On the local and global comparison of generalized Bajraktarević means

Given two continuous functions f,g:I→R such that g is positive and f/g is strictly monotone, a measurable space (T,A), a measurable family of d-variable means m:Id×T→I, and a probability measure μ on the measurable sets A, the d-variable mean Mf,g,m;μ:Id→I is defined byMf,g,m;μ(x):=(fg)−1(∫Tf(m(x1,…,xd,t))dμ(t)∫Tg(m(x1,…,xd,t))dμ(t))(x=(x1,…,xd)∈Id). The aim of this paper is to study the local and global comparison problem of these means, i.e., to find conditions for the generating functions (f,g) and (h,k), for the families of means m and n, and for the measures μ,ν such that the comparison inequalityMf,g,m;μ(x)≤Mh,k,n;ν(x)(x∈Id) be satisfied.