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

• Autorzy: Bakir Farhi
• 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.

Kategorie tematyczne

• 54E35: Metric spaces, metrizability
• 39B22: Equations for real functions
• 20K99: None of the above, but in this section

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2013
• Tom: 132
• Numer: 1
• Strony: 139-149

Daty

wydano

2013

Twórcy

autor

Bakir Farhi

• Department of Mathematics, University of Béjaia, Béjaia, Algeria

Identyfikatory

DOI

10.4064/cm132-1-11