On a Relation to Hilbert's Integral Inequality and a Hilbert-Type Inequality

  • Yang, Bicheng (Department of Mathematics, Guangdong Education Institute)
  • Received : 2008.05.13
  • Accepted : 2008.06.26
  • Published : 2009.09.30


In this paper, by introducing some parameters and using the way of weight function, a new integral inequality with a best constant factor is given, which is a relation between Hilbert's integral inequality and a Hilbert-type inequality. As applications, the equivalent form, the reverse forms and some particular inequalities are considered.


Hilbert's integral inequality;weight function;H-L-P inequality


  1. G. H. Hardy, J. E. Littlewood and G. Polya, Inequalities, Cambridge Univ. Press, Cambridge, 1952.
  2. D. S. Mintrinovic, J. E. Pecaric and A. M. Kink, Inequalities involving functions and their integrals and derivertives, Kluwer Academic Publishers, Boston, 1991.
  3. Bicheng Yang, On an extension of Hilbert's integral inequality with some parameters, The Australian Journal of Mathematical Analysis and Applications, 1(1)(2004), Article 11, 1-8.
  4. Bicheng Yang, On a generalization of the Hilbert's type inequality and its applications, Chinese Journal of Engineering Mathematics, 21(5)(2004), 821-824.
  5. Bicheng Yang, Ilko Brnetic, Mario Krnic and Josip Pecaric, Generalization of Hilbert and Hardy-Hilbert integral inequalities, Mathematical Inequalities and Applications, 8(2)(2005), 259-272.
  6. Yongjin Li and Bing He, A new Hilbert-type integral inequality and the equivalent form, Internat. J. Math. & Math. Soc., Vol. 2006, Arc. ID 457378, 1-6.
  7. Chune Xie, Best generalization of a new Hilbert-type inequality, Journal of Jinan University (Natural Science), 28(1)(2007), 24-27.
  8. Xiaokui Ge, New extension of a Hilbert-type integral inequality, Journal of Jinan University (Natural Science), 28(5)(2007), 447-450.
  9. Yongjin Li and Bing He, On inequalities of Hilbert's type, Bulletin of the Australian Mathematical Society, 76(1)(2007), 1-14.
  10. Jichang Kuang, Applied Inequalities, Shangdong Science Press, Jinan, 2004.
  11. Jichang Kuang, Introduction to real analysis, Hunan Education Press, Changsha, 1996.
  12. Laith Emil Azar, On some extensions of Hardy-Hilbert's inequality and applications, J. Ineq. Appl., 2008(2008), Article ID 546829, 14 pages.

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