Abstract
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.
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Research of Shahar Nevo was supported by the Israel Science Foundation, Grant No. 395/07.
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Grahl, J., Nevo, S. Exceptional functions wandering on the sphere and normal families. Isr. J. Math. 202, 21–34 (2014). https://doi.org/10.1007/s11856-014-1054-7
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DOI: https://doi.org/10.1007/s11856-014-1054-7