A necessary and sufficient condition for Bloch functions

Abstract:

A necessary and sufficient condition is given for a subset $E \subseteq {\mathbf {C}}$ to satisfy \[ \begin {array}{*{20}{c}} {\operatorname {Sup}\{ |f’(z)|(1 - |z{|^2})|z \in {f^{ - 1}}(E)\} < \infty } \\ { \Rightarrow \operatorname {Sup}\{ |f’(z)|(1 - |z{|^2})|z \in D\} < \infty } \\ \end {array} \] when $f:D \to {\mathbf {C}}$ is analytic. The condition is that the complement of E should not contain large discs.