Boundary behaviour of inner functions and holomorphic mappings

José L. Fernändez, Artur Nicolau

Research output: Contribution to journalArticleResearchpeer-review

3 Citations (Scopus)

Abstract

Let f be a holomorphic function in the unit disk omitting a set A of values of the complex plane. If A has positive logarithmic capacity, R. Nevanlinna proved that f has a radial limit at almost every point of the unit circle. If A is any infinite set, we show that f has a radial limit at every point of a set of Hausdorff dimension 1. A localization technique reduces this result to the following theorem on inner functions. If I is an inner function omitting a set of values B in the unit disk, then for any accumulation point b of B in the disk, there exists a set of Hausdorff dimension 1 of points in the circle where I has radial limit b.
Original languageEnglish
Pages (from-to)423-445
JournalMathematische Annalen
Volume310
Issue number3
Publication statusPublished - 1 Mar 1998

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