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

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.