The Inequality of Milne and Its Converse Ii

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 withw1+···+wn=1. The inequalities α ∑n j=1wj/(1−pj )≤ ∑n j=1wj/(1− pj) ∑n j=1wj/(1+ pj) ≤ β ∑n j=1wj/(1− pj ) 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−min1≤ 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.