On an Inequality of Ostrowski Type via Variant of Pompeiu's Mean Value Theorem
1Department of Mathematics, Faculty of Science and Arts, Düzce University, Düzce, Turkey
The main of this paper is to establish an Ostrowski type inequality for two variables functions by using a mean value theorem.
Keywords: Ostrowski inequality, Pompeiu's mean value theorem
Turkish Journal of Analysis and Number Theory, 2014 2 (3),
Received June 17, 2014; Revised June 26, 2014; Accepted June 29, 2014Copyright © 2014 Science and Education Publishing. All Rights Reserved.
Cite this article:
- SARIKAYA, Mehmet Zeki, and Hüseyin BUDAK. "On an Inequality of Ostrowski Type via Variant of Pompeiu's Mean Value Theorem." Turkish Journal of Analysis and Number Theory 2.3 (2014): 80-84.
- SARIKAYA, M. Z. , & BUDAK, H. (2014). On an Inequality of Ostrowski Type via Variant of Pompeiu's Mean Value Theorem. Turkish Journal of Analysis and Number Theory, 2(3), 80-84.
- SARIKAYA, Mehmet Zeki, and Hüseyin BUDAK. "On an Inequality of Ostrowski Type via Variant of Pompeiu's Mean Value Theorem." Turkish Journal of Analysis and Number Theory 2, no. 3 (2014): 80-84.
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The inequality of Ostrowski  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
In , Pecaric and Ungar proved a general estimate with the -norm, , which will for give the Dragomir  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
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
Proof. Define by . The function is continuously differentiable on , and for all we get
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
where , and for all
Proof From Lemma 2.1, we have
Integrating with respect to on and dividing by we get
Firstly, we will consider the case By using Hölder's inequality, the sum in the last line (2.5) is
The first factor in (2.6) equals
and for the second factor, for we get
For instead of (2.8), we obtain
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
Putting (2.10) into (2.5) and dividing by gives
Finally, we consider the case then, the last line of (2.5) is
Appending (2.11) to (2.5) and dividing by gives
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
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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