Research Article | Open Access
Yan Sun, Hai-Tao Yang, Feng Qi, "Some Inequalities for Multiple Integrals on the -Dimensional Ellipsoid, Spherical Shell, and Ball", Abstract and Applied Analysis, vol. 2013, Article ID 904721, 7 pages, 2013. https://doi.org/10.1155/2013/904721
Some Inequalities for Multiple Integrals on the -Dimensional Ellipsoid, Spherical Shell, and Ball
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).
In , 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.
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 3 (see ). 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
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
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 .
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
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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