Pexider invariance equation for embeddable mean-type mappings

We prove that whenever \(M_1,\dots ,M_n:I^k \rightarrow I\), (\(n,k \in \mathbb {N}\)) are symmetric, continuous means on the interval I and \(S_1,\dots ,S_m:I^k \rightarrow I\) (\(m < n\)) satisfy a sort of embeddability assumptions then for every continuous function \(\mu :I^n \rightarrow \mathbb {R}\) which is strictly monotone in each coordinate, the functional equation

$$ \mu (S_1(v),\dots ,S_m(v),\underbrace{F(v),\dots ,F(v)}_{(n-m)\text { times}})=\mu (M_1(v),\dots ,M_n(v)) $$

has the unique solution \(F=F_\mu :I^k \rightarrow I\) 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 aim of this research is to provide a broad overview of the family of Beta-type means introduced in (Himmel and Matkowski, 2018).