Boundary correspondence of Nevanlinna counting functions for self-maps of the unit disc

Authors:
Pekka J. Nieminen and Eero Saksman

Journal:
Trans. Amer. Math. Soc. **356** (2004), 3167-3187

MSC (2000):
Primary 30D35, 30D50; Secondary 47B33

DOI:
https://doi.org/10.1090/S0002-9947-03-03487-1

Published electronically:
October 29, 2003

MathSciNet review:
2052945

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a holomorphic self-map of the unit disc . For every , there is a measure on (sometimes called Aleksandrov measure) defined by the Poisson representation . Its singular part measures in a natural way the ``affinity'' of for the boundary value . The affinity for values inside is provided by the Nevanlinna counting function of . We introduce a natural measure-valued refinement of and establish that the measures are obtained as boundary values of the refined Nevanlinna counting function . More precisely, we prove that is the weak limit of whenever converges to non-tangentially outside a small exceptional set . We obtain a sharp estimate for the size of in the sense of capacity.

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Additional Information

**Pekka J. Nieminen**

Affiliation:
Department of Mathematics, University of Helsinki, P.O. Box 4 (Yliopistonkatu 5), FIN-00014 University of Helsinki, Finland

Email:
pekka.j.nieminen@helsinki.fi

**Eero Saksman**

Affiliation:
Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD), FIN-40014 University of Jyväskylä, Finland

Email:
saksman@maths.jyu.fi

DOI:
https://doi.org/10.1090/S0002-9947-03-03487-1

Keywords:
Nevanlinna counting function,
Aleksandrov measure,
multiplicity,
boundary value,
angular derivative

Received by editor(s):
February 3, 2003

Published electronically:
October 29, 2003

Additional Notes:
The first author was supported by the Academy of Finland, project 49077

Article copyright:
© Copyright 2003
American Mathematical Society