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  • Research Article
  • Open Access

On a Multiple Hilbert-Type Integral Inequality with the Symmetric Kernel

Journal of Inequalities and Applications20072007:027962

Received: 26 April 2007

Accepted: 29 August 2007

Published: 13 November 2007


We build a multiple Hilbert-type integral inequality with the symmetric kernel and involving an integral operator . For this objective, we introduce a norm , two pairs of conjugate exponents and , and two parameters. As applications, the equivalent form, the reverse forms, and some particular inequalities are given. We also prove that the constant factors in the new inequalities are all the best possible.


  • Integral Operator
  • Constant Factor
  • Equivalent Form
  • Integral Inequality
  • Reverse Form


Authors’ Affiliations

Department of Mathematics, Guangdong Institute of Education, Guangzhou, Guangdong, China


  1. Hardy GH, Littlewood JE, Pólya G: Inequalities. 2nd edition. Cambridge University Press, Cambridge, UK; 1952:xii+324.MATHGoogle Scholar
  2. Yang B: On the norm of an integral operator and applications. Journal of Mathematical Analysis and Applications 2006,321(1):182–192. 10.1016/j.jmaa.2005.07.071MathSciNetView ArticleMATHGoogle Scholar
  3. Brnetić I, Pečarić J: Generalization of inequalities of Hardy-Hilbert type. Mathematical Inequalities & Applications 2004,7(2):217–225.MathSciNetView ArticleMATHGoogle Scholar
  4. Zhong W, Yang B: A best extension of Hilbert inequality involving seveial parameters. Jinan University Journal (Natural Science and Medical Edition) 2007,28(1):20–23.Google Scholar
  5. Yang B, Debnath L: On the extended Hardy-Hilbert's inequality. Journal of Mathematical Analysis and Applications 2002,272(1):187–199. 10.1016/S0022-247X(02)00151-8MathSciNetView ArticleMATHGoogle Scholar
  6. Yang B, Gao MZ: An optimal constant in the Hardy-Hilbert inequality. Advances in Mathematics 1997,26(2):159–164.MathSciNetMATHGoogle Scholar
  7. Zhao C-J, Debnath L: Some new inverse type Hilbert integral inequalities. Journal of Mathematical Analysis and Applications 2001,262(1):411–418. 10.1006/jmaa.2001.7595MathSciNetView ArticleMATHGoogle Scholar
  8. Yang B: A reverse of the Hardy-Hilbert's type inequality. Journal of Southwest China Normal University (Natural Science) 2005,30(6):1012–1015.Google Scholar
  9. Zhong W: A reverse Hilbert's type integral inequality. International Journal of Pure and Applied Mathematics 2007,36(3):353–360.MathSciNetMATHGoogle Scholar
  10. Zhong W, Yang B: On the extended forms of the reverse Hardy-Hilbert's integral inequalities. Journal of Southwest China Normal University (Natural Science) 2007,29(4):44–48.Google Scholar
  11. Yang B: A multiple Hardy-Hilbert integral inequality. Chinese Annals of Mathematics 2003,24(6):743–750.MathSciNetMATHGoogle Scholar
  12. Brnetić I, Pečarić J: Generalization of Hilbert's integral inequality. Mathematical Inequalities & Applications 2004,7(2):199–205.MathSciNetView ArticleMATHGoogle Scholar
  13. Brnetić I, Krnić M, Pečarić J: Multiple Hilbert and Hardy-Hilbert inequalities with non-conjugate parameters. Bulletin of the Australian Mathematical Society 2005,71(3):447–457. 10.1017/S0004972700038454MathSciNetView ArticleMATHGoogle Scholar
  14. Yang B, Rassias TM: On the way of weight coefficient and research for the Hilbert-type inequalities. Mathematical Inequalities & Applications 2003,6(4):625–658.MathSciNetView ArticleMATHGoogle Scholar
  15. Hong Y: On multiple Hardy-Hilbert integral inequalities with some parameters. Journal of Inequalities and Applications 2006, 2006: 11 pages.Google Scholar
  16. Kuang J: Applied Inequalities. Shangdong Science and Technology Press, Jinan, China; 2004.Google Scholar
  17. Fichtingoloz GM: A Course in Differential and Integral Calculus. Renmin Education, Beijing, China; 1957.Google Scholar


© W. Zhong and B. Yang 2007

This article is published under license to BioMed Central Ltd. 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.