A generalization of Ky Fan's inequality

Let Pn,r (x) be the generalized weighted means. Let F(x) be a C1 function, y = y(x) an implicit decreasing function defined by f(x,y) = 0 and 0 < m < M ≤ m′, n ≥ 2, xi ∈ [m,M], yi ∈ [m′,M′]. Then for −1 ≤ r ≤ 1, if f ′ x/f ′ y ≤ 1, |(F(Pn,1(y))− F(Pn,r (y)))/(F(Pn,1(x)) − F(Pn,r (x)))| < (maxm′≤ξ≤M′ |F ′(ξ)|)/(minm≤η≤M |F ′(η)|) · M/m′. A similar result exists for f ′ x/f ′ y ≥ 1. By specifying f(x,y) and F(x), we get various generalizations of Ky Fan’s inequality. We also present some results on the comparison of Pα n,s(y)−P n,r (y) and Pα n,s(x)−P n,r (x) for s ≥ r , α∈R. 2000 Mathematics Subject Classification. 26D15, 26D20.