Open Access

Ostrowski Type Inequalities for Higher-Order Derivatives

Journal of Inequalities and Applications20092009:162689

Received: 12 February 2009

Accepted: 14 July 2009

Published: 4 August 2009


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.


Taylor ExpansionReal RootType InequalityIntegral InequalityOstrowski Type Inequality

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 [19] 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.


(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


When , then
When , then

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

Lemma 2.2.

Suppose , then for we have


It is obvious that (2.4) is true for . When , let
The only real root of is . Notice

Therefore we get the inequality (2.4).

Now, we give the proof of Theorem 1.1.


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.


(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


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


For any , we have
Consequently, (3.1) gives

Corollary 3.3.

With the assumptions in Theorem 3.1, we have


For any , we have

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

Theorem 3.4.


(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


Then we have
Using inequality (1.2) to gives
we have
Using Ostrowski's integral inequality (1.1) one gets

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



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

Department of Applied Mathematics, Jiangsu Polytechnic University, Changzhou, China
Department of Mechanical and Electrical Engineering, Hebi College of Vocation and Technology, Hebi, China


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© M.Wang and X. Zhao. 2009

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