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

Bakir Farhi¹

¹ Department of Mathematics University of Béjaia Béjaia, Algeria

Colloquium Mathematicum, Tome 132 (2013) no. 1, pp. 139-149

Résumé

We present new structures and results on the set ${\mathcal {M}}_\mathscr {D}$ of mean functions on a given symmetric domain $\mathscr {D}$ in $\mathbb {R}^2$. First, we construct on ${\mathcal {M}}_\mathscr {D}$ a structure of abelian group in which the neutral element is the arithmetic mean; then we study some symmetries in that group. Next, we construct on ${\mathcal {M}}_\mathscr {D}$ a structure of metric space under which ${\mathcal {M}}_\mathscr {D}$ is the closed ball with center the arithmetic mean and radius $1/2$. We show in particular that the geometric and harmonic means lie on the boundary of ${\mathcal {M}}_\mathscr {D}$. Finally, we give two theorems generalizing the construction of the ${\rm AGM}$ mean. Roughly speaking, those theorems show that for any two given means $M_1$ and $M_2$, which satisfy some regularity conditions, there exists a unique mean $M$ satisfying the functional equation $M(M_1 , M_2) = M$.

Détail

DOI : 10.4064/cm132-1-11

Keywords: present structures results set mathcal mathscr mean functions given symmetric domain mathscr mathbb first construct mathcal mathscr structure abelian group which neutral element arithmetic mean study symmetries group construct mathcal mathscr structure metric space under which mathcal mathscr closed ball center arithmetic mean radius particular geometric harmonic means lie boundary mathcal mathscr finally theorems generalizing construction agm mean roughly speaking those theorems given means which satisfy regularity conditions there exists unique mean satisfying functional equation