Skip to content

Advertisement

Journal of Inequalities and Applications
Journal of Inequalities and Applications Cover Image
  • Research Article
  • Open Access

Inverses of new Hilbert-Pachpatte-type inequalities

Journal of Inequalities and Applications20062006:97860

https://doi.org/10.1155/JIA/2006/97860

©  C.-J. Zhao and W.-S. Cheung. 2006

  • Received: 7 February 2006
  • Accepted: 5 June 2006
  • Published:

Abstract

We establish a new inequality of Hilbert type for a finite double number of nonnegative sequences of real numbers and some interrelated results, which are inverse and general forms of Pachpatte's and Handley's results. An integral version and some interrelated results are also obtained. These results provide some new estimates on such types of inequalities.

Keywords

  • Real Number
  • Integral Version
  • Double Number
  • Nonnegative Sequence
  • Hilbert Type

[123456789101112131415161718]

Authors’ Affiliations

(1)
Department of Information and Mathematics Sciences, College of Science, China Jiliang University, Hangzhou, 310018, China
(2)
Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong

References

  1. Cheung W-S, Hanjš Ž, Pečarić J: Some Hardy-type inequalities. Journal of Mathematical Analysis and Applications 2000,250(2):621–634. 10.1006/jmaa.2000.7006MathSciNetView ArticleMATHGoogle Scholar
  2. Cheung W-S, Ma Q-H: On certain new Gronwall-Ou-Iang type integral inequalities in two variables and their applications. Journal of Inequalities and Applications 2005,2005(4):347–361. 10.1155/JIA.2005.347MathSciNetView ArticleMATHGoogle Scholar
  3. Gao M, Yang B: On the extended Hilbert's inequality. Proceedings of the American Mathematical Society 1998,126(3):751–759. 10.1090/S0002-9939-98-04444-XMathSciNetView ArticleMATHGoogle Scholar
  4. Handley GD, Koliha JJ, Pečarić JE: A Hilbert type inequality. Tamkang Journal of Mathematics 2000,31(4):311–315.MathSciNetMATHGoogle Scholar
  5. Handley GD, Koliha JJ, Pečarić JE: New Hilbert-Pachpatte type integral inequalities. Journal of Mathematical Analysis and Applications 2001,257(1):238–250. 10.1006/jmaa.2000.7350MathSciNetView ArticleMATHGoogle Scholar
  6. Hardy GH, Littlewood JE, Pólya G: Inequalities. Cambridge University Press, Cambridge; 1934.MATHGoogle Scholar
  7. Jichang K: On new extensions of Hilbert's integral inequality. Journal of Mathematical Analysis and Applications 1999,235(2):608–614. 10.1006/jmaa.1999.6373MathSciNetView ArticleMATHGoogle Scholar
  8. Okpoti CA, Persson L-E, Wedestig A: Weight characterizations for the discrete Hardy inequality with kernel. Journal of Inequalities and Applications 2006, 2006: 14 pages.MathSciNetView ArticleMATHGoogle Scholar
  9. Pachpatte BG: A note on Hilbert type inequality. Tamkang Journal of Mathematics 1998,29(4):293–298.MathSciNetMATHGoogle Scholar
  10. Pachpatte BG: On some new inequalities similar to Hilbert's inequality. Journal of Mathematical Analysis and Applications 1998,226(1):166–179. 10.1006/jmaa.1998.6043MathSciNetView ArticleMATHGoogle Scholar
  11. Řehák P: Hardy inequality on time scales and its application to half-linear dynamic equations. Journal of Inequalities and Applications 2005,2005(5):495–507. 10.1155/JIA.2005.495MATHMathSciNetGoogle Scholar
  12. Yang B: On new generalizations of Hilbert's inequality. Journal of Mathematical Analysis and Applications 2000,248(1):29–40. 10.1006/jmaa.2000.6860MathSciNetView ArticleMATHGoogle Scholar
  13. 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
  14. Yaotian S, Zhihui C: General Hardy inequalities with optimal constants and remainder terms. Journal of Inequalities and Applications 2005,2005(3):207–219. 10.1155/JIA.2005.207MathSciNetView ArticleMATHGoogle Scholar
  15. Yuan J, Leng GS: Inequalities for dual affine quermassintegrals. Journal of Inequalities and Applications 2006, 2006: 7 pages.MathSciNetMATHGoogle Scholar
  16. Zhao C-J: Generalization on two new Hilbert type inequalities. Journal of Mathematics 2000,20(4):413–416.MathSciNetMATHGoogle Scholar
  17. Zhao C-J: Inverses of disperse and continuous Pachpatte's inequalities. Acta Mathematica Sinica 2003,46(6):1111–1116.MathSciNetGoogle Scholar
  18. 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

Copyright

© C.-J. Zhao and W.-S. Cheung. 2006

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.

Advertisement