International Journal of Mathematics and Mathematical Sciences
Volume 2008 (2008), Article ID 165089, 12 pages
Research Article

On Hilbert's Inequality for Double Series and Its Applications

Zhou Yu and Gao Mingzhe

Department of Mathematics and Computer Science, Normal College of Jishou University, Jishou, Hunan 416000, China

Received 2 September 2007; Accepted 26 March 2008

Academic Editor: Feng Qi

Copyright © 2008 Zhou Yu and Gao Mingzhe. 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.


This study shows that a refinement of the Hilbert inequality for double series can be established by introducing a real function u(x) and a parameter λ. In particular, some sharp results of the classical Hilbert inequality are obtained by means of a sharpening of the Cauchy inequality. As applications, some refinements of both the Fejer-Riesz inequality and Hardy inequality in Hp function are given.