- 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

(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 .

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

(1) to be continuous on ;

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

(3) .

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

(1) to be continuous on ;

(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

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## Copyright

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