Abstract :
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.
