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
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{displaymath}\begin{array}{*{20}{c}} {\operatorname{Sup}\{ \vert f'(z)\ver... ...1 - \vert z{\vert^2})\vert z \in D\} < \infty } \\ \end{array} \end{displaymath}

when $ f:D \to {\mathbf{C}}$ is analytic. The condition is that the complement of E should not contain large discs.

References [Enhancements On Off] (What's this?)

  • [1] J. M. Anderson, J. Clunie and Ch. Pommerenke, On Bloch functions and normal functions, J. Reine Angew. Math. 270 (1974), 12-37. MR 0361090 (50:13536)
  • [2] W. K. Haymann, Meromorphic functions, Clarendon Press, Oxford, 1964. MR 0164038 (29:1337)
  • [3] P. Lappan, A criterion for a meromorphic function to be normal, Comment. Math. Helv. 49 (1974), 492-495. MR 0379850 (52:755)
  • [4] A. J. Lohwater and Ch. Pommerenke, On normal meromorphic functions, Ann. Acad. Sci. Fenn. Ser. A I No. 550 (1973), 12 pp. MR 0338381 (49:3146)
  • [5] W. Rudin, Real and complex analysis, 2nd ed., McGraw-Hill Series in Higher Mathematics, McGraw-Hill, New York, 1974. MR 0344043 (49:8783)
  • [6] L. Zalcman, A heuristic principle in complex function theory, Amer. Math. Monthly 82 (1975), no. 8, 813-817. MR 0379852 (52:757)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1978-0481012-7
Keywords: Bloch function, normal function
Article copyright: © Copyright 1978 American Mathematical Society