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## Colloquium Mathematicum

2013 | 132 | 1 | 139-149
Tytuł artykułu

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

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Języki publikacji
EN
Abstrakty
EN
We present new structures and results on the set $𝓜 _𝒟$ of mean functions on a given symmetric domain 𝒟 in ℝ². First, we construct on $𝓜 _𝒟$ 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 $𝓜 _𝒟$ a structure of metric space under which $𝓜 _𝒟$ 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 $𝓜 _𝒟$. 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.
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Tom
Numer
Strony
139-149
Opis fizyczny
Daty
wydano
2013
Twórcy
autor
• Department of Mathematics, University of Béjaia, Béjaia, Algeria
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