Chinese Annals of Mathematics, Series B

, Volume 33, Issue 6, pp 823–830 | Cite as

On univalence of the power deformation \( z(\frac{{f(z)}} {z})^c \)

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Abstract

The authors mainly concern the set U f of c ∈ ℂ such that the power deformation \( z(\frac{{f(z)}} {z})^c \) is univalent in the unit disk |z| < 1 for a given analytic univalent function f(z) = z + a 2 z 2 + … in the unit disk. It is shown that U f is a compact, polynomially convex subset of the complex plane ℂ unless f is the identity function. In particular, the interior of U f is simply connected. This fact enables us to apply various versions of the λ-lemma for the holomorphic family \( z(\frac{{f(z)}} {z})^c \) of injections parametrized over the interior of U f . The necessary or sufficient conditions for U f to contain 0 or 1 as an interior point are also given.

Keywords

Univalent function Holomorphic motion Quasiconformal extension Grunsky inequality Univalence criterion 

2000 MR Subject Classification

30C55 30C62 
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Copyright information

© Fudan University and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of Mathematics EducationYeungnam UniversityYeungnamKorea
  2. 2.Graduate School of Information SciencesTohoku UniversityAoba-ku, SendaiJapan

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