Exceptional functions wandering on the sphere and normal families

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• 作者：Jürgen Grahl ; Shahar Nevo
• 刊名：Israel Journal of Mathematics
• 出版年：2014
• 出版时间：July 2014
• 年：2014
• 卷：202
• 期：1
• 页码：21-34
• 全文大小：198 KB
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• 作者单位：Jürgen Grahl (1)
  Shahar Nevo (2)
  
  1. Department of Mathematics, University of Würzburg, 97074, Würzburg, Germany
  2. Department of Mathematics, Bar-Ilan University, Ramat-Gan, 52900, Israel
  
• ISSN：1565-8511

文摘

We extend Carathéodory’s generalization of Montel’s fundamental normality test to “wandering-exceptional functions (i.e., depending on the respective function in the family under consideration), and we give a corresponding result on shared functions. Furthermore, we prove that if we have a family of pairs (a, b) of functions meromorphic in a domain such that a and b uniformly “stay away from each other,-then the families of the functions a resp. b are normal. The proofs are based on a “simultaneous rescaling-version of Zalcman’s lemma.