## Abstract

We establish some new Ostrowski-type integral inequalities involving higher-order partial derivatives. As applications, we get some interrelated results. Our results provide new estimates on inequalities of this type.

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# On Ostrowski-Type Inequalities for Higher-Order Partial Derivatives

## Abstract

## 1. Introduction

## 2. Main Results

## References

## Acknowledgments

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*Journal of Inequalities and Applications*
**volume 2010**, Article number: 960672 (2010)

We establish some new Ostrowski-type integral inequalities involving higher-order partial derivatives. As applications, we get some interrelated results. Our results provide new estimates on inequalities of this type.

The following inequality is well known in the literature as Ostrowski's integral inequality (see [1, page 468]).

Theorem 1.1.

Let be a differentiable mapping on whose derivative is bounded on that is, then

(1.1)

for all .

Many generalizations, extensions and variations of this inequality have appeared in the literature; see [1–10] and the references given therein. In particular, in 2009, Wang and Zhao [11] established a new Ostrowski-type inequality for higher-order derivatives as follows (see [11] for definitions and notations):

(1.2)

The main purpose of the present paper is to establish the following Ostrowski-type inequality involving higher-order partial derivatives (see next section for definitions and notations):

(1.3)

where is a constant, , and

This is a generalization of inequality (1.2).

Moreover, as applications, we get some interrelated results. Our results provide new estimates on such type of inequalities.

Theorem 2.1.

Suppose that

(1) is continuous on

(2) is differentiable in up to order , with bounded th-order mixed partial derivatives ( are natural numbers, and ), that is,

(2.1)

(3)there exists such that , ;

then for any ,

(2.2)

where

Proof.

From the hypotheses and using the -dimensional Taylor expansion of at we have, for some

(2.3)

Dividing both sides of (2.3) by , then integrating over from to first, and then integrating the resulting inequality over from to , we observe that

(2.4)

Note that we have replaced the dummy variables by , respectively. From (2.3) and (2.4), we have

(2.5)

Hence

(2.6)

On the other hand, by applying the following two elementary inequalities [11]:

(2.7)

to the right-hand side of (2.6), we obtain

(2.8)

This completes the proof.

Remark.

With suitable modifications, it is easy to see that (2.2) reduces to the following inequality in the 1-dimensional situation:

(2.9)

where is continuous on , with th-order derivative bounded on , that is, , and

Observe that this is a recent result of Wang and Zhao [11].

Theorem.

Suppose that

(1) is continuous on

(2) is twice differentiable in with bounded second-order partial derivatives that is,

(2.10)

(3)there exists such that

Then, one has

(2.11)

Proof.

From the hypotheses and in view of the -dimensional Taylor expansion, it easily follows that for some ,

(2.12)

Hence,

(2.13)

This proves Theorem 2.3.

Let and change to and , respectively, and with suitable modifications, Theorem 2.3 reduces to the following.

Theorem.

Suppose that

(1) is continuous on

(2) is twice differentiable in with bounded second -order derivative, that is,

(3)there exists such that (or ),

then, one has

(2.14)

This is another recent result of Wang and Zhao in [11].

Mtrinović DS, Pečarić JE, Fink AM:

*Inequalities for Functions and Their Integrals and Dervatives*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1994.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.2000.6913Pachpatte BG: On a new Ostrowski type inequality in two independent variables.

*Tamkang Journal of Mathematics*2001, 32(1):45–49.Anastassiou GA: Multivariate Ostrowski type inequalities.

*Acta Mathematica Hungarica*1997, 76(4):267–278. 10.1023/A:1006529405430Dragomir SS, Wang S: A new inequality of Ostrowski's type in norm and applications to some special means and to some numerical quadrature rules.

*Tamkang Journal of Mathematics*1997, 28(3):239–244.Dragomir SS, Wang S: An inequality of Ostrowski-Grüss' type and its applications to the estimation of error bounds for some special means and for some numerical quadrature rules.

*Computers & Mathematics with Applications*1997, 33(11):15–20. 10.1016/S0898-1221(97)00084-9Dragomir SS, Wang S: Applications of Ostrowski's inequality to the estimation of error bounds for some special means and for some numerical quadrature rules.

*Applied Mathematics Letters*1998, 11(1):105–109. 10.1016/S0893-9659(97)00142-0Barnett NS, Dragomir SS: An Ostrowski type inequality for double integrals and applications for cubature formulae.

*RGMIA Research Report Collection*1998, 1: 13–23.Baĭnov D, Simeonov P:

*Integral Inequalities and Applications, Mathematics and Its Applications*.*Volume 57*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1992:xii+245.Mitrinović DS, Pečarić JE, Fink AM:

*Inequalities Involving Functions and Their Integrals and Derivatives, Mathematics and Its Applications*.*Volume 53*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1991:xvi+587.Wang M, Zhao X: Ostrowski type inequalities for higher-order derivatives.

*Journal of Inequalities and Applications*2009, 2009:-8.

The research is supported by the National Natural Sciences Foundation of China (10971205). It is partially supported by the Research Grants Council of the Hong Kong SAR, China (Project no. HKU7016/07P) and an HKU Seed Grant for Basic Research.

**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.

Changjian, Z., Cheung, WS. On Ostrowski-Type Inequalities for Higher-Order Partial Derivatives.
*J Inequal Appl* **2010**, 960672 (2010). https://doi.org/10.1155/2010/960672

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DOI: https://doi.org/10.1155/2010/960672

- Partial Derivative
- Natural Number
- Recent Result
- Taylor Expansion
- Differentiable Mapping