On an additive analogue of Ky Fan's inequality

AbstractWe prove: If An and Gn (respectively, A′n and G′n) denote the weighted arithmetic and geometric means of x1,…, xn (respectively, 1 − x1,…,1 − xn), where xi ϵ (0, 12] (i = 1,…, n; n ≥ 2) are real numbers which are not all equal, then we have min1≤i≤nxi1-xi<A′n−G′nAn