Research Problems in Function Theory — Problem 5.14

Definitions and mathematical statement

The extracted statement supplies the differential-value-omission condition \[ f'(z)-f(z)^n\ne a, \] where $a\in{\mathbb{C}}$ is fixed, $n\geq5$ in the meromorphic case, and $n\geq3$ in the entire case. Here $n$ is an integer and $f(z)^n$ is an ordinary power, not an iterate. On a specified domain $D$, the condition means that the meromorphic differential expression $f'-f^n$ omits the finite value $a$: it is unequal to $a$ at every point where it is finite.

No requested conclusion, domain, family of functions, or preceding common hypotheses are included in the supplied statement. Thus the displayed omission condition alone is not a complete conjecture. The research summary says that the intended target is a disc analogue of established plane results, but the record does not say whether the disc conclusion is normality of a family, a growth estimate, or another property. None is invented here. If a disc setting is intended, the distinction between functions entire on ${\mathbb{C}}$ and functions merely holomorphic on the disc also needs the missing source context.

The statement says that the corresponding results for functions in the "place" are proved by Hayman; the summary interprets this as the plane. It separately attributes the case $n=1$ of Problem 5.12 to Clunie. That cross-reference is retained as literature context, not merged with the thresholds $n\geq5$ and $n\geq3$ of the displayed condition. The supplied summary reports no definitive resolution of the disc question, but the exact disc assertion cannot be recovered from this extraction.

Short English statement

The record gives a condition that $f'-f^n$ omit a fixed complex value, with separate meromorphic and entire exponent thresholds. It omits the conclusion being asked for. The summary identifies a disc analogue of known plane results, but a complete problem requires the missing surrounding text.

Sources

• Local UnsolvedMath registry, permanent ID 2305014, exact source code catalogue entry, "Research Problems in Function Theory --- Problem 5.14"; status open, classification OPEN-TRIAGE.
• Hayman, Research Problems in Function Theory (2018), Problem 5.14, 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 statement credits Hayman with corresponding plane results and Clunie with the $n=1$ case of Problem 5.12, but supplies no complete bibliographic entries or statements. The supplied research summary says the disc analogue is open and no definitive resolution was located; no further reference is given.