The inequality of Milne and its converse II

Abstract

We prove the following let α,β,a > 0, and b < 0 be real numbers, and let Wj (j = 1,...,n; n ≥2) be positive real numbers with w1+ ⋯+wn= 1. The inequalities α ∑j=1n wj/(1- pja) ≤ ∑j=1n wj/(1 - pj) ∑ j=1n wj/(1+pj) ≤ β ∑j=1n wj/(1-pjb) hold for all real numbers pj ∈ [0,1) (j = 1,...,n) if and only if α ≤ min(1,a/2) and β ≥ max(1,(1 -min 1≤j≤nwj/2)b). Furthermore, we provide a matrix version. The first inequality (with α = 1 and a = 2) is a discrete counterpart of an integral inequality published by E. A. Milne in 1925.