On analytic families of conformal maps

Let Λ be a domain in C and let fλ(z) = z + a0(λ) + a1(λ)z −1 + ... be meromorphic in D∗ := {z ∈ C : |z| 1} ∪ {∞}. We assume that fλ(z) is holomorphic in λ ∈ Λ for fixed z.The main theorem states: Let Λ0 be a subdomain of Λ such that fλ is univalent in D∗ for λ ∈ Λ0. If fλ0 has a quasiconformal exten...

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Autores:: Becker, Jochen
Pommerenke, Christian

Tipo de recurso:: Article of journal

Fecha de publicación:: 2017

Institución:: Universidad Nacional de Colombia

Repositorio:: Universidad Nacional de Colombia

Idioma:: spa

Description
Summary:	Let Λ be a domain in C and let fλ(z) = z + a0(λ) + a1(λ)z −1 + ... be meromorphic in D∗ := {z ∈ C : \|z\| 1} ∪ {∞}. We assume that fλ(z) is holomorphic in λ ∈ Λ for fixed z.The main theorem states: Let Λ0 be a subdomain of Λ such that fλ is univalent in D∗ for λ ∈ Λ0. If fλ0 has a quasiconformal extension to the closure of D∗ for one λ0 ∈ Λ0 then fλ has a quasiconformal extension for all λ ∈ Λ0.This result is related to a theorem of Mañé, Sad and Sullivan (1983) where the assumptions are however different. The main tool of our proof is the Grunsky inequality for univalent functions.

On analytic families of conformal maps

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