# Extending means to several variables

## TL;DR

This paper explores methods to extend and reduce means across multiple variables, establishing that inequalities between two-variable quasi-arithmetic means suffice to infer inequalities among multiple variables, with applications to symmetrization and compounding.

## Contribution

It introduces a technique linking two-variable mean inequalities to multi-variable cases, simplifying analysis and connecting to Markov chains.

## Key findings

- A reduction to two-variable inequalities suffices for multi-variable mean inequalities.
- The technique relates to Markov chains and applies to symmetrization and compounding of means.
- Provides a new approach to extending and shrinking means across multiple variables.

## Abstract

We begin the study of how to extend few variable means to several variable ones and how to shrink means of several variables to less variables. With the help of one of the techniques we show that it is enough to check an inequality between two quasi-arithmetic means in 2-variables and that simply implies the inequality in m-variables. The technique has some relation to Markov chains. This method can be applied to symmetrization and compounding means as well.