Invariant means, complementary averages of means, and a characterization of the beta-type means

Abstract

We prove that whenever the selfmapping $(M_1,\dots ,M_p):I^p\to I^p$, ($p\in \mathbb{N}$ and $M_i$-s are $p$-variable means on the interval $I$) is invariant with respect to some continuous and strictly monotone mean $K:I^p\to I$ then for every nonempty subset $S\subseteq \{1,\dots ,p\}$ there exists a uniquely determined mean $K_S:I^p\to I$ such that the mean-type mapping $(N_1,\dots ,N_p):I^p\to I^p$ is $K$-invariant, where $N_i:=K_S$ for $i\in S$ and $N_i:=M_i$ otherwise. Moreover \begin{equation*} \min(M_i\colon i \in S)\le K_S\le \max(M_i\colon i \in S). \end{equation*} Later we use this result to: (1) construct a broad family of $K$-invariant mean-type mappings, (2) solve functional equations of invariant-type, and (3) characterize Beta-type means.

References in corpus (1)

• Mean-type mappings and invariance principle