Research Problems in Function Theory — Problem 7.17

Definitions and mathematical statement

Let $H^+=\{z\in{\mathbb{C}}:\operatorname{Re}z>0\}$, let $f$ be analytic and bounded on $H^+$, and fix $\alpha\in{\mathbb{R}}$ with $|\alpha|<\pi/2$. The whole ray $re^{i\alpha}$, $r>0$, lies in $H^+$. With the signed-logarithm directional-type convention, define \[ h_f(\alpha)=\limsup_{r\to\infty}\frac{\log|f(re^{i\alpha})|}{r}, \] using $\log0=-\infty$. For a sample sequence tending to infinity define \[ h_f((r_n),\alpha)=\limsup_{n\to\infty} \frac{\log|f(r_ne^{i\alpha})|}{r_n}. \] These may be negative or $-\infty$ despite boundedness; replacing $\log$ by $\log^+$ would lose the intended decay information.

The source assumes positive integer sample radii with \[ \sum_{n=1}^{\infty}\frac1{r_n}=\infty \] and asks to prove \[ h_f((r_n),\alpha)=h_f(\alpha). \] It does not explicitly require that $r_n$ increase, be distinct, or tend to infinity. Those conditions are essential to interpreting sampling type and the density hypothesis: repetitions can make the reciprocal sum diverge without providing additional sampling radii. A standard conditional intended version takes $r_1<r_2<\cdots$ positive integers and retains the divergent reciprocal sum, but that is not silently substituted for the literal statement. The source also does not define its exponential-type convention, so the meaningful signed-logarithm reading above should be confirmed.

The source attributes proofs for $\alpha=0$ to Boas and Levinson and says their methods do not seem to work when $|\alpha|>\pi/4$. This is a reported limitation of those proofs, not a statement that all intermediate angles are settled.

Short English statement

Does a sufficiently dense integer-radius sample recover the full exponential decay/growth type of a bounded analytic function along any ray inside the right half-plane? The precise intended sampling condition needs confirmation because the printed sequence assumptions allow repetitions. The supplied triage reports the question open in Hayman's 2018 update and no definitive resolution located; no independent status review is asserted.

Sources

• UnsolvedMath permanent ID 2307017, source code catalogue entry, "Research Problems in Function Theory --- Problem 7.17." 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 7.17, attributed to J. Korevaar: 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.
• Boas and Levinson are named for proofs at angle zero without complete titles, dates or locators. No further theorem or reference is inferred.