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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Bounds on the ratio $ n(r,\,a)/S(r)$ for meromorphic functions


Author: Joseph Miles
Journal: Trans. Amer. Math. Soc. 162 (1971), 383-393
MSC: Primary 30.61
DOI: https://doi.org/10.1090/S0002-9947-1971-0285711-X
MathSciNet review: 0285711
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Abstract: Let f be a meromorphic function in the plane. We prove the existence of an absolute constant K such that if $ {a_1},{a_2}, \ldots ,{a_q}$ are distinct elements of the Riemann sphere then $ \lim {\inf _{r \to \infty }}\;(\Sigma _{j = 1}^q\vert n(r,{a_j})/S(r) - 1\vert) < K$. We show by example that in general no such bound exists for the corresponding upper limit. These results involving the unintegrated functionals of Nevanlinna theory are related to previous work of Ahlfors, Hayman and Stewart, and the author.


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DOI: https://doi.org/10.1090/S0002-9947-1971-0285711-X
Keywords: Meromorphic function, Nevanlinna theory, counting function, mean covering number
Article copyright: © Copyright 1971 American Mathematical Society

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