On omitted tuples for univalent functions

Abstract

In the family S of normalized, univalent functions, an omitted point in F⊂S is a complex number w 0, such that there is at least one function f∈ F , satisfying f(z)≠w 0 for all |z|<1. Let a set of m distinct complex numbers w 1, w 2,…, w m all ≠0, be given such that 0⩽ arg w 1< arg w 2<⋯< arg w m<2 π . The tuple ( w 1, w 2,…, w m ) shall be called an omitted tuple for F if there exists at least one f∈ F such that f( z)≠ w i , ∀ i=1,2,…, m and all | z|<1. In this paper we shall be concerned with the question whether ( tw 1, tw 2,…, tw m ), t an arbitrary positive number, is an omitted tuple in S or not, more precisely the number of functions omitting the tuple for different values of t. An answer to this question in full generality will not be offered, but some partial results are given. Moreover, two subfamilies where complete solutions are obtained, are briefly mentioned.