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Transactions of the American Mathematical Society

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



Functions of uniformly bounded characteristic on Riemann surfaces

Author: Shinji Yamashita
Journal: Trans. Amer. Math. Soc. 288 (1985), 395-412
MSC: Primary 30D50; Secondary 30F99
MathSciNet review: 773067
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Abstract: A characteristic function $ T(D,w,f)$ of Shimizu and Ahlfors type for a function $ f$ meromorphic in a Riemann surface $ R$ is defined, where $ D$ is a regular subdomain of $ R$ containing a reference point $ w \in R$. Next we suppose that $ R$ has the Green functions. Letting $ T(w,f) = {\lim _{D \uparrow R}}T(D,w,f)$, we define $ f$ to be of uniformly bounded characteristic in $ R$, $ f \in {\text{UBC}}(R)$ in notation, if $ {\sup _{w \in R}}T(w,f) < \infty $. We shall propose, among other results, some criteria for $ f$ to be in $ {\text{UBC}}(R)$ in various terms, namely, Green's potentials, harmonic majorants, and counting functions. They reveal that $ {\text{UBC}}(\Delta )$ for the unit disk $ \Delta $ coincides precisely with that introduced in our former work. Many known facts on $ {\text{UBC}}(\Delta )$ are extended to $ {\text{UBC}}(R)$ by various methods. New proofs even for $ R = \Delta $ are found. Some new facts, even for $ \Delta $, are added.

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Article copyright: © Copyright 1985 American Mathematical Society