Functions of Uniformly Bounded Characteristic on Riemann Surfaces

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.