A problem on spaces of holomorphic maps and the geometry of image domains

We present an old problem in Geometric Function Theory: characterizing the domains <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>⊂</m:mo> <m:mi mathvariant="double-struck">C</m:mi> </m:math> Ω⊂ 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">B</m:mi> </m:math> 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msubsup> <m:mrow> <m:mi>A</m:mi> </m:mrow> <m:mrow> <m:mi>α</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msubsup> </m:math> Aαᵖ . This work synthesizes a number of seminal results by several authors, aiming to provide a unified introduction to this classical but active topic.