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On an Inequality of Ostrowski Type via Variant of Pompeiu's Mean Value Theorem

Mehmet Zeki SARIKAYA*, Hüseyin BUDAK

Department of Mathematics, Faculty of Science and Arts, Düzce University, Düzce, Turkey

Abstract: 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

Received June 17, 2014; Revised June 26, 2014; Accepted June 29, 2014

Citation: Mehmet Zeki SARIKAYA, 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). 80-84. January 2014. doi:10.12691/tjant-2-3-5

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

[picture] | (2.1)

Proof. Define by . The function is continuously differentiable on , and for all we get

[picture] | (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

[picture] | (2.3)

where , and for all

Proof From Lemma 2.1, we have

[picture] | (2.4)

Integrating with respect to on and dividing by we get

and therefore

[picture] | (2.5)

Firstly, we will consider the case By using Hölder's inequality, the sum in the last line (2.5) is

[picture] | (2.6)

The first factor in (2.6) equals

[picture] | (2.7)

and for the second factor, for we get

[picture] | (2.8)

For instead of (2.8), we obtain

[picture] | (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

[picture] | (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

[picture] | (2.11)

Appending (2.11) to (2.5) and dividing by gives

[picture] | (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

[picture] | (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).

References

[1]   | A. M. Acu, A. Babos and F. D. Sofonea, The mean value theorems and inequalities of Ostrowski type. Sci. Stud. Res. Ser. Math. Inform. 21 (2011), no. 1, 5-16.
[2]   | S.S. Dragomir, An inequality of Ostrowski type via Pompeiu's mean value theorem, J. of Inequal. in Pure and Appl. Math., 6(3) (2005), Art. 83.
[3]   | I. Muntean, Extensions of some mean value theorems, Babes-Bolyai University, Faculty of Mathematics, Research Seminars on Mathematical Analysis, Preprint Nr. 7, 1991, 7-24.
[4]   | P.P Pecaric and S. Ungar, On an inequality of Ostrowski type, J. of Inequal. in Pure and Appl. Math., 7(4) (2006), Art. 151.
[5]   | E. C. Popa, An inequality of Ostrowski type via a mean value theorem, General Mathematics Vol. 15, No. 1, 2007, 93-100.
[6]   | D. Pompeiu, Sur une proposition analogue au théorème des accroissements finis, Mathematica (Cluj, Romania), 22 (1946), 143-146.
[7]   | A. Ostrowski, Uber die Absolutabweichung einer differentierbaren Funktionen von ihrem Integralmittelwert, Comment. Math. Helv., 10(1938), 226-227.
[8]   | F. Ahmad, N. A. Mir and M.Z. Sarikaya, An inequality of Ostrowski type via variant of Pompeiu's mean value theorem, J. Basic. Appl. Sci. Res., 4(4)204-211, 2014.