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

Abstract

We describe the region V ( z 0 ) of values of f ( z 0 ) for all normalized bounded univalent functions f in the unit disk D at a fixed point z 0 ∈ D . The proof is based on the radial Loewner differential equation. We also prove an analogous result for the upper half-plane using the chordal Loewner equation.