Nevanlinna Theory - Further Development

Further Development

A substantial part of the research in functions of one complex variable in the 20th century was focused on Nevanlinna theory. One direction of this research was to find out whether the main conclusions of Nevanlinna theory are best possible. For example, the Inverse Problem of Nevanlinna theory consists in constructing meromorphic functions with pre-assigned deficiencies at given points. This was solved by David Drasin in 1975. Another direction was concentrated on the study of various subclasses of the class of all meromorphic functions in the plane. The most important subclass consists of functions of finite order. It turns out that for this class, deficiencies are subject to several restrictions, in addition to the defect relation (Norair Arakelyan, David Drasin, Albert Edrei, Alexandre Eremenko, Wolfgang Fuchs, Anatolii Goldberg, Walter Hayman, Joseph Miles, Daniel Shea, Oswald Teichmüller, Alan Weitsman and others).

Henri Cartan, Joachim and Hermann Weyl and Lars Ahlfors extended Nevanlinna theory to holomorphic curves. This extension is the main tool of Complex Hyperbolic Geometry. Intensive research in the classical one-dimensional theory still continues.