Rogosinski's lemma for univalent functions, hyperbolic Archimedean spirals and the Loewner equation

Abstract

We describe the region $\mathcal{V}(z_0)$ of values of $f(z_0)$ for all normalized bounded univalent functions $f$ in the unit disk $\mathbb{D}$ at a fixed point $z_0\in \mathbb{D}$. The proof is based on identifying $\mathcal{V}(z_0)$ as the reachable set of the radial Loewner differential equation. We also prove an analogous result for the upper half-plane using the chordal Loewner equation.