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 $\zeta$, either (i) $f$ has a finite nontangential limit at $\zeta$, or (ii) the image $f(S)$ of any Stolz angle $S$ at $\zeta$ 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.

arXiv commentIn press, Advances in Mathematics (open access)

References in corpus (1)

• Boundary behaviour of Dirichlet series with applications to universal series