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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A heuristic principle for a nonessential isolated singularity


Author: David Minda
Journal: Proc. Amer. Math. Soc. 93 (1985), 443-447
MSC: Primary 30D45; Secondary 30C80
MathSciNet review: 773999
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Abstract: A heuristic principle in function theory claims that a family of holomorphic (meromorphic) functions which share a property $ P$ in a region $ \Omega $ is likely to be normal in $ \Omega $ if $ P$ cannot be possessed by nonconstant entire (meromorphic) functions in the finite plane. L. Zalcman established a rigorous version of this principle. An analogous principle for a nonessential singularity is plausible: If a holomorphic (meromorphic) function $ f$ has an isolated singularity at $ {z_0}$, and in a deleted neighborhood of $ {z_0}$ the function $ f$ has a property $ P$ which cannot be possessed by nonconstant entire (meromorphic) functions in the finite plane, then $ {z_0}$ is a nonessential singularity. We establish a rigorous version of the principle for holomorphic functions that is very similar to Zalcman's precise statement of the other principle. However, this rendition of the heuristic principle for a nonessential singularity fails for meromorphic functions in contrast to Zalcman's solution.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1985-0773999-3
PII: S 0002-9939(1985)0773999-3
Keywords: Heuristic principle, nonessential isolated singularity, spherical derivative
Article copyright: © Copyright 1985 American Mathematical Society