Value Ranges of Univalent Self-Mappings of the Unit Disc

TL;DR

This paper characterizes the set of possible values of univalent self-maps of the unit disc with fixed initial conditions, using Pontryagin's maximum principle applied to the radial Loewner equation.

Contribution

It introduces a novel application of Pontryagin's maximum principle to describe value ranges of specific univalent functions.

Findings

Explicit description of the value set for given initial conditions

Application of Pontryagin's maximum principle to Loewner equation

Advancement in understanding univalent function value ranges

Abstract

We describe the value set $\{f(z_0)\,:\, f:\mathbb{D}\to\mathbb{D} \text{ univalent}, f(0)=0, f'(0)= e^{-T} \},$ 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.