A strong form of Plessner's theorem

Abstract

Let f be a holomorphic, or even meromorphic, function on the unit disc. Plessner's theorem then says that, for almost every boundary point ζ, either (i) f has a finite nontangential limit at ζ, or (ii) the image f(S) of any Stolz angle S at ζ is dense in the complex plane. This paper shows that statement (ii) can be replaced by a much stronger assertion. This new theorem and its analogue for harmonic functions on halfspaces also strengthen classical results of Spencer, Stein and Carleson.