Value Ranges of Univalent Self-Mappings of the Unit Disc

Abstract

We describe the value set $\{f(z_0):f:\mathbb{D}\to \mathbb{D}\text{ univalent},f(0)=0,f^{\prime }(0)=e^{-T}\},$ where $\mathbb{D}$ denotes the unit disc and $z_0\in \mathbb{D}\setminus \{0\}$, $T>0,$ by applying Pontryagin's maximum principle to the radial Loewner equation.

arXiv commentv2: Some typos corrected