Invariance in a class of operations related to weighted quasi-geometric means

TL;DR

This paper investigates a class of operations related to weighted quasi-geometric means, exploring their invariance properties, iterative limits, and functional equations, with applications to determining limits and invariant functions.

Contribution

It characterizes invariance in operations connected to weighted quasi-geometric means, linking to iterative means and solving related functional equations.

Findings

Invariance leads to iterative-type means.

Identifies the form of functions invariant under these operations.

Provides methods to compute limits of iterated means.

Abstract

Let $I\subset (0,\infty )$ be an interval that is closed with respect to the multiplication. The operations $C_{f,g}\colon I^{2}\rightarrow I$ of the form \begin{equation*} C_{f,g}\left( x,y\right) =\left( f\circ g\right) ^{-1}\left( f\left( x\right) \cdot g\left( y\right) \right) \text{,} \end{equation*} where $f,g$ are bijections of $I$ are considered. Their connections with generalized weighted quasi-geometric means is presented. It is shown that invariance question within the class of this operations leads to means of iterative type and to a problem on a composite functional equation. An application of the invariance identity to determine effectively the limit of the sequence of iterates of some generalized quasi-geometric mean-type mapping, and the form of all continuous functions which are invariant with respect to this mapping are given. The equality of two considered operations is also discussed.