In Theorem 2.7,(i)if we choose , , and , then we get (ii)if we choose , , and , then we get (iii)if we choose , , , and , then we get where

Theorem 2.9. Let be a partial differentiable mapping on . If is in and is a coordinated -convex mapping in the second sense on , then, for and with and , the following inequality holds: where and are as given in Theorem 2.3.

Proof. From Lemma 2.1, we can write
By the simple calculations, we have
Since is a coordinated -convex mapping in the second sense on , we have that, for ,
By (2.41)â€“(2.44), the assertion (2.40) holds.

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