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On the Simpson’s Inequality for Convex Functions on the Co-Ordinates

M. EMIN ÖZDEMIR, AHMET OCAK AKDEMIR*, HAVVA KAVURMACI

Ağri İbrahim Çeçen University, Faculty of Science and Letters, Department of Mathematics, AĞRI, Turkey

Abstract: In this paper, a new lemma is proved and inequalities of Simpson type are established for convex functions on the co-ordinates and bounded functions.

Keywords: Simpson’s inequality, co-ordinates, convex functions, bounded functions

Received July 28, 2014; Revised September 19, 2014; Accepted September 27, 2014

Citation: M. EMIN ÖZDEMIR, AHMET OCAK AKDEMIR, HAVVA KAVURMACI, On the Simpson’s Inequality for Convex Functions on the Co-Ordinates. Turkish Journal of Analysis and Number Theory, 2 (5). 165-169. January 2014. doi:10.12691/tjant-2-5-2

1. Introduction

The following inequality is well-known in the literature as Simpson.s inequality:

Theorem 1. Let be a four times continuously differentiable mapping on and Then the following inequality holds:

For recent results on Simpson.s type inequalities see the papers [11-19].

Convexity on the co-ordinates can be given as following (see [10]);

Let us consider the bidimensional interval in with and A function will be called convex on the co-ordinates if the partial mappings and are convex where defined for all and

Recall that the mapping is convex on , if the following inequality;

holds for all and

In [10], Dragomir proved the following inequalities:

Theorem 2. Suppose that is convex on the co-ordinates on Then one has the inequalities;

[picture] | (1.1)

The above inequalities are sharp.

Recently, several papers have been written on the convex functions on the co-ordinates. Similar results can be found in [1-9] and [20,21,22,23].

In this paper, we will give Simpson-type inequalities for convex functions on the co-ordinates and bounded functions on the basis of the following lemma.

2. Main Results

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

Lemma 1. Let be a partial differentiable mapping on If then the following equality holds:

[picture] | (2.1)

where

and

Proof. Integrating by parts, we can write

By integrating the right hand side of equality, we get

Computing these integrals and using the change of the variable and for then multiplying both sides with we get the desired result.

Theorem 3. Let be a partial differentiable mapping If is a convex function on the co-ordinates on and ,then the following inequality holds:

where

Proof. By using Lemma 1, we can write

Since is co-ordinated convex on ,we get

Computing the integral in the right hand side of above inequality, we have

We obtain

[picture] | (2.2)

By a similar argument for the above integral, we have

[picture] | (2.3)

If we use (2.3) in (2.2), we get the required result.

Theorem 4. Let be a partial differentiable mapping on If is bounded, i.e.,

for all and Then the following inequality holds:

where A is as in Theorem 3.

Proof. From Lemma 1 and using the property of modulus, we have

Since is bounded, we have

[picture] | (2.4)

By a simple calculation,

[picture] | (2.5)

If we use (2.5) in (2.4), we have

This completes the proof.

References

[1]   | Latif, M.A. and Alomari, M., On Hadamard-type inequalities for h-convex functions on the co-ordinates, International Journal of Math. Analysis, 3 (2009), no: 33, 1645-1656.
[2]   | Latif, M.A. and Alomari, M., Hadamard-type inequalities for product two convex functions on the co-ordinates, International Mathematical Forum, 4 (2009), no: 47, 2327-2338.
[3]   | Bakula, M.K. and Peµcari´c, J., On the Jensen.s inequality for convex functions on the co-ordinates in a rectangle from the plane, Taiwanese Journal of Math., 5 (2006), 1271-1292.
[4]   | Alomari, M. and Darus, M., The Hadamard.s inequality for s-convex function of 2-variables on the co-ordinates, International Journal of Math. Analysis, 2 (2008) no: 13, 629-638.
[5]   | Alomari, M. and Darus, M., Hadamard-type inequalities for s-convex functions, Interna-tional Mathematical Forum, 3 (2008), no: 40, 1965-1975.
[6]   | Alomari, M. and Darus, M., Co-ordinated s-convex function in the first sense with some Hadamard-type inequalities, Int. Journal Contemp. Math. Sciences, 3 (2008), no: 32, 1557-1567.
[7]   | Hwang, D.Y., Tseng, K.L. and Yang, G.S., Some Hadamard.s inequalities for co-ordinated convex functions in a rectangle from the plane, Taiwanese Journal of Mathematics, 11 (2007), 63-73.
[8]   | Özdemir, M.E., Set, E. and Sarkaya, M.Z., Some new Hadamard.s type inequalities for co-ordinated m-convex and (α, m)-convex functions, Hacettepe J. of. Math. and St., 40, 219-229, (2011).
[9]   | Sarkaya, M.Z., Set, E., Özdemir, M.E. and Dragomir, S. S., New some Hadamard’s type inequalities for co-ordinated convex functions, Tamsui Oxford Journal of Information and Mathematical Sciences, 28 (2), (2012), 137-152.
[10]   | Dragomir, S.S., On the Hadamard’s inequality for convex functions on the co-ordinates in a rectangle from the plane, Taiwanese Journal of Mathematics, 5 (2001), no: 4, 775-788.
[11]   | Sarkaya, M.Z., Set, E. and Özdemir, M.E., On new Inequalities of Simpson’s type for functions whose second derivatives absolute values are convex, RGMIA Res. Rep. Coll., 13 (1) (2010), Article 1.
[12]   | Sarkaya, M.Z., Set, E. and Özdemir, M.E., On new inequalities of Simpson.s type for convex functions, RGMIA Res. Rep. Coll., 13 (2) (2010), Article 2.
[13]   | Set, E., Özdemir, M.E. and Sarkaya, M.Z., On new inequalities of Simpson.s type for quasiconvex functions with applications, RGMIA Res. Rep. Coll., 13 (1) (2010), Article 6.
[14]   | Sarkaya, M.Z., Set, E. and Özdemir, M.E., On new inequalities of Simpson.s type for s-convex functions, Computers & Mathematics with Applications, 60, 8 (2010).
[15]   | Liu, B.Z., An inequality of Simpson type, Proc. R. Soc. A, 461 (2005), 2155-2158.
[16]   | Dragomir, S.S., Agarwal, R.P. and Cerone, P., On Simpson’s inequality and applications, J. of Ineq. and Appl., 5 (2000), 533-579.
[17]   | Alomari, M., Darus, M. and Dragomir, S.S., New inequalities of Simpson.s type for s-convex functions with applications, RGMIA Res. Rep. Coll., 12 (4) (2009), Article 9.
[18]   | Ujević, N., Double integral inequalities of Simpson type and applications, J. Appl. Math. and Computing, 14 (2004), no: 1-2, p. 213-223.
[19]   | Zhongxue, L., On sharp inequalities of Simpson type and Ostrowski type in two independent variables, Comp. and Math. with Appl., 56 (2008), 2043-2047.
[20]   | Özdemir, M.E., Tunç, M. and Akdemir, A.O., On some new Hadamard-like inequalities for co-ordinated s-convex Functions, Facta Universitatis Series Mathematics and Informatics, Vol 28 No 3 (2013).
[21]   | Özdemir, M.E., Akdemir, A.O. and Yldz, Ç., On co-ordinated quasi-convex functions, Czechoslovak Mathematical Journal, 62(137) (2012), 889-900.
[22]   | Özdemir, M.E., Kavurmac, H., Akdemir, A.O. and Avc, M., Inequalities for convex and s-convex functions on Δ = [a b]×[c,d], Journal of Inequalities and Applications, 2012, Published: 1 February 2012.
[23]   | Özdemir, M.E., Yldz, Ç. and Akdemir, A.O., On some new Hadamard-type inequalities for co-ordinated quasi-convex functions, Hacettepe Journal of Mathematics and Statistics, 41(5) (2012), 697-707.
[24]   | İşcan, İ., A new generalization of some integral inequalities for (α,m)-convex functions, Mathematical Sciences, 7(1) (2013), 1-8.