IJPAM: Volume 57, No. 4 (2009)

A NEW HILBERT TYPE INEQUALITY FOR DOUBLE SERIES

Xiaozhou Shang$^1$, Mingzhe Gao$^2$
$^{1,2}$Department of Mathematics and Computer Science
Normal College
Jishou University
Jishou, Hunan, 416000, P.R. CHINA
$^1$e-mail: jdsysxz003@163.com
$^2$e-mail: mingzhegao@163.com


Abstract.A new Hilbert type inequality for double series can be established by introducing a proper logarithm function. The weight function is estimated by means of the Euler-Maclaurin summation formula. And the constant factor $\pi ^{2r
+ 1}E_r $ is proved to be the best possible, where $E_0 = 1$úČand $E_{r's} $ are the Euler numbers, viz. $E_1 = 1,\;\;E_2 = 5,\;E_3 =
61$, etc. As applications, some equivalent inequalities are considered.

Received: November 8, 2009

AMS Subject Classification: 26D15

Key Words and Phrases: Hilbert-type inequality, double series, monotonous, sequence, Euler-Maclaurin

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 57
Issue: 4