On Ostrowski-Type Inequalities for Higher-Order Partial Derivatives

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.

1. Introduction

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.

2. Main Results

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].