参量化的Hilbert不等式.pdf

495 Vol.49, No.5 20069 ACTA MATHEMATICA SINICA, Chinese Series Sept., 2006 : 0583-1431(2006)05-1121-06 : A Hilbert Æ Æ 510303 E-mail: bcyang@; /yangbicheng/index.htm1 Hilbert β . , . Hilbert ; ; β MR(2000) 26D15 O178 On Hilbert’s Inequality with Some Parameters Bi Cheng YANG Department of Mathematics, Guangdong Institute of Education, Guangzhou 510303, P. R. China E-mail: bcyang@; /yangbicheng/index.htm1 Abstract In this paper, by introducing some parameters and estimating the weight coefficient, we give a generalization of Hilbert’s double series inequality with a best constant factor, which involves the β function. As its applications, we consider the equivalent form and some particular results. Keywords Hilbert’s inequality; weight coefficient; β function MR(2000) Subject Classification 26D15 Chinese Library Classification O178 1 {an }, {bn } , 0 ∞ a2 ∞ 0 ∞ b2 ∞, [1] n=1 n n=1 n 1 ∞ ∞ ∞ ∞ 2 am bn 2 2 π an bn . (1.1) m + n n=1 m=1 n=1 n=1 , Æπ . (1.1) Hilbert , ( Mitrinovic [2]), ∞ ∞ 2 ∞ am π2 a2 . (1.2) n m + n