Research Problems in Function Theory — Problem 2.12a

Definitions and mathematical statement

The source imposes the same conditions as its Problem 2.12, but neither that problem nor the definitions of $\rho$ and $\Delta$ are included. The two questions must therefore retain those unresolved inherited hypotheses.

A finite asymptotic value of a function defined near infinity is a number $a\in{\mathbb{C}}$ for which there is a curve $\gamma:[0,\infty)\to{\mathbb{C}}$ with $|\gamma(t)|\to\infty$ and $f(\gamma(t))\to a$; the curve is taken in the function's domain. The first question is whether the inherited conditions together with \[ \rho\Delta<1 \] force $f$ to have no finite asymptotic value. The source attributes to Kövari a result under the stronger condition $\rho\Delta<1/\pi^2$.

The second question uses the minimum and maximum moduli \[ m_0(r,f)=\min_{|z|=r}|f(z)|,\qquad M(r,f)=\max_{|z|=r}|f(z)|, \] where these expressions are defined in the inherited function class, and asks whether \[ \limsup_{r\to\infty} \frac{\log m_0(r,f)}{\log M(r,f)} \geq\cos(\pi\rho\Delta). \] At a radius containing a zero, $\log m_0=-\infty$ in the extended-real sense; a nondegenerate interpretation of the denominator and any other restrictions must follow from the missing hypotheses. The second question does not separately repeat the threshold $\rho\Delta<1$, so whether that restriction is intended for it cannot be silently decided.

No meaning for $\Delta$, ambient function class, order convention for $\rho$, or additional zero/growth assumptions is guessed from the formulas. Recovering Problem 2.12 is essential to completing either precise assertion.

Short English statement

Under the missing hypotheses of Problem 2.12, improve the reported $1/\pi^2$ threshold to one for exclusion of finite asymptotic values, and establish the stated cosine lower bound for the minimum-to-maximum logarithmic growth ratio. The supplied triage says both were open in Hayman's 2018 update and reports no recent resolution located. Its claim that only the $1/\pi^2$ partial result is known is a dated assessment rather than independent verification in this draft.

Sources

• UnsolvedMath permanent ID 2302512, source code catalogue entry, "Research Problems in Function Theory --- Problem 2.12a." Registry curation and metadata: Ulam AI and UnsolvedMath Contributors, CC BY 4.0; updated 2026-08-06.
• Hayman, Research Problems in Function Theory (2018 source update), Problem 2.12a, referring to Problem 2.12: https://arxiv.org/abs/1809.07200. Supplied source-TeX extraction accessed 2026-07-29. The metadata retains the update for status triage and marks status/release and rights review NEEDS_REVIEW.
• Kövari, as rendered from the supplied source-TeX name, is cited for the threshold result and the related minimum-modulus question. No complete title, date, theorem number or locator is supplied; none is inferred.