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

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

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