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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
.
AMS Subject Classifications:
Notes
1 The second author is supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) [grant number 2009-0084583].