Abstract and Applied Analysis

Volume 2013 (2013), Article ID 904721, 7 pages

http://dx.doi.org/10.1155/2013/904721

Research Article

Some Inequalities for Multiple Integrals on the -Dimensional Ellipsoid, Spherical Shell, and Ball

Yan Sun,¹ Hai-Tao Yang,¹ and Feng Qi2,3

¹College of Mathematics, Inner Mongolia University for Nationalities, Inner Mongolia Autonomous Region, Tongliao City 028043, China

²Department of Mathematics, School of Science, Tianjin Polytechnic University, Tianjin City 300387, China

³School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo City, Henan Province 454010, China

Received 11 January 2013; Accepted 28 February 2013

Academic Editor: Josip E. Pečarić

Copyright © 2013 Yan Sun et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The authors establish some new inequalities of Pólya type for multiple integrals on the -dimensional ellipsoid, spherical shell, and ball, in terms of bounds of the higher order derivatives of the integrands. These results generalize the main result in the paper by Feng Qi, Inequalities for a multiple integral, Acta Mathematica Hungarica (1999).

1. Introduction

In [1], it was obtained that if is differentiable and if , then

for a certain between and . This inequality can be found in [2–4] and many other textbooks. It can be reformulated as follows. If is differentiable and not identically constant, such that and on , then

In the literature, the inequalities (1) or (2) is called the Pólya integral inequality.

In [5], the inequality (1), or say (2), was generalized as

where is a differentiable function and .

In [6–9], the above inequalities were refined and generalized as follows.

Theorem 1 (see [9, Proposition 1]). Let be continuous on and differentiable in . Suppose that , and that in . If is not identically zero, then and

Theorem 2 (see [6, 7, 9]). Let be continuous on and differentiable in . Suppose that is not identically constant, and that in . Then,

where

Theorem 3 (see [8]). For and with for , denote the -rectangles by

where for and . Let be a multi-index; that is, is a nonnegative integer, with . Let be a function of variables on , and let its partial derivatives of th order remain between and in ; that is,

where and

Let

for . Then, for any ,

(1) when is even, one has

(2) When is odd, one has

We remark that Theorem 2 has been applied in [10] to give bounds for the complete elliptic integrals of the first and second kinds.

For more information on this topic, please refer to [11–18] and [19, pp. 558–561], especially to the preprint [20].

In what follows, we will continue to use some notations from Theorem 3. Assume that , for and with , and adopt the following notations:

Moreover, let be an -times differentiable function, and let

In this paper, we will establish some new inequalities of Pólya type for multiple integrals of the composition function on the -dimensional ellipsoid , of the composition function on the spherical shell , and of the composition function on the -dimensional ball . We also obtain a general inequality for the multiple integral .

2. A Lemma

In order to establish some new inequalities of Pólya type for multiple integrals, we need the following lemma.

Lemma 4. For , , and , one has

where

is the classical Euler gamma function.

Proof. Using the spherical coordinates on the region yields

where and , and

We note that when , the empty product in (18) is understood to be . It is clear that the expressions in (17) are solutions of (18), and that

A straightforward computation gives

Since

we obtain

The proof of Lemma 4 is complete.

3. Main Results

Now, we start out to state and prove our main results.

Theorem 5. Let be an -times differentiable function satisfying

Then, one has

Proof. Using the transformation in (17) on and letting for yield the Jacobian determinant

Because

we have

By integration by parts, one has

Choosing and in the above equality shows that

Further utilizing the condition (23) leads to the inequality (24). The proof of Theorem 5 is completed.

Theorem 6. Let be an -times differentiable function satisfying the inequality (23). Then, one has

Proof. Using the transformation in (17) on and choosing , , and for yield

Further letting and in (29) gives

Hence, by virtue of the condition (23), the inequality (31) follows immediately. The proof of Theorem 6 is completed.

Theorem 7. Let be an -times differentiable function satisfying (23). Then, one has

Proof. Similar to the proof of Theorem 5, by choosing and , we obtain the inequality (34). The proof is complete.

Corollary 8. Under the conditions of Theorem 7, if for , then

4. A More General Inequality

Let be an -tuple index; that is, the numbers are nonnegative and denote . Let be a function which has an times continuous derivative on , and let

and .

Theorem 9. Let satisfy

Then

Proof. By Taylor’s formula, we obtain

where

Using

we have

Integrating on both sides of the above equality leads to

where

By Lemma 4 and (44), one has

From (37) and

we have

Consequently, the proof of Theorem 9 is complete.

Corollary 10. Let , and let with (37). Then, for one has

where

5. An Application

Now, we list some special cases of as follows.

(1)If we take , the body becomes a closed region between the -dimensional pyramid and the rectangle for . (2)If we take , the body is a closed region between the -dimensional ellipsoid and the rectangle for . (3)If we take and , the body is a closed region between the -dimensional ball and the rectangle for .

In the calculation of the uniform -dimensional volume, static moment, the moment of inertia, the centrifugal moment, and so on, have important applications. See [21, 22].

To show the applicability of the above main results, we now estimate the value of a triple integral

where is the ellipsoid

Choosing , , , and in (25), the Jacobian determinant is

Using Taylor’s formula, it follows that

Specially, we have

where and . Therefore,

By (54) and the above inequality, we have

Acknowledgments

The authors appreciate the anonymous referees for their very careful suggestions and their greatly valuable comments on the original version of this paper. This work was partially supported by the Foundation of the Research Program of Science and Technology at Universities of Inner Mongolia Autonomous Region under Grant no. NJZY13159, China.

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