BOUNDARY-BEHAVIOR OF INNER FUNCTIONS AND HOLOMORPHIC MAPPINGS

Let f be a holomorphic function in the unit disk omitting a set A of v alues of the complex plane, VA has positive logarithmic capacity, R. N evanlinna proved that f has a radial limit at almost every point of th e unit circle. If A is any infinite set, we show that f has a radial l imit at every point of a set of Hausdorff dimension 1. A localization technique reduces this result to the following theorem on inner functi ons. 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 ra dial limit b.