Keywords: Ostrowski inequality, Pompeiu's mean value theorem
Turkish Journal of Analysis and Number Theory, 2014 2 (3),
pp 8084.
DOI: 10.12691/tjant235
Copyright © 2013 Science and Education Publishing. All Rights Reserved.
1. Introduction
The inequality of Ostrowski ^{[7]} gives us an estimate for the deviation of the values of a smooth function from its mean value. More precisely, if is a differentiable function with bounded derivative, then
for every . Moreover the constant is the best possible. For a differentiable function ,, Dragomir has in ^{[2]} proved, using Pompeiu's mean value theorem ^{[6]}, the following Ostrowski type inequality:
where and
In ^{[4]}, Pecaric and Ungar proved a general estimate with the norm, , which will for give the Dragomir ^{[2]} result. The interested reader is also referred to (^{[1, 2, 3, 4, 5, 8]}) for integral inequalities by using Pompeiu's mean value theorem. In this paper, we establish some new integral inequalities similar to that of the Ostrowski type integral inequality for two variables functions via Pompeiu's mean value theorem.
2. Main Results
First we give the following notations used to simplify the details of presentation
and
To prove our theorems, we need the following lemma:
Lemma 2.1. be an absolutely continuous function such that the partial derivative of order exists for all with Then for any we have
 (2.1) 
Proof. Define by . The function is continuously differentiable on , and for all we get
 (2.2) 
Using the change of the variable in last integrals with and we get
Denote and Then for all from (2.2), we have
which gives (2.1) and completes the proof.
Theorem 2.1 be an absolutely continuous function such that the partial derivative of order exists for all with Then for with any we have
 (2.3) 
where , and for all
Proof From Lemma 2.1, we have
 (2.4) 
Integrating with respect to on and dividing by we get
and therefore
 (2.5) 
Firstly, we will consider the case By using Hölder's inequality, the sum in the last line (2.5) is
 (2.6) 
The first factor in (2.6) equals
 (2.7) 
and for the second factor, for we get
 (2.8) 
For instead of (2.8), we obtain
 (2.9) 
which is easily shown to be equal to the limit of the right hand side of (2.8) for , i.e.
Now, consider the case Then, the last line in (2.5) is
 (2.10) 
Putting (2.10) into (2.5) and dividing by gives
Finally, we consider the case then, the last line of (2.5) is
 (2.11) 
Appending (2.11) to (2.5) and dividing by gives
 (2.12) 
It is not too difficult to show that
so (2.12) proves formula (2.3) for proving the theorem.
heorem 2.2 be an absolutely continuous function such that the partial derivative of order exists for all with and let be a nonnegative integrable function. Then for with any we have
 (2.13) 
Proof Multiplying (2.4) by and integrating with respect to on , we have
and as in the proof of Theorem 2.1, we get
which gives (2.13).
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