Abstract We present an old problem in Geometric Function Theory: characterizing the domains Ω ⊂ C C for which every holomorphic map from the unit disk into Ω must belong to a specific function space X. We will provide some generalities about this problem, although the main aim is to delve into the geometric characterizations for several classical spaces, such as the Bloch space B B, the spaces of analytic functions of bounded mean oscillation BMOA, the Nevanlinna class N, the Smirnov class N +, the Hardy spaces H p, and the weighted Bergman spaces A α p A ^p. This work synthesizes a number of seminal results by several authors, aiming to provide a unified introduction to this classical but active topic.
Francisco J. Cruz-Zamorano (Thu,) studied this question.