ON A CHARACTERIZATION OF FINITE BLASCHKE PRODUCTS

Jacques Cartier

Description

Niveau: Supérieur, Doctorat, Bac+8
ON A CHARACTERIZATION OF FINITE BLASCHKE PRODUCTS EMMANUEL FRICAIN, JAVAD MASHREGHI Abstract. We study the convergence of a sequence of finite Blaschke products of a fix order toward a rotation. This would enable us to get a better picture of a characterization theorem for finite Blaschke products. 1. Introduction Let (zk)1≤k≤n be a finite sequence in the open unit disc D. Then the rational function B(z) = ? n∏ k=1 zk ? z 1? z¯k z , where ? is a unimodular constant, is called a finite Blaschke product of order n for D [8]. There are various results characterizing these functions. For example, one of the oldest ones is due to Fatou. Theorem A (Fatou [5]). Let f be analytic in the open unit disc D and suppose that lim |z|?1 |f(z)| = 1. Then f is a finite Blaschke product. For an analytic function f : ?1 ?? ?2, the number of solutions of the equation f(z) = w, (z ? ?1, w ? ?2), counting multiplicities, is called the valence of f at w and is denoted by vf (w). It is well-known that a finite Blaschke product of order n has the constant valence n for each w ? D.

• hyperbolic convex


• ?? ?0 ?


• ei? ?


• all hyperbolic convex


• let z1


• b?


• ?1 ??


• convex hull

Sujets

• Cartier
• Convex hull

Informations

• Publié par: mijec
• Nombre de lectures: 38
• Langue: English

Extrait

ON A CHARACTERIZATION OF FINITE BLASCHKE PRODUCTS

EMMANUEL FRICAIN, JAVAD MASHREGHI

Abstract.We study the convergence of a sequence of finite Blaschke products of a fix order toward a rotation. This would enable us to get a better picture of a characterization theorem for finite Blaschke products.

1.Introduction Let (z k ) 1≤k≤n be a finite sequence in the open unit discDthe rational. Then function n Y z k −z B(z) =γ , 1−z¯ k z k=1 whereγis a unimodular constant, is called a finite Blaschke product of ordernfor D[8]. There For example, one ofare various results characterizing these functions. the oldest ones is due to Fatou.

Theorem A(Fatou [5]).Letfbe analytic in the open unit discDand suppose that lim|f(z)|= 1. |z|→1 Thenfis a finite Blaschke product.

For an analytic functionf: Ω 1 −→Ω 2 , the number of solutions of the equation f(z) =w,(z∈Ω 1 , w∈Ω 2 ), counting multiplicities, is called thevalenceoffatwand is denoted byv f (w). It is well-known that a finite Blaschke product of ordernhas the constant valencen for eachw∈D. But, this property in fact characterizes finite Blaschke products of ordern.

2000Mathematics Subject Classification.Secondary: 32A70.Primary: 30D50, Key words and phrases.Blaschke products, zero sets, convergence, automorphism. This work was supported by NSERC (Canada), Jacques Cartier Center and ANR project FRAB (France). 1