Comparison of means generated by two functions and a measure ✩

Given two continuous functions f , g : I →R such that g is positive and f /g is strictly monotone, and a probability measure μ on the Borel subsets of [0, 1], the two variable mean Mf ,g;μ : I2→I is defined by Mf ,g;μ(x, y) := f g −1 1 0 f (tx+ (1 −t)y)dμ(t) 1 0 g(tx + (1 −t)y)dμ(t) (x, y ∈ I). The aim of this paper is to study the comparison problem of these means, i.e., to find conditions for the generating functions ( f , g) and (h,k) and for the measures μ, ν such that the comparison inequality Mf ,g;μ(x, y) Mh,k;ν (x, y) (x, y ∈ I) holds.