On the radial limits of analytic functions

Abstract

1. Examples have been given [5, p. 185] of functions f(z), analytic in the unit-circle K: j z j <1, and not identically constant, for which the radial limit f(ei) = lim7r_f(rei) is zero for all e on |z = except for a set of linear measure zero. In view of the Riesz-Nevanlinna theorem [6, p. 197], such functions cannot be bounded, or even of bounded characteristic, in I z I < 1. Functions of this sort appear again whenever we have an analytic function f(z) whose radial limits coincide almost everywhere with the radial limits of a bounded analytic function g(z), for the difference F(z) =f(z) -g(z) has a radial limit zero almost everywhere on I zI = 1. The Riesz-Nevanlinna theorem shows that, if f(z) is bounded, or of bounded characteristic, and if the radial limit values of f(z) coincide almost everywhere on an arc of I z = 1 with the radial limit values of g(z), then F(z) must be identically zero in Iz <1. The object of this note is to discuss certain aspects of the behavior of nonconstant analytic functions whose radial limits vanish almost everywhere on an arc A(01<0<02) Of IzI =1. One result of such a study (which the author plans as a sequel to this note) will be to give some idea of the way in which a function f(z), whose radial limits coincide almost everywhere with the radial limits of a function g(z) of bounded characteristic, can differ from g(z). We shall say that a nonconstant function f(z), analytic in j z j <1, is of class (LP) on an arc A of I zI = 1, if lim, jf(rei) =f(e ) =0 for almost all ei? belonging to the arc A. If the arc A is the whole circumference jzj =1, we shall say simply that the function f(z) is of class (LP). One property of functions which are of class (LP) on an arc A is immediate: the cluster set of f(z) at each point eiGo of A (i.e., the set of all values a with the property that there exists a sequence { z,n }z I <1, lim,0 Zn = eio, such that limn-,f(zn) = a) is the whole plane. For, if there is a point eio on A and a complex number oz which does not belong to the cluster set of f(z) at eiGo, then there is a circular neighborhood V(ei?o) of ei?o such that, in V(efo)qK, the function g(z) = [f(z) -a]-' is analytic and bounded. Since the function g(z) has the constant limit 1/a along almost all normal segments drawn to that arc of I z| =1 which bounds VnK, it follows from a simple corollary of the Riesz-Nevanlinna theorem that g(z), and hence f(z),