On univalence of the power deformation Z(f(z)/z) c

Yong Chan Kim, Toshiyuki Sugawa

Research output: Contribution to journalArticlepeer-review


The authors mainly concern the set U f of c ∈ ℂ such that the power deformation z(f(z)/z) c is univalent in the unit disk {pipe}z{pipe} < 1 for a given analytic univalent function f(z) = z + a 2z 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(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.

Original languageEnglish
Pages (from-to)823-830
Number of pages8
JournalChinese Annals of Mathematics. Series B
Issue number6
Publication statusPublished - 2012 Nov


  • Grunsky inequality
  • Holomorphic motion
  • Quasiconformal extension
  • Univalence criterion
  • Univalent function

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics


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