THE INEQUALITY OF MILNE AND ITS CONVERSE, III

摘要

The discrete version of Milne's inequality and its converse states that (*) ∑_(j=1)~n (w_j)/(1-p_j~2)≤∑_(j=1)~n (w_j)/(1-p_j)∑_(j=1)~n (w_j)/(1+p_j)≤(∑_(j=1)~n (w_j)/(1-p_j~2))~2 is valid for all w_j > 0 (j = 1.....n) with w_1+…+w_n = 1 and p_j ∈ (- 1.1) (j = 1..... n). We present new upper and lower bounds for the product ∑~w/(1-p) ∑ w/(1 + p). In particular, we obtain an improvement of the right-hand side of (*). Moreover, we prove a matrix analogue of our double-inequality.