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

TL;DR

This paper investigates the set of complex exponents for which a power deformation of a univalent function remains univalent, revealing its geometric properties and conditions for specific interior points.

Contribution

It characterizes the set of parameters ensuring univalence of the power deformation, showing it is compact, polynomially convex, and simply connected, with implications for holomorphic families.

Findings

$U_f$ is compact and polynomially convex unless $f$ is the identity.

The interior of $U_f$ is simply connected.

Conditions for $0$ or $1$ to be interior points of $U_f$.

Abstract

In this note, we mainly concern the set $U_f$ of $c\in\mathbb{C}$ such that the power deformation $z(f(z)/z)^c$ is univalent in the unit disk $|z|<1$ for a given analytic univalent function $f(z)=z+a_2z^2+\cdots$ in the unit disk. We will show 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 $\lambda$-lemma for the holomorphic family $z(f(z)/z)^c$ of injections parametrized over the interior of $U_f.$ We also give necessary or sufficient conditions for $U_f$ to contain $0$ or $1$ as an interior point.