Equality and comparison of generalized quasiarithmetic means

TL;DR

This paper extends quasiarithmetic means by relaxing the continuity requirement on the generating function, analyzes their properties, compares them to standard means, and solves related equality and comparability problems.

Contribution

It introduces a broader class of generalized quasiarithmetic means with strictly monotone but not necessarily continuous generators, and studies their fundamental properties and relations.

Findings

Established properties of generalized quasiarithmetic means

Solved comparability and equality problems for these means

Provided examples illustrating representability issues

Abstract

The purpose of this paper is to extend the definition of quasiarithmetic means by taking a strictly monotone generating function instead of a strictly monotone and continuous one. We establish the properties of such means and compare them to the analogous properties of standard quasiarithmetic means. The comparability and equality problems of generalized quasiarithmetic are also solved. We also provide an example of a mean which, depending on the underlying interval or on the number of variables, could be or could not be represented as a generalized quasiarithmetic mean.