Equality and homogeneity of generalized integral means

Abstract

Given two continuous functions $f,g:I\to\mathbb{R}$ such that $g$ is positive and $f/g$ is strictly monotone, a measurable space $(T,A)$, a measurable family of $d$-variable means $m: I^d\times T\to I$, and a probability measure $\mu$ on the measurable sets $A$, the $d$-variable mean $M_{f,g,m;\mu}:I^d\to I$ is defined by $$ M_{f,g,m;\mu}(\pmb{x}) :=\left(\frac{f}{g}\right)^{-1}\left( \frac{\int_T f\big(m(x_1,\dots,x_d,t)\big)\,d\mu(t)} {\int_T g\big(m(x_1,\dots,x_d,t)\big)\,d\mu(t)}\right) \qquad(\pmb{x}=(x_1,\dots,x_d)\in I^d). $$ The aim of this paper is to solve the equality and homogeneity problems of these means, i.e., to find conditions for the generating functions $(f,g)$ and $(h,k)$, for the family of means $m$, and for the measure $\mu$ such that the equality $$ M_{f,g,m;\mu}(\pmb{x})=M_{h,k,m;\mu}(\pmb{x}) \qquad(\pmb{x}\in I^d) $$ and the homogeneity property $$ M_{f,g,m;\mu}(\lambda\pmb{x})=\lambda M_{f,g,m;\mu}(\pmb{x}) \qquad(\lambda>0,\,\pmb{x},\lambda\pmb{x}\in I^d), $$ respectively, be satisfied.