Research Problems in Function Theory — Problem 6.92

Definitions and mathematical statement

Identify ${\mathbb{R}}^2$ with ${\mathbb{C}}$, and put \[ \mathbb H=\{z:\operatorname{Im}z>0\},\qquad B^2=\{z:\operatorname{Im}z>0,\ |z|<1\}. \] The target is the upper half of the unit disk, exactly as printed. Let $\mathcal F$ be the source's class of analytic conformal maps $f:\mathbb H\to B^2$; whether conformal explicitly requires global injectivity should be confirmed. An angular limit $\alpha$ at $0$ means that $f(z)\to\alpha$ as $z\to0$ in each nontangential region $|\operatorname{Re}z|\le C\operatorname{Im}z$, $C>0$.

The source first reports a sufficient condition on an approach set $E\subset\mathbb H$: convergence $f(z)\to\alpha$ along $E$ forces the same angular limit if $E$ is a curve terminating at zero, or under the weaker radial-density condition \[ \liminf_{r\downarrow0}\frac{m(A\cap(0,r))}{r}>0, \qquad A=\{|z|:z\in E\}, \] where $m$ is one-dimensional Lebesgue measure, subject to the missing measurability convention for arbitrary $E$.

The converse paragraph introduces a set $K$ but repeats convergence along $E$, leaving $K$ absent from its stated transfer property. A natural intended correction would require, for every $f\in\mathcal F$ and every candidate limit $\alpha$, that convergence along $K$ imply angular convergence to $\alpha$. The relation between $E$ and $K$ is not given, so this correction is explicit and conditional. An approach-set formulation also needs $0\in\overline K$ to avoid vacuous limits.

For a fixed positive hyperbolic width $\ell$, let \[ K_1=\{z\in\mathbb H:\rho(z,K)<\ell\},\qquad \rho(z,K)=\inf_{w\in K}\rho(z,w),\qquad B'=\{|z|:z\in K_1\}, \] where $\rho$ is the Poincaré distance on $\mathbb H$. The requested thickness conclusion is \[ \limsup_{r\downarrow0}\frac{m(B'\cap(0,r))}{r}>0. \] This is positive upper radial density, not the stronger lower-density condition used in the sufficient result. The question concerns whether it is forced by the intended universal transfer property of $K$. The source does not specify whether the assertion must hold for every $\ell>0$ or a particular width.

The source additionally attributes to conformality the extension of a given limit along $K$ to the thickened set $K_1$, but the printed hypothesis instead uses $E$; the same identification issue must be resolved there. It reports the desired density conclusion when $K$ lies in an angle with vertex zero whose closure stays inside $\mathbb H$; since the vertex itself is on $\partial\mathbb H$, this must exclude the vertex when describing the closure, or use a clarified angular convention. These are attributed background claims. The supplied triage labels the problem open as of 2018 and reports no definitive later resolution; no independent verification is claimed.

Short English statement

Must an approach set that forces angular limits for every conformal map have a hyperbolic thickening with positive upper radial density at zero? The printed converse switches from $E$ to $K$ without relating them, and leaves the thickening width and approach-set conventions unclear. The target half-disk and upper-versus-lower density distinction are retained exactly.

Sources

• Local source registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $2306092$, source code catalogue entry, including the supplied OPEN-TRIAGE assessment. Dataset attribution: Ulam AI and UnsolvedMath Contributors; original-source attribution and rights are retained.
• Hayman, Research Problems in Function Theory, source-list date 2018, Problem 6.92, attributed to M. Vuorinen: https://arxiv.org/abs/1809.07200. The requested density inequality has internal source label J6.92. Metadata reports source-TeX extraction and July 29, 2026 access, with status and rights review flags.
• The statement names Lindelöf's theorem for a terminal curve and cites Vuorinen for the weaker sufficient condition and thickening assertion, but the separate Vuorinen references are missing. No complete additional bibliography or independent source inspection is supplied here.