Pexider invariance equation for embeddable mean-type mappings

Abstract

We prove that whenever $M_1,\dots ,M_n:I^k\to I$, ($n,k\in \mathbb{N}$) are symmetric, continuous means on the interval $I$ and $S_1,\dots ,S_m:I^k\to I$ ($m<n$) satisfies a sort of embeddability assumptions then for every continuous function $\mu :I^n\to \mathbb{R}$ which is strictly monotone in each coordinate, the functional equation $$\mu (S_1(v),\dots ,S_m(v),\underset{(n-m)\text{ times}}{\underbrace{F(v),\dots ,F(v)}})=\mu (M_1(v),\dots ,M_n(v))$$ has the unique solution $F=F_{\mu }:I^k\to I$ which is a mean. We deliver some sufficient conditions so that $F_{\mu }$ is well-defined (in particular uniquely determined) and study its properties. The background of this research is to provide a broad overview of the family of Beta-type means introduced in (Himmel and Matkowski, 2018).