- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 904721, 7 pages
Some Inequalities for Multiple Integrals on the -Dimensional Ellipsoid, Spherical Shell, and Ball
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.
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.
- G. Pólya, “Ein mittelwertsatz für funktionen mehrerer veränderlichen,” Tohoku Mathematical Journal, vol. 19, pp. 1–3, 1921.
- G. Pólya and G. Szegö, Aufgaben und Lehrsätze aus der Analysis, vol. 1, Springer, Berlin, Germany, 1925, German.
- G. Pólya and G. Szegö, Problems and Theorems in Analysis, vol. 1 of Classics in Mathematics, Springer, Berlin, Germany, 1972.
- G. Pólya and G. Szego, Problems and Theorems in Analysis, vol. 1, 1984, Chinese Edition.
- K. S. K. Iyengar, “Note on an inequality,” Math Students, vol. 6, pp. 75–76, 1938.
- R. P. Agarwal and S. S. Dragomir, “An application of Hayashi's inequality for differentiable functions,” Computers & Mathematics with Applications, vol. 32, no. 6, pp. 95–99, 1996.
- P. Cerone and S. S. Dragomir, “Lobatto type quadrature rules for functions with bounded derivative,” Mathematical Inequalities & Applications, vol. 3, no. 2, pp. 197–209, 2000.
- F. Qi, “Inequalities for a multiple integral,” Acta Mathematica Hungarica, vol. 84, no. 1-2, pp. 19–26, 1999.
- F. Qi, “Inequalities for an integral,” The Mathematical Gazette, vol. 80, no. 488, pp. 376–377, 1996.
- B.-N. Guo and F. Qi, “Some bounds for the complete elliptic integrals of the first and second kinds,” Mathematical Inequalities & Applications, vol. 14, no. 2, pp. 323–334, 2011.
- B.-N. Guo and F. Qi, “Estimates for an integral in norm of the -th derivative of its integrand,” in Inequality Theory and Applications, pp. 127–131, Nova Science Publishers, Hauppauge, NY, USA, 2003.
- B.-N. Guo and F. Qi, “Some estimates of an integral in terms of the -norm of the st derivative of its integrand,” Analysis Mathematica, vol. 29, no. 1, pp. 1–6, 2003.
- V. N. Huy and Q. A. Ngô, “On an Iyengar-type inequality involving quadratures in knots,” Applied Mathematics and Computation, vol. 217, no. 1, pp. 289–294, 2010.
- F. Qi, “Further generalizations of inequalities for an integral,” Univerzitet u Beogradu Publikacije Elektrotehničkog Fakulteta, Serija: Matematika, vol. 8, pp. 79–83, 1997.
- F. Qi, “Inequalities for a weighted multiple integral,” Journal of Mathematical Analysis and Applications, vol. 253, no. 2, pp. 381–388, 2001.
- F. Qi and Y.-J. Zhang, “Inequalities for a weighted integral,” Advanced Studies in Contemporary Mathematics, vol. 4, no. 2, pp. 93–101, 2002.
- F. Qi, Z. L. Wei, and Q. Yang, “Generalizations and refinements of Hermite-Hadamard's inequality,” The Rocky Mountain Journal of Mathematics, vol. 35, no. 1, pp. 235–251, 2005.
- Y. X. Shi and Z. Liu, “On Iyengar type integral inequalities,” Journal of Anshan University of Science and Technology, vol. 26, no. 1, pp. 57–60, 2003, Chinese.
- J. C. Kuang, Chángyòng Bùdĕng Shì (Applied Inequalities), Shandong Science and Technology Press Shandong Province, Jinan, China, 3rd edition, 2004, Chinese.
- F. Qi, “Polya type integral inequalities: origin, variants, proofs, refinements, generalizations, equivalences, and applications,” RGMIA Research Report Collection, 32 pages, 2013, http://rgmia.org/papers/v16/v16a20.pdf.
- A. O. Akdemir, M. E. Özdemir, and S. Varošanec, “On some inequalities for -concave functions,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 746–753, 2012.
- C. P. Niculescu and L. E. Persson, Convex Functions and Their Applications, CMS Books in Mathematics, Springer, New York, NU, USA, 2006, A contemporary approach.