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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.

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Authors and Affiliations

  1. Department of Mathematics Education, Yeungnam University, 214-1 Daedong Gyongsan, Yeungnam, 712-749, Korea

    Yong Chan Kim

  2. Graduate School of Information Sciences, Tohoku University, Aoba-ku, Sendai, 980-8579, Japan

    Toshiyuki Sugawa

Authors
  1. Yong Chan Kim
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  2. Toshiyuki Sugawa
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Corresponding author

Correspondence to Yong Chan Kim.

Additional information

Project supported by Yeungnam University (2011) (No. 211A380226) and the JSPS Grant-in-Aid for Scientific Research (B) (No. 22340025).

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Kim, Y.C., Sugawa, T. On univalence of the power deformation \( z(\frac{{f(z)}} {z})^c \) . Chin. Ann. Math. Ser. B 33, 823–830 (2012). https://doi.org/10.1007/s11401-012-0750-z

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  • DOI: https://doi.org/10.1007/s11401-012-0750-z

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