Keywords: Ostrowski type inequalities, coordinated convex functions, Hölder's inequality
Turkish Journal of Analysis and Number Theory, 2014 2 (5),
pp 176182.
DOI: 10.12691/tjant254
Received September 01, 2014; Revised October 03, 2014; Accepted October 12, 2014
Copyright © 2013 Science and Education Publishing. All Rights Reserved.
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 coordinates 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 HermiteHadamard's inequality for some convex function on the coordinates on a rectangle from the planewe refer the reader to ([1,2,3,4,812,18,19]).
Also, in ^{[3]}, Dragomir establish the following HermiteHadamard'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 coordinates 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 coordinates 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 coordinates 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 coordinates 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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