IJPAM: Volume 53, No. 2 (2009)


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

Abstract.In this paper it is shown that some new Hardy-Hilbert type inequalities for double series can be established by introducing a logarithm function of the form ln($n/m)$ and a parameter $s$. And the weight function is estimated by means of the Euler-Maclaurin summation formula. At the same time the constant factor is proved to be the best possible. In particular, for case $p=2$, some new Hilbert type inequalities are built. As applications, some equivalent inequalities are studied.

Received: April 25, 2009

AMS Subject Classification: 26D15

Key Words and Phrases: Hardy-Hilbert type inequality, double series, logarithm function, Riemann zeta function, Euler number, Euler-Maclaurin summation formula

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 53
Issue: 2