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Some New Ostrowski Type Inequalities for Co-Ordinated Convex Functions

Mehmet Zeki SARIKAYA*, Hüseyin BUDAK, Hatice YALDIZ

Department of Mathematics, Faculty of Science and Arts, Düzce University, Düzce-TURKEY

Abstract: In this paper, we obtain new identity for function of two variables and apply them to give new Ostrowski type integral inequality for double integrals involving functions whose derivatives are co-ordinates convex function on in R² with .

Keywords: Ostrowski type inequalities, coordinated convex functions, Hölder's inequality

Received September 10, 2014; Revised October 01, 2014; Accepted October 12, 2014

Citation: Mehmet Zeki SARIKAYA, Hüseyin BUDAK, Hatice YALDIZ, Some New Ostrowski Type Inequalities for Co-Ordinated Convex Functions. Turkish Journal of Analysis and Number Theory, 2 (5). 176-182. January 2014. doi:10.12691/tjant-2-5-4

1. Introduction

Let be a differentiable mapping on whose derivative is baunded on i.e. Then, we have the inequality

[picture] | (1.1)

for all (see, [13]). The constant is the best possible. This inequality is well known in the literature as the Ostrowski inequality. For some results which generalize, improve and extend the inequality (1.1) see ([5,6,7,14,15,16,17]) and the references therein.

Let us consider now a bidimensional interval in with and a mapping is said to be convex on if the inequality

holds for all and The mapping is said to be concave on the co-ordinates on if the above inequality holds in reverse direction, for all and

A formal definition for coordinated convex function may be stated as follows:

Definition 1. A function will be coordinated convex on , for all and if the following inequality holds:

Clearly, every convex function is coordinated convex. Furthermore, there exist coordinated convex function which is not convex, (see, [3]). For several recent results concerning Hermite-Hadamard's inequality for some convex function on the co-ordinates on a rectangle from the planewe refer the reader to ([1,2,3,4,8-12,18,19]).

Also, in [3], Dragomir establish the following Hermite-Hadamard's type inequality for coordinated convex mapping on a rectangle from the plane

Theorem 1. Suppose that is coordinated convex on Then one has the inequalities:

[picture] | (1.2)

The above inequalities are sharp.

In a recent paper [5], Barnett and Dragomir proved the following Ostrowski type inequality for double integrals:

Theorem 2. Let be continuous on exists on and is boundedi.e.,

Then, we have the inequality:

[picture] | (1.3)

for all

The main aim of this paper is to establish some new Ostrowski type inequalities for double integrals involving functions whose derivatives are co-ordinates convex function on in with .

2. Main Results

To establish our main results, we need the following identity:

Lemma 1. Let be a partial differentiable mapping on . If then for any we have the equality:

[picture] | (2.1)

Proof For any and , we have

and

For and we obtain

[picture] | (2.2)

By integrating (2.2) with respect to on and divide by we get the desired equality (2.1).

Theorem 3. Let be a partial differentiable mapping on and is co-ordinates convex function on .

(i) If then for any

(ii) If then for any

(iii) If then for any

Proof (i). Using (2.1), convexity of and taking the modulus, it follows that

Since we get

(ii). As above, we can write

Using Hölder's ineguality for , we obtain

(iii). As above, we obtain the following inequality

Using convexity of we obtain

This completes the proof.

Corollary 1. With the assumptions of Theorem 3 with and , we have the inequality

provided

If we have

If then

Theorem 4. Let be a partial differentiable mapping on and is co-ordinates convex function on .

(i) If then for any

(ii) If then for any

(iii) If then for any

Proof As in the proof of Theorem 3, we can write

for any From Hölder's inequality, we get

for any .

Since is a co-ordinates convex function on we get

for any Therefore

[picture] | (2.3)

(i). Now, if then

for any

(ii). If by using Hölder's inequality in (2.3), we have

for any

(iii) If , then, by Hölder’s inequality, we have

Corollary 2. With the assumptions of Theorem 4 with and , we have the inequality

where

If then we have

If , then we have

References

[1]   | M. Alomari and M. Darus, Co-ordinated s-convex function in the first sense with some Hadamard-type inequalities, Int. J. Contemp. Math. Sciences, 3 (32) (2008), 1557-1567.
[2]   | M. Alomari and M. Darus, On the Hadamard's inequality for log -convex functions on the coordinates, J. of Inequal. and Appl, Article ID 283147, (2009), 13 pages.
[3]   | S.S. Dragomir, On Hadamard's inequality for convex functions on the co-ordinates in a rectangle from the plane, Taiwanese Journal of Mathematics, 4 (2001), 775-788.
[4]   | M.E. Özdemir, E. Set and M.Z. Sarikaya, New some Hadamard's type inequalities for co-ordinated m-convex and (α,m) -convex functions, RGMIA, Res. Rep. Coll., 13 (2010), Supplement, Article 4.
[5]   | N. S. Barnett and S. S. Dragomir, An Ostrowski type inequality for double integrals and applications for cubature formulae, Soochow J. Math., 27 (1), (2001), 109-114.
[6]   | P. Cerone and S.S. Dragomir, Ostrowski type inequalities for functions whose derivatives satisfy certain convexity assumptions, Demonstratio Math., 37 (2004), no. 2, 299-308.
[7]   | S. S. Dragomir, Ostrowski type inequalities for functions whose derivatives are h-convex in absolute value, RGMIA Research Report Collection, 16 (2013), Article 71, 15 pp.
[8]   | M. A. Latif and M. Alomari, Hadamard-type inequalities for product two convex functions on the co-ordinetes, Int. Math. Forum, 4 (47), 2009, 2327-2338.
[9]   | M. A. Latif and M. Alomari, On the Hadamard-type inequalities for h-convex functions on the co-ordinetes, Int. J. of Math. Analysis, 3 (33), 2009, 1645-1656.
[10]   | M. A. Latif, S. Hussain and S. S. Dragomir, New Ostrowski type inequalities for co-ordinated convex functions, TJMM, 4 (2012), No. 2, 125-136.
[11]   | M. A. Latif, S. S. Dragomir, A. E. Matouk, New inequalites of Ostrowski type for co-ordinated s -convex functions via fractional integrals, Journal of Fractional Calculus and Applications,Vol. 4 (1) Jan. 2013, pp. 22-36.
[12]   | M. A. Latif and S. S. Dragomir, New Ostrowski type inequalites for co-ordinated S-convex functions in the second sense,Le Matematiche Vol. LXVII (2012), pp. 57-72.
[13]   | A. M. Ostrowski, Über die absolutabweichung einer differentiebaren funktion von ihrem integralmitelwert, Comment. Math. Helv. 10 (1938), 226-227.
[14]   | B. G. Pachpatte, On an inequality of Ostrowski type in three independent variables, J. Math. Anal. Appl., 249 (2000), 583-591.
[15]   | M. Z. Sarikaya, On the Ostrowski type integral inequality, Acta Math. Univ. Comenianae, Vol. LXXIX, 1 (2010), pp. 129-134.
[16]   | M. Z. Sarikaya On the Ostrowski type integral inequality for double integrals, Demonstratio Mathematica, Vol. XLV No 3 2012.
[17]   | M. Z. Sarikaya and H. Ogunmez, On the weighted Ostrowski type integral inequality for double integrals, The Arabian Journal for Science and Engineering (AJSE)-Mathematics, (2011) 36: 1153-1160.
[18]   | M. Z. Sarikaya, E. Set, M. E. Ozdemir and S. S. Dragomir, New some Hadamard's type inequalities for co-ordinated convex functions, Tamsui Oxford Journal of Information and Mathematical Sciences, 28 (2) (2012) 137-152.
[19]   | M. Z. Sarikaya and H. Yaldiz, On the Hadamard's type inequalities for L-Lipschitzian mapping, Konuralp Journal of Mathematics, Volume 1, No. 2, pp. 33-40 (2013).