A strong form of Plessner's theorem

TL;DR

This paper strengthens Plessner's theorem by replacing the density condition with a stronger assertion, also extending the result to harmonic functions and improving classical boundary behavior theorems.

Contribution

It introduces a stronger boundary behavior theorem for holomorphic and harmonic functions, enhancing classical results of Spencer, Stein, and Carleson.

Findings

Replaces density condition with a stronger assertion in Plessner's theorem

Extends the theorem to harmonic functions on halfspaces

Strengthens classical boundary behavior results

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.