Functions of uniformly bounded characteristic on Riemann surfaces

Abstract

A characteristic function T ( D , w , f ) T(D,w,f) of Shimizu and Ahlfors type for a function f f meromorphic in a Riemann surface R R is defined, where D D is a regular subdomain of R R containing a reference point w ∈ R w \in R . Next we suppose that R R has the Green functions. Letting T ( w , f ) = lim D ↑ R T ( D , w , f ) T(w,f) = {\lim _{D \uparrow R}}T(D,w,f) , we define f f to be of uniformly bounded characteristic in R R , f ∈ UBC ( R ) f \in {\text {UBC}}(R) in notation, if sup w ∈ R T ( w , f ) > ∞ {\sup _{w \in R}}T(w,f) > \infty . We shall propose, among other results, some criteria for f f to be in UBC ( R ) {\text {UBC}}(R) in various terms, namely, Green’s potentials, harmonic majorants, and counting functions. They reveal that UBC ( Δ ) {\text {UBC}}(\Delta ) for the unit disk Δ \Delta coincides precisely with that introduced in our former work. Many known facts on UBC ( Δ ) {\text {UBC}}(\Delta ) are extended to UBC ( R ) {\text {UBC}}(R) by various methods. New proofs even for R = Δ R = \Delta are found. Some new facts, even for Δ \Delta , are added.