# Algebraic and topological structures on the set of mean functions and generalization of the AGM mean - Mathematics > Number Theory

Algebraic and topological structures on the set of mean functions and generalization of the AGM mean - Mathematics > Number Theory - Download this document for free, or read online. Document in PDF available to download.

Abstract: In this paper, we present new structures and results on the set $\M \D$ ofmean functions on a given symmetric domain $\D$ of $\mathbb{R}^2$. First, weconstruct on $\M \D$ a structure of abelian group in which the neutral elementis simply the {\it Arithmetic} mean; then we study some symmetries in thatgroup. Next, we construct on $\M \D$ a structure of metric space under which$\M \D$ is nothing else the closed ball with center the {\it Arithmetic} meanand radius 1-2. We show in particular that the {\it Geometric} and {\itHarmonic} means lie in the border of $\M \D$. Finally, we give two importanttheorems generalizing the construction of the $\AGM$ mean. Roughly speaking,those theorems show that for any two given means $M 1$ and $M 2$, which satisfysome regular conditions, there exists a unique mean $M$ satisfying thefunctional equation: $MM 1, M 2 = M$.

Author: ** Bakir Farhi**

Source: https://arxiv.org/