## Abstract

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 _{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(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 language | English |
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Pages (from-to) | 823-830 |

Number of pages | 8 |

Journal | Chinese Annals of Mathematics. Series B |

Volume | 33 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2012 Nov |

## Keywords

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

## ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics