We present new structures and results on the set of mean functions on a given symmetric domain in ℝ². First, we construct on 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 a structure of metric space under which 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 . Finally, we give two theorems generalizing the construction of the AGM mean. Roughly speaking, those theorems show that for any two given means M₁ and M₂, which satisfy some regularity conditions, there exists a unique mean M satisfying the functional equation M(M₁,M₂) = M.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm132-1-11,
author = {Bakir Farhi},
title = {Algebraic and topological structures on the set of mean functions and generalization of the AGM mean},
journal = {Colloquium Mathematicae},
volume = {131},
year = {2013},
pages = {139-149},
zbl = {1282.26047},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm132-1-11}
}
Bakir Farhi. Algebraic and topological structures on the set of mean functions and generalization of the AGM mean. Colloquium Mathematicae, Tome 131 (2013) pp. 139-149. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm132-1-11/