1. Introduction

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

(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:

(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:

(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:

(2.1)

Proof For any and , we have

and

For and we obtain

(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

(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

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