Theorem of Hayman-Wu

Abstract

[en] This work consists in the statement and proof of the Hayman-Wu theorem:

Let $\varphi$ be a conformal mapping from de unit disk $\mathbb{D}$ to a simply connected domain $\Omega$ in the complex plane and let $L$ be any line. Then lenght $(\varphi^{-1}(L\cap\Omega))\leq 4 \pi$.

We will present the elementary proof based on an idea of Knut $\O$yma, following the sketch in the first chapter of the book by John B. Garnett and Donald E. Marshall named Harmonic Measure.

To state and prove this theorem we study various notions and previous results in the fields of complex analysis and potential theory. Examples of these are: automorphisms of the disk (or of a simply connected domain in general), pseudohiperbolic distance, the Schwarz and Schwarz-Pick lemmas, Riemann’s theorem on conformal mapping, harmonic functions, the Dirichlet problem and harmonic measure.