Some unsolved problems on meromorphic functions of uniformly bounded characteristic

Abstract

The family UBC(R) of meromorphic functions of uniformly bounded characteristic in a Rieman surface R is defined in terms of the Shimizu-Ahlfors characteristic function. There are some natural parallels between UBC(R) and BMOA(R), the family of holomorphic functions of bounded mean oscillation in R. After a survey some open problems are proposed in contrast with BMOA(R).

Keywords

• Riemann surface
• Shimizu-Ahlfors' characteristic function
• UBC(R)
• BMOA(R)
• harmonic majorant
• F. Riesz' decomposition
• Carleson measure
• classification of Riemann surfaces.