On Univalence of the Power Deformation z(f(z)/z)c

O1; The authors mainly concern the set Uf of c ∈( C) such that the power deformationz(f(z/z))c is univalent in the unit disk |z| ＜ 1 for a given analytic univalent function f(z) =z + a2z2 +...in the unit disk.It is shown that Uf is a compact,polynomially convex subset of the complex plane (C) unless...

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Bibliographic Details
Published in:数学年刊B辑（英文版） 2012-11, Vol.33 (6), p.823-830
Main Authors: Yong Chan KIM, Toshiyuki SUGAWA
Format: Article
Language:English
Online Access:Get full text
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Summary:O1; The authors mainly concern the set Uf of c ∈( C) such that the power deformationz(f(z/z))c is univalent in the unit disk |z| ＜ 1 for a given analytic univalent function f(z) =z + a2z2 +...in the unit disk.It is shown that Uf is a compact,polynomially convex subset of the complex plane (C) unless f is the identity function.In particular,the interior of Uf is simply connected.This fact enables us to apply various versions of the λ-lemma for the holomorphic family z(f(z)/z)c of injections parametrized over the interior of Uf.The necessary or sufficient conditions for Uf to contain 0 or 1 as an interior point are also given.
ISSN:0252-9599
DOI:10.1007/s11401-012-0750-z