The new method of determining Koebe domains for the class of typically real functions under Montel's normalization

Abstract

We consider the class T(r) of typically real functions with the normalization f(0) = 0 and f(r) = r for a fixed r ∈ (0, 1). In the limiting case, when r tends to 0, the class T(r) coincides with the class T of typically real functions normalized by f(0) = f ′ (0) − 1 = 0. In 1980, Lewandowski and Miazga determined the Koebe domain for T(r), i.e. the set of the form ∩ f∈T (r) f(∆). They used the method applied earlier by Goodman. In this paper we present a new, complete method of determining this set. As a corollary, we obtain the Koebe set for T .