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

DOI:
https://doi.org/10.1090/S0002-9939-1985-0773999-3

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 in a region is likely to be normal in if 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 has an isolated singularity at , and in a deleted neighborhood of the function has a property which cannot be possessed by nonconstant entire (meromorphic) functions in the finite plane, then 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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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1985-0773999-3

Keywords:
Heuristic principle,
nonessential isolated singularity,
spherical derivative

Article copyright:
© Copyright 1985
American Mathematical Society