Small perturbations on means and quasi-means

TL;DR

This paper explores how minor modifications to the definitions of means influence their fundamental properties, proposing weaker notions and analyzing their characteristics with numerous examples.

Contribution

It introduces weakened definitions of means and investigates their properties, expanding the understanding of mean-like functions.

Findings

Weaker notions of means are characterized and exemplified.

Small perturbations can significantly alter mean properties.

New classes of functions related to means are identified.

Abstract

We start to investigate how small changes on the definition of ordinary means affect their properties. Especially the property of being a mean. In that direction we are looking for weakenings of the basic defining property of means. Hence we introduce weaker notions in two directions. We investigate such functions and provide many examples as well.