  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.
