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

TL;DR

This paper characterizes beta-type means by exploring invariant means and their complements, establishing a framework for constructing invariant mean mappings, solving related functional equations, and providing a new perspective on mean theory.

Contribution

It introduces a novel characterization of beta-type means through invariance properties and constructs a broad family of invariant mean-type mappings.

Findings

Existence of unique means $K_S$ satisfying invariance conditions

Construction of a broad family of $K$-invariant mean mappings

Characterization of Beta-type means using invariance properties

Abstract

We prove that whenever the selfmapping $(M_1,\dots,M_p)\colon I^p \to I^p$, ($p \in \mathbb{N}$ and $M_i$-s are $p$-variable means on the interval $I$) is invariant with respect to some continuous and strictly monotone mean $K \colon I^p \to I$ then for every nonempty subset $S \subseteq\{1,\dots,p\}$ there exists a uniquely determined mean $K_S \colon I^p \to I$ such that the mean-type mapping $(N_1,\dots,N_p) \colon I^p \to I^p$ 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.