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# Ostrowski Type Inequalities for Higher-Order Derivatives

*Journal of Inequalities and Applications*
**volume 2009**, Article number: 162689 (2009)

## Abstract

This paper has shown some new Ostrowski type inequalities involving higher-order derivatives. The results generalized the Ostrowski type inequalities. Applications of the inequalities are also given.

## 1. Main Result and Introduction

The following inequality is well known in literature as Ostrowski's integral inequality.

Let be continuous on and differentiable on whose derivative is bounded on , that is, . Then

Moreover the constant 1/4 is the best possible. Because Ostrowski's integral inequality is useful in some fields, many generalizations, extensions, and variants of this inequality have appeared in the literature; see [1–9] and the references given therein. The main aim of this paper is to establish some new Ostrowski type inequalities involving higher-order derivatives. The analysis used in the proof is elementary. The main result of this paper is the following inequality.

Theorem 1.1.

Suppose

(1) to be continuous on ;

(2) to be nth order differentiable on whose nth order derivative is bounded on , that is, ;

(3)there exists such that .

Then for any , we have

As applications of the inequality (1.2), we give more Ostrowski type inequalities.

## 2. The Proof of Theorem 1.1

In this section, we use the Taylor expansion to prove Theorem 1.1. Before the proof, we need the following lemmas.

Lemma 2.1.

Suppose and , then we have

Proof.

When , then

When , then

From (2.2) and (2.3), we know that (2.1) holds.

Lemma 2.2.

Suppose , then for we have

Proof.

It is obvious that (2.4) is true for . When , let

then

The only real root of is . Notice

Therefore we get the inequality (2.4).

Now, we give the proof of Theorem 1.1.

Proof.

Using the Taylor expansion of at gives

Taking the integral on both sides of (2.8) with respect to variable over , we have

where the parameter is not a constant but depends on . From (2.8) and (2.9) one gets

So we have

Using Lemmas 2.1 and 2.2 gives (1.2). Thus, we complete the proof.

## 3. Some Applications

In this section, we show some applications of the inequality (1.2). In fact, we can use (1.2) to derive some new Ostrowski type inequalities.

Theorem 3.1.

Suppose

(1) to be continuous on ;

(2) to be second order differentiable on whose second derivative is bounded on , that is, ;

(3).

Then for any , we have

Proof.

From Rolle's mean value theorem, we know that there exists such that . Let in the inequality (1.2), then we have (3.1).

Corollary 3.2.

With the assumptions in Theorem 3.1, we have

Proof.

For any , we have

Consequently, (3.1) gives

Corollary 3.3.

With the assumptions in Theorem 3.1, we have

Proof.

For any , we have

Substituting (3.6) into (3.1) gives (3.5).

Theorem 3.4.

Suppose

(1) to be continuous on ;

(2) to be nth order differentiable on whose nth order derivative is bounded on , that is, .

Then for any and , we have

Proof.

Let

Then we have

Using inequality (1.2) to gives

Since

we have

Using Ostrowski's integral inequality (1.1) one gets

Notice

Substituting (3.13) and (3.14) into (3.12) gives (3.7).

It is easy to see that (3.7) is the generalization of (1.2). If we let in (3.7) and use (3.6), we get the following inequality.

Corollary 3.5.

With the assumptions in Theorem 3.4, we have

## References

- 1.
Ostrowski A:

**Über die absolutabweichung einer differentienbaren Funktionen von ihren integralmittelwert.***Commentarii Mathematics Helvetici*1938,**10:**226–227. - 2.
Anastassiou GA:

**Multivariate Ostrowski type inequalities.***Acta Mathematica Hungarica*1997,**76**(4):267–278. 10.1023/A:1006529405430 - 3.
Barnett NS, Dragomir SS:

**An Ostrowski type inequality for double integrals and applications for cubature formulae.***RGMIA Research Report Collection*1998,**1**(1):13–22. - 4.
Dragomir SS, Barnett NS, Cerone P:

**An n-dimensional version of Ostrowski's inequality for mappings of the Hölder type.***RGMIA Research Report Collection*1999,**2**(2):169–180. - 5.
Dragomir SS, Agarwal RP, Cerone P:

**On Simpson's inequality and applications.***Journal of Inequalities and Applications*2000,**5**(6):533–579. 10.1155/S102558340000031X - 6.
Dragomir SS:

**Ostrowski type inequalities for isotonic linear functionals.***Journal of Inequalities in Pure and Applied Mathematics*2002,**3**(5, article 68):1–13. - 7.
Florea A, Niculescu CP:

**A note on Ostrowski's inequality.***Journal of Inequalities and Applications*2005, (5):459–468. - 8.
Pachpatte BG:

**On an inequality of Ostrowski type in three independent variables.***Journal of Mathematical Analysis and Applications*2000,**249**(2):583–591. 10.1006/jmaa.2000.6913 - 9.
Pachpatte BG:

**On a new Ostrowski type inequality in two independent variables.***Tamkang Journal of Mathematics*2001,**32**(1):45–49.

## Acknowledgments

The author would like to express deep appreciation to the referees for the helpful suggestions. Mingjin Wang was supported by STF of Jiangsu Polytechnic University.

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**Open Access** This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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### Cite this article

Wang, M., Zhao, X. Ostrowski Type Inequalities for Higher-Order Derivatives.
*J Inequal Appl* **2009, **162689 (2009). https://doi.org/10.1155/2009/162689

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### Keywords

- Taylor Expansion
- Real Root
- Type Inequality
- Integral Inequality
- Ostrowski Type Inequality