Comparison of means generated by two functions and a measure

Given two continuous functions f,g:I→Rf,g:I→R such that g   is positive and f/gf/g is strictly monotone, and a probability measure μ   on the Borel subsets of [0,1][0,1], the two variable mean Mf,g;μ:I2→IMf,g;μ:I2→I is defined byMf,g;μ(x,y):=(fg)−1(∫01f(tx+(1−t)y)dμ(t)∫01g(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)(f,g) and (h,k)(h,k) and for the measures μ,νμ,ν such that the comparison inequalityMf,g;μ(x,y)⩽Mh,k;ν(x,y)(x,y∈I) holds.