A new invariance identity and means

TL;DR

This paper introduces a new invariance identity involving three operations related to means, explores their connections, and investigates conditions under which these operations admit three means, posing an open problem on continuous solutions.

Contribution

It proposes a novel invariance identity involving three operations and explores their relationship with means, including conditions for admitting three means and an open problem on continuous solutions.

Findings

Introduced a new invariance identity involving operations D_{f,g}

Connected the invariance identity with mean functions

Posed an open problem on continuous solutions to a functional equation

Abstract

The invariance identity involving three operations $D_{f,g}:X\times X\rightarrow X$ of the form \begin{equation*} D_{f,g}\left( x,y\right) =\left( f\circ g\right) ^{-1}\left( f\left( x\right) \oplus g\left( y\right) \right) \text{,} \end{equation*} is proposed. The connections of these operations with means is investigated. The question when the invariance equality admits three means leads to a composite functional equation. Problem to determine its continuous solutions is posed.