IJPAM: Volume 19, No. 2 (2005)

A HARDY-HILBERT'S TYPE INTEGRAL
INEQUALITY WITH WEIGHTS AND
ITS APPLICATION

Yang Qiaoshun$^1$, Gao Mingzhe$^2$
$^{1,2}$Department of Mathematics and Computer Science
Normal College
Jishou University
Jishou Hunan, 416 000, P.R. CHINA
$^1$e-mail: yqs244@163.com
$^2$e-mail: mingzhegao@163.com


Abstract.In this paper, it is shown that a Hardy-Hilbert's type integral inequality with weights can be established by introducing a power-exponent function of the form $x^{1 + x}(x \in [0,+ \infty ))$, and the coefficient $\textstyle{\pi \over {\sin \;\pi \mathord{\left/ {\vphantom {\pi
p}} \right. \kern-\nulldelimiterspace} p}}$ is proved to be best possible. In particular, for case $p = 2$, some extensions of the classical Hilbert integral inequality are obtained. As application, some generalizations of Hardy-Littlewood's integral inequality are given.

Received: January 3, 2005

AMS Subject Classification: 26D15

Key Words and Phrases: power-exponent function, weight function, Hardy-Hilbert's integral inequality, Hilbert's integral inequality, Hardy-Littlewood's integral inequality

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2005
Volume: 19
Issue: 2