Abstract

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′⋅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 Pn,sα(y)−Pn,rα(y) and Pn,sα(x)−Pn,rα(x) for s≥r, α∈ℝ.