Complete Nevanlinna counting functions of boundary-preserving Nevanlinna functions

Abstract

We start with an identity for a Nevanlinna class function :where is the usual Nevanlinna counting function, and are the numerator singular measure for and the denominator singular measure for . We study potential theoretic properties for the complete Nevanlinna counting function for . For example, is shown to be subharmonic in and the behaviour at the boundary point of is related to the singular measure and that at infinity to . When , meaning the nontangential boundary values a.e. , is shown to be a subharmonic extension of the Green's function for with pole at , which is unique in an appropriate sense. This property of gives new descriptions for the Green's function and harmonic measures for the Green domain , as well as a classification of boundary points of . Finally, in the case of covering map , has no numerator singular factor for any.