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

The local range set of a meromorphic function


Authors: Leon Brown and P. M. Gauthier
Journal: Proc. Amer. Math. Soc. 41 (1973), 518-524
MSC: Primary 30A72
MathSciNet review: 0325970
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Abstract: Let $ f$ be a function meromorphic in a domain $ G$ of the Riemann sphere. The global range set of $ f$ is the set of values assumed infinitely often by $ f$, and similarly the local range set of $ f$ at a boundary point $ p$ is the set of values assumed infinitely often in every neighborhood of $ p$. Obviously any range set is a $ {G_\delta }$ set. In this paper we show that every $ {G_\delta }$ set is the local range set of some meromorphic function. This contrasts with the situation for the global range set. Our methods rely on prime end theory and Arakélian's approximation theorems.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1973-0325970-5
PII: S 0002-9939(1973)0325970-5
Keywords: Range set, cluster sets, tangential approximation
Article copyright: © Copyright 1973 American Mathematical Society