On function with Weierstrass boundary points everywhere
D.M. Campbell, J. S. Hwang · Rocky Mountain Journal of Mathematics · 1983
In this paper we answer a conjecture of the second author by proving that any meromorphic function in the unit disc with radial limits on at most a set of measure zero must have Weierstrass points at each point of \z\ = 1.Given any tangential set D and any closed proper subset E of \z\ = 1, an analytic function /is constructed such that a) if E has measure zero, then / has no finite or infinite radial limits and the restricted cluster set C DCd ^(A exp(/0)) is uniformly bounded for all exp(/0) in E % b) if E has capacity zero then/has no finite or infinite radial limits on the complement of E but CflcnO; exp(/0)) = {0} for all exp(/0) in E. Introduction.While answering some open questions due to J. L. Doob [3] the second author introduced the class k(0), the set of Bloch functions which have no finite radial limits at any point of the unit disc.The second author conjectured [5] that for any / in k(Q) and for any Ô in [0, 2K] the cluster set of/at e id is the entire complex plane, that is, that every point of |z| = 1 is a Weierstrass point.In this paper we prove that any meromorphic function with radial limits on at most a set of measure zero must have Weierstrass points at every point of \z\ = 1.This answers the conjecture in the affirmative for a class of functions much more general than k(0).This result is a corollary to the more general theorem 1 in which we prove that the presence of a dense set of Weierstrass points implies that every point of \z\ = 1 is a Weierstrass point which extends the claims of the conjecture to functions which may have radial limits on sets whose measure is greater than zero.In theorems 3 and 4 we show how to construct analytic functions which have no finite or infinite radial limits but for which the restricted cluster set is bounded or degenerate.The technique makes use of Mergelyan's theorem, Blaschke products, and a construction due to A. Lohwater and G. Piranian.DEFINITIONS.If/is an arbitrary complex valued function (not necessarily even continuous) defined in \z\ < 1, then the cluster set of/at the