IJPAM: Volume 14, No. 2 (2004)

ON A WEIGHTED HARDY-HILBERT'S
TYPE INEQUALITY

Jia Weijian$^1$, Gao Mingzhe$^2$, Gao Xuemei$^3$
$^{1, 2, 3}$Department of Mathematics and Computer Science
Normal College
Jishou University
Jishou Hunan 416000, P.R. CHINA
$^1$e-mail: jwj1959@163.com
$^2$e-mail: mingzhegao1940@yahoo.com.cn
$^3$e-mail: xuemeigao1971@163.com


Abstract.In this paper, it is shown that Generalized Hardy-Hilbert's double series inequality with weights can be established by introducing a parameter $ \lambda $ ( $ 1 - \textstyle{q \over p} < \lambda \le 2) $ and two positive and differentiable functions $ u\left( x \right) $ and $ v\left( x \right) $ in interval (0, +  $ \infty ) $. In particular, for case $ p = 2 $, the various new extensions of the classical Hilbert's inequality for double series are obtained. As applications, some important inequalities are built, when $ u\left( x \right) $ and $ v\left( x \right) $ are power function, logarithm function, inverse trigonometric function and exponent function.

Received: April 28, 2004

AMS Subject Classification: 26D15, 33B15

Key Words and Phrases: generalized Hardy-Hilbert's inequality, Hilbert's inequality, double series, weight function, beta function

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2004
Volume: 14
Issue: 2