Skip to main content
Journal of Inequalities and Applications
Download PDF

Inverses of new Hilbert-Pachpatte-type inequalities

Journal of Inequalities and Applications volume 2006, Article number: 97860 (2006) Cite this article

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.

[123456789101112131415161718]

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.2000.7006

    MathSciNet  Article  MATH  Google 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.2005.347

    MathSciNet  Article  MATH  Google 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-X

    MathSciNet  Article  MATH  Google Scholar 

  4. Handley GD, Koliha JJ, Pečarić JE: A Hilbert type inequality. Tamkang Journal of Mathematics 2000,31(4):311–315.

    MathSciNet  MATH  Google 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.2000.7350

    MathSciNet  Article  MATH  Google Scholar 

  6. Hardy GH, Littlewood JE, Pólya G: Inequalities. Cambridge University Press, Cambridge; 1934.

    MATH  Google 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.1999.6373

    MathSciNet  Article  MATH  Google 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.

    MathSciNet  Article  MATH  Google Scholar 

  9. Pachpatte BG: A note on Hilbert type inequality. Tamkang Journal of Mathematics 1998,29(4):293–298.

    MathSciNet  MATH  Google 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.1998.6043

    MathSciNet  Article  MATH  Google 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.2005.495

    MATH  MathSciNet  Google 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.2000.6860

    MathSciNet  Article  MATH  Google 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.

    MathSciNet  Article  MATH  Google 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.2005.207

    MathSciNet  Article  MATH  Google Scholar 

  15. Yuan J, Leng GS: Inequalities for dual affine quermassintegrals. Journal of Inequalities and Applications 2006, 2006: 7 pages.

    MathSciNet  MATH  Google Scholar 

  16. Zhao C-J: Generalization on two new Hilbert type inequalities. Journal of Mathematics 2000,20(4):413–416.

    MathSciNet  MATH  Google Scholar 

  17. Zhao C-J: Inverses of disperse and continuous Pachpatte's inequalities. Acta Mathematica Sinica 2003,46(6):1111–1116.

    MathSciNet  Google 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.2001.7595

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Information and Mathematics Sciences, College of Science, China Jiliang University, Hangzhou, 310018, China

    Chang-Jian Zhao

  2. Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong

    Wing-Sum Cheung

Authors
  1. Chang-Jian Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wing-Sum Cheung
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chang-Jian Zhao.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Zhao, CJ., Cheung, WS. Inverses of new Hilbert-Pachpatte-type inequalities. J Inequal Appl 2006, 97860 (2006). https://doi.org/10.1155/JIA/2006/97860

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

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

Keywords

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