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Ostrowski Type Inequalities for Higher-Order Derivatives

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

(1.1)

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.

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

(1.2)

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

(2.1)

Proof.

When , then

(2.2)

When , then

(2.3)

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

Lemma 2.2.

Suppose , then for we have

(2.4)

Proof.

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

(2.5)

then

(2.6)

The only real root of is . Notice

(2.7)

Therefore we get the inequality (2.4).

Now, we give the proof of Theorem 1.1.

Proof.

Using the Taylor expansion of at gives

(2.8)

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

(2.9)

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

(2.10)

So we have

(2.11)

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

(3.1)

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

(3.2)

Proof.

For any , we have

(3.3)

Consequently, (3.1) gives

(3.4)

Corollary 3.3.

With the assumptions in Theorem 3.1, we have

(3.5)

Proof.

For any , we have

(3.6)

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

(3.7)

Proof.

Let

(3.8)

Then we have

(3.9)

Using inequality (1.2) to gives

(3.10)

Since

(3.11)

we have

(3.12)

Using Ostrowski's integral inequality (1.1) one gets

(3.13)

Notice

(3.14)

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

(3.15)

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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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Correspondence to Mingjin Wang.

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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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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