Research Problems in Function Theory — Problem 5.45

Definitions and mathematical statement

Let $\mathbb D=\{z\in\mathbb C:|z|<1\}$ and $\widehat{\mathbb C}=\mathbb C\cup\{\infty\}$ with its spherical topology. An analytic subset of this sphere is understood in the descriptive-set-theoretic sense (a Suslin set), namely the empty set or a continuous image of the Baire space $\mathbb N^{\mathbb N}$, not a complex-analytic subvariety.

For a holomorphic or meromorphic $f$ on $\mathbb D$, an asymptotic value $a\in\widehat{\mathbb C}$ is a spherical limit $f(\gamma(t))\to a$ along a continuous path $\gamma:[0,1)\to\mathbb D$ tending to a boundary point $\zeta\in\partial\mathbb D$. Write $\operatorname{AV}(f)$ for the set of all such values. The exact boundary-path convention should be matched to the source; merely requiring $|\gamma(t)|\to1$ without convergence to one boundary point is a potentially different convention.

Problem (K. F. Barth). Find a simpler characterization of the class \[ \mathcal A=\{A\subseteq\widehat{\mathbb C}:\text{there exists }f\in\operatorname{Hol}(\mathbb D) \text{ with }\operatorname{AV}(f)=A\}. \] The equality is exact, not equality after closure. The supplied source says Ryan has already characterized this class; the request is for a simpler necessary-and-sufficient criterion, not a claim that no characterization exists.

As background, the source attributes to Kierst that every analytic subset $A$ is exactly the asymptotic-value set of a meromorphic function on $\mathbb D$, and that every such $A$ containing $\infty$ is realizable by a holomorphic function. It also says some analytic sets are not holomorphically realizable. These reported results explain why replacing the desired criterion by analyticity alone would not match the source.

Short English statement

Simplify Ryan's characterization of exactly which sphere subsets occur as all asymptotic values of a holomorphic disk function. Meromorphic realization and the case containing infinity are supplied background results.

Sources

• Hayman, Research Problems in Function Theory (2018), Problem 5.45, attributed to K. F. Barth, https://arxiv.org/abs/1809.07200; source-TeX extraction, accessed 2026-07-29. The 2018 update was retained for status triage.
• Kierst, realization theorems for meromorphic disk functions and for holomorphic disk functions when $\infty\in A$; Ryan, characterization of the holomorphically realizable class. No titles, dates, or theorem identifiers are supplied for these attributions.
• Local registry data/unsolvedmath/problems.json, permanent ID $2305045$, source code catalogue entry, Research Problems in Function Theory --- Problem 5.45. The record labels the simplification request open and OPEN-TRIAGE, with NEEDS_REVIEW source-status and rights qualifications.