International Journal of Mathematics and Mathematical Sciences

VolumeÂ 2012Â (2012), Article IDÂ 715751, 16 pages

http://dx.doi.org/10.1155/2012/715751

## 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

Academic Editor: FengÂ Qi

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.

#### References

- M. Alomari, M. Darus, and S. S. Dragomir, â€śNew inequalities of Simpson’s type for
*s*-convex functions with applications,â€ť*RGMIA Research Report Collection*, vol. 12, no. 4, article 9, 2009. View at Google Scholar - S. S. Dragomir, R. P. Agarwal, and P. Cerone, â€śOn Simpson's inequality and applications,â€ť
*Journal of Inequalities and Applications*, vol. 5, no. 6, pp. 533â€“579, 2000. View at Publisher Â· View at Google Scholar Â· View at Zentralblatt MATH Â· View at MathSciNet - M. E. Özdemir, A. O. Akdemir, H. Kavurmaci, and M. Avci, â€śOn the Simpson’s inequality for co-ordinated convex functions,â€ť
*Classical Analysis and ODEs*, http://arxiv.org/abs/1101.0075. - J. Park, â€śGeneralization of some Simpson-like type inequalities via differentiable
*s*-convex mappings in the second sense,â€ť*International Journal of Mathematics and Mathematical Sciences*, vol. 2011, Article ID 493531, 13 pages, 2011. View at Google Scholar - M. Z. Sarikaya, E. Set, and M. E. Ozdemir, â€śOn new inequalities of Simpson's type for
*s*-convex functions,â€ť*Computers & Mathematics with Applications*, vol. 60, no. 8, pp. 2191â€“2199, 2010. View at Publisher Â· View at Google Scholar - M. Alomari and M. Darus, â€śCo-ordinated s-convex function in the first sense with some Hadamard-type inequalities,â€ť
*International Journal of Contemporary Mathematical Sciences*, vol. 3, no. 29-32, pp. 1557â€“1567, 2008. View at Google Scholar - S. S. Dragomir, â€śOn the Hadamard's inequality for convex functions on the co-ordinates in a rectangle from the plane,â€ť
*Taiwanese Journal of Mathematics*, vol. 5, no. 5, pp. 775â€“788, 2001. View at Google Scholar Â· View at Zentralblatt MATH - J. Park, â€śGeneralizations of the Simpson-like type inequalities for coordinated s-convex mappings,â€ť
*Far East Journal of Mathematical Sciences*, vol. 54, no. 2, pp. 225â€“236, 2011. View at Google Scholar - M. Z. Sarikaya, E. Set, M. E. Ozdemir, and S. S. Dragomir, â€śNew some Hadamard’s type inequalities for co-ordinated convex functions,â€ť
*Classical Analysis and ODEs*, http://arxiv.org/abs/1005.0700. - E. Set, M. E. Ozdemir, and M. Z. Sarikaya, â€śInequalities of Hermite-Hadamard’s type for functions whose derivatives absolute values are mconvex,â€ť
*AIP Conference Proceedings*, vol. 1309, no. 1, pp. 861â€“873, 2010. View at Google Scholar - J. Park, â€śNew generalizations of Simpson's type inequalities for twice differentiable convex mappings,â€ť
*Far East Journal of Mathematical Sciences*, vol. 52, no. 1, pp. 43â€“55, 2011. View at Google Scholar - J. Park, â€śSome Hadamard's type inequalities for co-ordinated (
*s, m*)-convex mappings in the second sense,â€ť*Far East Journal of Mathematical Sciences*, vol. 51, no. 2, pp. 205â€“216, 2011. View at Google Scholar - D.-Y. Hwang, K.-L. Tseng, and G.-S. Yang, â€śSome Hadamard's inequalities for co-ordinated convex functions in a rectangle from the plane,â€ť
*Taiwanese Journal of Mathematics*, vol. 11, no. 1, pp. 63â€“73, 2007. View at Google Scholar Â· View at Zentralblatt MATH - M. Alomari and M. Darus, â€śThe Hadamard's inequality for
*s*-convex function of 2-variables on the co-ordinates,â€ť*International Journal of Mathematical Analysis*, vol. 2, no. 13-16, pp. 629â€“638, 2008. View at Google Scholar - M. A. Latif and M. Alomari, â€śHadamard-type inequalities for product two convex functions on the co-ordinates,â€ť
*International Mathematical Forum*, vol. 4, no. 45-48, pp. 2327â€“2338, 2009. View at Google Scholar Â· View at Zentralblatt MATH