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

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.