On Boundary Values of an Analytic Transformation of a Circle into a Riemann Surface

Abstract

Let f(z) be a nonconstant analytic transformation of the unit circle U : ∣z∣ < 1 into a Riemann surface ℜ. As an extension of a classical theorem of F. and M. Riesz, the author proved in Theorem 3.4 of [2] that if the image of U is relatively compact in ℜ and has universal covering surface of hyperbolic type, and if, at every point of a set on ∣z∣ = 1 of positive inner linear measure, there terminates a curve along which f(z) has limit, then the set of such limits has positive inner logarithmic capacity. This theorem was followed by the first proposition in Kuramochi [1], which asserts that, if ℜ has a null boundary and the image of U excludes a set of positive logarithmic capacity on ℜ and if, at every point of a set E on ∣z∣ = 1, there terminates a curve along which f(z) has limit in the union of a set of inner logarithmic capacity zero on ℜ and the boundary components of ℜ then the inner linear measure of E is zero.