Equality and homogeneity of generalized integral means

TL;DR

This paper investigates conditions under which generalized integral means are equal or homogeneous, focusing on the relationships between their generating functions, the family of means, and the measure involved.

Contribution

It provides a characterization of when two generalized integral means are equal or homogeneous, extending the understanding of mean functions in measure-theoretic contexts.

Findings

Conditions for equality of generalized integral means derived.

Criteria for homogeneity of these means established.

Results applicable to various families of means and measures.

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.