A correction to “Some boundary properties of the Riemann mapping function”

Abstract

The paper [1I, dealing with applications of boundary properties of the Riemann mapping function, makes essential use of a rectifiability hypothesis on the boundary. Michael R. Cullen has kindly called the author's attention to the fact that the form in which this hypothesis is stated differs from that actually used in the derivations. However, a simple rephrasing of the underlying conventions on the region Q suffices to set matters straight, and we present the necessary changes here, along with some related comments. A generalization of the classical Osgood-Taylor-Caratheodory theorem serves as the starting point in [1]. Let Q be a bounded simply connected plane region for which (iQ can be parametrized as a closed curve. Then any Riemann mapping function X for Q, i.e. any function mapping the open unit disc co conformally onto Q, can be extended to a continuous mapping of co onto Q. In stating the rectifiability hypothesis in [1] a parallel wording was used, namely that aQ be parametrizable as a rectifiable closed curve. The condition actually employed in the derivations, however, is that of rectifiability of disc-induced parametrizations of the prime-end boundary. While the two conditions are probably equivalent, a proof does not appear to be obvious. On the other hand, the whole question can be regarded as peripheral to the analytical results of [1], since these would normally be applied (as in the case of the disc slit along a radius) by tracing out the prime-end boundary in the natural way. As discussed in [1], the generalized Osgood-Taylor-Caratheodory theorem ensures that the prime-end boundary can be parametrized as a Jordan curve, the so-called Jordan-Caratheodory boundary curve A. Since any two Riemann mapping functions X are connected by a linear fractional transformation, it is clear that different choices of X yield equivalent parametrizations of A. Thus, the notions of JordanCaratheodory boundary curve and arc length along such a curve are intrinsic. In the light of these remarks we revise Theorem 1 of [1] as