- Research Article
- Open Access

# Ostrowski Type Inequalities for Higher-Order Derivatives

- Mingjin Wang
^{1}Email author and - Xilai Zhao
^{2}

**2009**:162689

https://doi.org/10.1155/2009/162689

© M.Wang and X. Zhao. 2009

**Received: **12 February 2009

**Accepted: **14 July 2009

**Published: **4 August 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.

## Keywords

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

## 1. Main Result and Introduction

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

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

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

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.

Proof.

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

Lemma 2.2.

Proof.

Therefore we get the inequality (2.4).

Now, we give the proof of Theorem 1.1.

Proof.

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

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

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.

Proof.

Corollary 3.3.

Proof.

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

Theorem 3.4.

Suppose

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

Proof.

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.

## Declarations

### 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.

## Authors’ Affiliations

## References

- Ostrowski A:
**Über die absolutabweichung einer differentienbaren Funktionen von ihren integralmittelwert.***Commentarii Mathematics Helvetici*1938,**10:**226–227.MathSciNetView ArticleMATHGoogle Scholar - Anastassiou GA:
**Multivariate Ostrowski type inequalities.***Acta Mathematica Hungarica*1997,**76**(4):267–278. 10.1023/A:1006529405430MATHMathSciNetView ArticleGoogle Scholar - 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.MathSciNetGoogle Scholar - 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.MATHGoogle Scholar - Dragomir SS, Agarwal RP, Cerone P:
**On Simpson's inequality and applications.***Journal of Inequalities and Applications*2000,**5**(6):533–579. 10.1155/S102558340000031XMATHMathSciNetGoogle Scholar - Dragomir SS:
**Ostrowski type inequalities for isotonic linear functionals.***Journal of Inequalities in Pure and Applied Mathematics*2002,**3**(5, article 68):1–13.MathSciNetMATHGoogle Scholar - Florea A, Niculescu CP:
**A note on Ostrowski's inequality.***Journal of Inequalities and Applications*2005, (5):459–468.MathSciNetMATHGoogle Scholar - 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.2000.6913MATHMathSciNetView ArticleGoogle Scholar - Pachpatte BG:
**On a new Ostrowski type inequality in two independent variables.***Tamkang Journal of Mathematics*2001,**32**(1):45–49.MATHMathSciNetGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.