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Bulletin of the American Mathematical Society

Published by the American Mathematical Society, the Bulletin of the American Mathematical Society (BULL) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.47.

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Normal families: New perspectives
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by Lawrence Zalcman PDF
Bull. Amer. Math. Soc. 35 (1998), 215-230 Request permission

Abstract:

This paper surveys some surprising applications of a lemma characterizing normal families of meromorphic functions on plane domains. These include short and efficient proofs of generalizations of (i) the Picard Theorems, (ii) Gol’dberg’s Theorem (a meromorphic function on $\mathbb {C}$ which is the solution of a first-order algebraic differential equation has finite order), and (iii) the Fatou-Julia Theorem (the Julia set of a rational function of degree $d\ge 2$ is the closure of the repelling periodic points). We also discuss Bloch’s Principle and provide simple solutions to some problems of Hayman connected with this principle.
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Additional Information
  • Lawrence Zalcman
  • Affiliation: Department of Mathematics and Computer Science, Bar-Ilan University, Ramat-Gan 52900, Israel
  • Email: zalcman@macs.biu.ac.il
  • Received by editor(s): October 15, 1997
  • Received by editor(s) in revised form: May 26, 1998
  • © Copyright 1998 American Mathematical Society
  • Journal: Bull. Amer. Math. Soc. 35 (1998), 215-230
  • MSC (1991): Primary 30D45; Secondary 30D35, 34A20, 58F23
  • DOI: https://doi.org/10.1090/S0273-0979-98-00755-1
  • MathSciNet review: 1624862