# IJPAM: Volume 57, No. 4 (2009)

A NEW HILBERT TYPE INEQUALITY FOR DOUBLE SERIES

Xiaozhou Shang, Mingzhe Gao
Department of Mathematics and Computer Science
Normal College
Jishou University
Jishou, Hunan, 416000, P.R. CHINA
e-mail: jdsysxz003@163.com
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  is proved to be the best possible, where £¬and  are the Euler numbers, viz. , etc. As applications, some equivalent inequalities are considered.