Algebraic and topological structures on the set of mean functions and generalization of the AGM mean

Details Algebraic and topological structures on the set of mean functions and generalization of the AGM mean Algebraic and topological structures on the set of mean functions and generalization of the AGM mean

2010-02-25

arxiv.org/abs/1002.4757v1

Mathematics

Number Theory

23 pages. To appear

Scientific paper

In this paper, we present new structures and results on the set $\M_\D$ of mean functions on a given symmetric domain $\D$ of $\mathbb{R}^2$. First, we construct on $\M_\D$ a structure of abelian group in which the neutral element is simply the {\it Arithmetic} mean; then we study some symmetries in that group. 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} mean and radius 1/2. We show in particular that the {\it Geometric} and {\it Harmonic} means lie in the border of $\M_\D$. Finally, we give two important theorems generalizing the construction of the $\AGM$ mean. Roughly speaking, those theorems show that for any two given means $M_1$ and $M_2$, which satisfy some regular conditions, there exists a unique mean $M$ satisfying the functional equation: $M(M_1, M_2) = M$.

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