Research Problems in Function Theory — Problem 5.79

Definitions and mathematical statement

Let $\mathbb D=\{z\in{\mathbb{C}}:|z|<1\}$ and $\mathbb T=\partial\mathbb D$. A finite asymptotic value at $\zeta\in\mathbb T$ is a limit in ${\mathbb{C}}$ of $f$ along a curve in $\mathbb D$ tending to $\zeta$. MacLane's class $A$ is the source's class of nonconstant analytic functions with asymptotic values at a dense set of boundary points; the finite-versus-infinite-value convention should be confirmed, since the theorem's first alternative specifically requires finite values.

For a nonconstant analytic $f$ and an integer $m\geq1$, put $\psi=f^mf'$. The source attributes to Sons the theorem that, when $f,\psi\in A$, at least one of the following holds:

1. $f$ has finite asymptotic values at a dense subset of $\mathbb T$;
2. for every $a\in{\mathbb{C}}$, there are infinitely many distinct $z\in\mathbb D$ such that $\psi(z)=a$.

The first task asks for other meaningful hypotheses/conclusions in place of alternative (i) that yield a correct dichotomy; it specifies no candidate replacement or optimality criterion.

The second task asks whether $\psi$ can instead be a differential monomial \[ \psi=\prod_{j=0}^{k}(f^{(j)})^{m_j},\qquad k,m_0,\ldots,m_k\in{\mathbb{N}}\cup\{0\},\quad f^{(0)}=f. \] The natural contextual membership assumptions are again $f,\psi\in A$, but their scope should be confirmed. The all-zero exponent choice gives $\psi=1$, and other choices can also give constant output, so these cannot satisfy the nonconstant MacLane-class requirement. No unstated nontriviality restriction is silently inserted into the printed exponent range.

The source also reports the same dichotomy for \[ \psi=f^{(l)}+\sum_{\nu=0}^{i-1}a_\nu f^{(\nu)},\qquad l\in{\mathbb{N}}, \] with coefficients analytic in $|z|<1+s$ for some $s>0$. The printed upper index is $i-1$, but $i$ is undefined; a possible intended $l-1$ is not silently substituted. The extension-radius parameter is renamed $s$ to distinguish it from a radial variable. It asks both for replacements of alternative (i) in this linear-differential setting and whether the coefficients may instead be analytic only in $\mathbb D$ with \[ T(r,a_\nu)=o(T(r,f))\qquad(r\uparrow1), \] where for analytic $g$, $T(r,g)=(2\pi)^{-1}\int_0^{2\pi}\log^+|g(re^{it})|\,{\,\mathrm{d}} t$. The exact coefficient range and inherited class assumptions require recovery of the source context.

Short English statement

Extend Sons's boundary-asymptotic-value/value-distribution dichotomy to differential monomials and to linear differential expressions with small analytic coefficients, and find alternative first branches that still give meaningful theorems. The source's index mismatch and degenerate exponent choices are retained as explicit issues. The supplied triage reports the question open in Hayman's 2018 update and no definitive later resolution located; this draft makes no independent status determination.

Sources

• UnsolvedMath permanent ID 2305079, source code catalogue entry, "Research Problems in Function Theory --- Problem 5.79." 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 5.79, attributed to L. R. Sons, comparing Problem 1.38: 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.
• Sons and MacLane are cited in the source without complete bibliographic entries. The comparison Problem 1.38 is not supplied. No missing reference or comparison statement is inferred.