Research Problems in Function Theory — Problem 1.30

Definitions and mathematical statement

Let $f$ be a nonconstant meromorphic function on ${\mathbb{C}}$, with Nevanlinna characteristic $T(r,f)$ and order \[ \rho(f)=\limsup_{r\to\infty}\frac{\log T(r,f)}{\log r}. \] A finite asymptotic value is an $a\in{\mathbb{C}}$ for which there is a continuous curve $\gamma:[0,\infty)\to{\mathbb{C}}$ with $|\gamma(t)|\to\infty$ and $f(\gamma(t))\to a$. Let $A_{\mathrm{fin}}(f)$ be the set of distinct such values; values are counted once even if approached through several different tracts.

W. Al-Katifi asks for an upper bound on $\#A_{\mathrm{fin}}(f)$ that takes into account both $\rho(f)$ and the angular measures of the asymptotic tracts. For illustration, a tract over $a$ may be represented by compatible unbounded components of inverse images of small disks $\{w:|w-a|<\eta\}$, and a chosen planar tract $U$ has circular angular size \[ \theta_U(r)=\operatorname{meas}\{\vartheta\in[0,2\pi):re^{i\vartheta}\in U\}. \] The source does not specify this particular tract convention, how shrinking disk components are selected, or which functional of angular size is to enter a bound. The display therefore explains one common meaning of angular measure without prescribing the requested inequality.

A precise bound must specify the admissible angular data and its uniformity across tracts and radii. It must also distinguish all asymptotic tracts from direct or logarithmic tracts: the source does not restrict the singularity type, so such a restriction cannot be inserted silently. Finite order is not explicitly assumed, and no proposed numerical inequality or dependence on $\rho$ and the angular data is supplied. The reported question remains an open-ended quantitative-bound problem. The supplied summary says it was open in Hayman's 2018 edition and that no recent resolution was located.

Short English statement

Al-Katifi asks how to bound the number of distinct finite asymptotic values of a plane meromorphic function using both its growth order and the angular sizes of its tracts. The tract type, angular statistic, and exact form of the desired bound are unspecified.

Sources

• Local UnsolvedMath registry, permanent ID 2301030, exact source code catalogue entry, "Research Problems in Function Theory --- Problem 1.30"; status open, classification OPEN-TRIAGE.
• Hayman, Research Problems in Function Theory (2018), Problem 1.30, attributed to W. Al-Katifi, https://arxiv.org/abs/1809.07200. The record identifies extraction from public arXiv source TeX, accessed July 29, 2026; the 2018 update was retained for status triage.
• The supplied research summary reports openness as of the 2018 edition and no recent resolution located. It supplies no additional authoritative citation, partial bound, or precise recent search date.