Research Problems in Function Theory — Problem 2.57

Definitions and mathematical statement

For an entire function $f$, put $M(r,f)=\max_{|z|=r}|f(z)|$. For each real $\alpha>1$, the source defines $I_\alpha$ by the two simultaneous conditions \[ \log M(r,f)=O((\log r)^{1+\alpha}),\qquad (\log r)^\alpha=o(\log M(r,f))\quad(r\to\infty). \] Equivalently, the first has a constant upper bound for the displayed normalized growth eventually, and the second requires $(\log r)^\alpha/\log M(r,f)\to0$.

The source credits Hayman with the following result under the smaller-growth hypothesis $\log M(r,f)=O((\log r)^2)$: \[ \frac{\log|f(z)|}{\log M(|z|,f)}\longrightarrow1 \quad\text{as }|z|\to\infty\text{ outside }E, \] where $E$ consists of exceptional "circles" whose subtended angles at $0$ have finite sum. In particular, it records the same ratio limit for $z=re^{i\theta}$, $r\to\infty$, for Lebesgue-almost every $\theta$. The first question is whether these conclusions continue to hold for every $f\in I_\alpha$, for $\alpha>1$.

For meromorphic $f$, let $T(r,f)$ be the Nevanlinna characteristic and let \[ \delta(a,f)=1-\limsup_{r\to\infty}\frac{N(r,a)}{T(r,f)} \quad(a\in\widehat{\mathbb{C}}), \] where $N(r,a)$ is the integrated counting function with multiplicity, counting poles for $a=\infty$. The source credits Anderson--Clunie with: if $T(r,f)=O((\log r)^2)$, there is at most one deficient value, and every $a$ with $\delta(a,f)>0$ is asymptotic and satisfies \[ f(re^{i\theta})\longrightarrow a\quad(r\to\infty) \quad\text{for almost every }\theta. \] Limits to $\infty$ are understood spherically. An asymptotic value is a limit along some curve tending to infinity. The second question asks for the corresponding extension of these conclusions. A natural meromorphic analogue of $I_\alpha$ would require \[ T(r,f)=O((\log r)^{1+\alpha}),\qquad (\log r)^\alpha=o(T(r,f)), \] but the source explicitly defines $I_\alpha$ only for entire functions, so this substitution is a conditional interpretation rather than an unqualified source definition.

J. M. Anderson asks whether the results depend on small growth or instead on growth smoothness. The source calls Hayman's functions $I_1$, although its displayed definition would also impose a lower-growth condition; the precise intended inclusion requires checking. The exceptional "circles" also need a disk/covering convention for the stronger off-exceptional-set statement. The supplied OPEN-TRIAGE summary reports open status in the 2018 edition and no recent resolution located.

Short English statement

Do the maximum-modulus and deficient-value conclusions known at growth $O((\log r)^2)$ extend to the two-sided growth classes $I_\alpha$, $\alpha>1$? The meromorphic analogue and exceptional-circle convention need clarification.

Sources

• Local UnsolvedMath registry, permanent ID $2302057$, exact source code catalogue entry, "Research Problems in Function Theory --- Problem 2.57."
• Hayman, Research Problems in Function Theory (2018), Problem 2.57, attributed to J. M. Anderson; https://arxiv.org/abs/1809.07200. Source-TeX extraction; recorded access 2026-07-29.
• The statement attributes the cited small-growth results to Hayman and Anderson--Clunie without complete references. The supplied OPEN-TRIAGE summary gives no separate current-status citation.