Research Problems in Function Theory — Problem 5.57

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $\mathbb D=\{z\in\mathbb C:|z|<1\}$ and $\mathbb T=\partial\mathbb D$. A Stolz angle with vertex $\zeta\in\mathbb T$ is an open angular approach region inside $\mathbb D$ whose sides approach $\zeta$ strictly within the inward half-plane; it may be truncated arbitrarily close to its vertex. An angular limit is a common spherical limit through every such non-tangential region. A set $E\subset\mathbb C$ has zero logarithmic capacity if each compact $K\subset E$ has capacity zero; for compact $K$, \[\operatorname{cap}(K)=\exp\left(-\inf_{\mu\in\mathcal P(K)}\iint\log\frac1{|z-w|}\,d\mu(z)d\mu(w)\right),\] with value $0$ when the infimum is $+\infty$, where $\mathcal P(K)$ is the set of probability measures supported on $K$. Almost every boundary point means arc-length almost every point.

Statement recorded in the supplied source.

For every analytic $f:\mathbb D\to\mathbb C$, can the dense-image alternative in Plessner’s theorem be strengthened? In particular, is it true that for almost every $\zeta\in\mathbb T$, either $f$ has an angular limit at $\zeta$, or for every Stolz angle $S$ at $\zeta$ the omitted set $\mathbb C\setminus f(S)$ has zero logarithmic capacity? [S2]

Short English statement

Can Plessner’s almost-everywhere dichotomy be sharpened so that every Stolz-angle image omits only a set of logarithmic capacity zero?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2305057 (AMR-022-5057). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory (2018), arXiv:1809.07200. Problem 5.57, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.