On the Isomorphic Means and Comparison Inequalities

TL;DR

This paper introduces a unified framework for extending traditional means to 'isomorphic means' using bijections, explores their properties, and applies them to generate new bivariate means like quasi-Stolarsky means.

Contribution

It develops the concept of isomorphic means based on bijections, classifies their subclasses, and demonstrates their use in generating new families of means such as quasi-Stolarsky means.

Findings

Seven subclasses of isomorphic means are identified.

A subclass related to Cauchy mean value is used to generate bivariate means.

The framework unifies and extends existing means into a broader family.

Abstract

Based on collection of bijections, variable and function are extended into ``isomorphic variable'' and ``dual-variable-isomorphic function'', then mean values such as arithmetic mean and mean of a function are extended to ``isomorphic means''. 7 sub-classes of isomorphic mean of a function are distinguished. Comparison problems of isomorphic means are discussed. A sub-class(class V) of isomorphic mean of a function related to Cauchy mean value is utilized for generation of bivariate means e.g. quasi-Stolarsky means. Demonstrated as an example of math related to ``isomorphic frames'', this paper attempts to unify current common means into a better extended family of means.