On omitted tuples for univalent functions (Q1301991)

Let \(S\) be the usual class of normalized univalent functions and let \(W = \{w_1. w_2, \dots, w_n\}\) be a finite set of points in the complex plane. The question under consideration is: Does there exist a function \(f \in S\) which omits \(W\)? Clearly, if any \(|w_k|< 1/4\) then no such function exists, while if all \(|w_k|\geq 1\) then \(f(z) = z\) omits all of \(W\). The author discusses this problem when \(f \in S\) and when \(f\) is in a subclass of \(S\). Also considered are the associated sets \(tW = \{tw_1, tw_2, \dots, tw_n \}\). These problems are associated with the mapping radii of the various unbounded sets containing \(W\). Results of Grötzsch are used to show the uniqueness of certain functions and results of Roger Barnard are shown to be useful in searching for the extremal cases. This paper introduces an interesting set of problems and gives an excellent introduction to the basic ideas.