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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A necessary and sufficient condition for Bloch functions


Author: Richard M. Timoney
Journal: Proc. Amer. Math. Soc. 71 (1978), 263-266
MSC: Primary 30A74
DOI: https://doi.org/10.1090/S0002-9939-1978-0481012-7
MathSciNet review: 0481012
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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.


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Keywords: Bloch function, normal function
Article copyright: © Copyright 1978 American Mathematical Society