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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
DOI: https://doi.org/10.1090/S0002-9947-1985-0773067-5
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