Properties of the Ahlfors function

TL;DR

This paper explores the properties of the Ahlfors function across general domains, connecting it to analytic capacity and Riemann maps using advanced complex analysis and algebraic methods.

Contribution

It provides a comprehensive analysis of the Ahlfors function's properties in broad domains, linking it to analytic capacity and Riemann mapping theory.

Findings

Characterization of the Ahlfors function in general domains

Relation between Ahlfors function and analytic capacity

Connection to Riemann maps of simply connected domains

Abstract

The so called Ahlfors function is the solution to an extremal problem. In this thesis, we study several properties of this function in a most general domain Omega. We consider the Ahlfors function in the context of analytic capacity, and also relate it to Riemann maps of simply connected domains onto the unit disk. Our methods involve Gelfand representation of commutative Banach algebras and standard techniques from complex analysis.