On the Koebe domain for univalent functions with a fixed omitted value (Q1922964)

Let \(D=\{|z|<1\}\) denote the unit disk and \(S\) the usual class of analytic, univalent, normalized functions in \(D\). The family \(SM\) (for slit mappings) consists of all functions in \(S\) that map the unit disk onto the complement of a (single or double) slit supported by a straight line or for which the complement of \(f(D)\) is a half plane. For any family of functions \(F\subset S\), the author defines \(F_p=\{f\in F\): \(f\) omits \(p\}\). Further, the Koebe set of a family \(F\) is \(K(F)=\bigcap_{f\in F} f(D)\). In this paper, the author completely determines \(K(SM_p)\), the Koebe set of the class \(SM_p\). The author also finds upper and lower bounds for \(K(S_p)\). The results of this paper are based on the author's doctoral dissertation. Many of the proofs involve breaking the set of all possible values of \(p\) into separate intervals and looking at each of these intervals.