Correspondences Among Inner Functions, Functions with Non-Negative Real Parts and Conformal Mappings

Abstract

We study an interesting family of dynamical systems on the set of the singular inner functions (defined on the unit disk). Starting with an inner function \( S_0(z) \), we obtain new singular inner functions \( S_1(z), S_2(z), \ldots \). This sequence converges to a holomorphic self-map of the unit disk which we call \( S \). The convergence is proved with the aid of a fixed-point theorem, a special case of the Earle-Hamilton Theorem. The function \( S \) itself is not a singular inner function as \( zS(z) \) is a conformal map. This conformal map has the surprising property that its inverse (which is a priori defined on a proper subset of the disk) extends to the entire disk. The motivating question for this research is whether \( z \) times a singular inner function can have an omitted value in the unit disk. This question appears within a book on the Krzyz problem written by the author. This question is still open.