The Best Constants of Hardy Type Inequalities for p = -1  

作　　者：WEN Jia Jin GAO Chao Bang 

机构地区：[1]College of Mathematics and Information Science, Chengdu University, Sichuan 610106, China

出　　处：《Journal of Mathematical Research and Exposition》2008年第2期316-322,共7页数学研究与评论（英文版）

基　　金：Foundation item： the National Natural Science Foundation of China （No. 10671136）; the Natural Science Foundation of Sichuan Provincial Education Department （No. 2005A201）.

摘　　要：For p ：〉 1, many improved or generalized results of the well-known Hardy＇s inequality have been established： In this paper, by means of the weight coefficient method, we establish the following Hardy type inequality for p = -1：∑^n i=1（1/i∑^i j=1 aj）^-1〈∑^n i=1（1-π^2-9/3i）ai^-1,where ai 〉 0, i = 1,2,... ,n. For any fixed positive integer n 〉 2, we study the best constant Cn such that the inequality ∑^ni=1（1/i∑^ij=1aj）^-1≤cn∑^ni=1ai^-1holds. Moreover, by means ofthe Mathematica software, we givesome examples.For p>1,many improved or generalized results of the well-known Hardy's inequality have been established.In this paper,by means of the weight coefficient method,we establish the following Hardy type inequality for p=-1: sum from i=1 to n((1/i)sum from j=1 to i a_j)^(-1)<2sum from i=1 to n(1-(π~2-9)/(3i)a_i^(-1), where a_i>0,i=1,2,…,n.For any fixed positive integer n≥2,we study the best constant C_n such that the inequality∑_(i=1)~n(1/i∑_(j=1)~i=a_j)^(-1)≤C_n∑_i=1~n a_i^(-1) holds.Moreover,by means of the Mathematica software,we give some examples.

关 键 词：Hardy type inequalities weight coefficient the best constant. 

分 类 号：O178.1[理学—数学]