Research Problems in Function Theory — Problem 5.34

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.

Write $\mathbb D=\{z\in\mathbb C:|z|<1\}$ and $\mathbb T=\partial\mathbb D$. All Taylor coefficients below are taken at $0$. For $\zeta\in\mathbb T$, a Stolz angle means a nondegenerate Euclidean angular sector with vertex $\zeta$, whose two rays enter $\mathbb D$ nontangentially, restricted to a sufficiently small neighborhood of $\zeta$. A finite angular limit is a common finite limit as $z\to\zeta$ within every such sector.A Bloch function is holomorphic $h$ with $\sup_{z\in\mathbb D}(1-|z|^2)|h'(z)|<\infty$. Let $E_h\subset\mathbb T$ be the set where $h$ has an angular limit in $\widehat{\mathbb C}$, allowing $\infty$. A Hausdorff gauge is a nondecreasing function $\psi:[0,\infty)\to[0,\infty)$ with $\psi(0)=0$, $\psi(t)>0$ for $t>0$, and $\psi(t)\to0$ as $t\downarrow0$. Define \[\mathcal H^\psi(E)=\lim_{\varepsilon\downarrow0}\inf\left\{\sum_j\psi(\operatorname{diam}U_j):E\subseteq\bigcup_j U_j,\ \operatorname{diam}U_j\le\varepsilon\right\}. \] For the logarithmic gauge take $\psi_{\log}(t)=1/\log(1/t)$ for sufficiently small $t>0$; its values away from $0$ do not affect the resulting Hausdorff measure.

Statement recorded in the supplied source.

Is there a single Hausdorff gauge $\psi$, independent of the Bloch function, such that $\mathcal H^\psi(E_h)>0$ for every Bloch function $h$? In particular, does this hold for the logarithmic gauge $\psi_{\log}$? [S2]

Short English statement

Can angular-limit points of every Bloch function be guaranteed positive measure for one universal Hausdorff gauge, particularly a logarithmic gauge?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2305034 (AMR-022-5034). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), arXiv:1809.07200, Chapter 5, Problem 5.34, its update and the preceding shared notation. https://arxiv.org/abs/1809.07200.