Higher-order Rubel classes and geometric characterizations

Determine whether Rubel(3) = Rubel(4) = 3 = Rubel() or identify a further distinction among these classes, and determine whether any Rubel(k) admits a clean geometric characterization for k [3, ] .

Background

The paper defines Rubel(k) as the class of domains on which every unbounded analytic function admits a sequence along which the function and its first k derivatives tend to infinity. It proves Rubel(1) = Rubel(2), while also showing that Rubel_0(1) is not contained in Rubel_0(3). The unresolved issue is whether the hierarchy stabilizes from Rubel(3) onward, and whether the higher-order classes possess geometric descriptions analogous to the interior-path-metric characterization obtained for simply connected Rubel_0(1) and Rubel_0(2) domains.

References

Verify that $Rubel(3) = Rubel(4) = \cdots = Rubel(\infty)$ or find a further distinction. Do any of $Rubel(k)$ admit a clean geometric characterization, $k \in [3, \infty] \cap \Bbb{N}$?

— On Simply Connected Domains Supporting an Unbounded Analytic Function with Bounded Derivative  (2609.20607 - MacMahon, 17 Sep 2026) in Section 6, Open Questions

Is there a clean geometric description of $Rubel(S)$?

— On Simply Connected Domains Supporting an Unbounded Analytic Function with Bounded Derivative  (2609.20607 - MacMahon, 17 Sep 2026) in Section 6, Open Questions

Is there a $Rubel(2)$ domain $\Omega$ and an unbounded analytic function $f \in \mathcal{O}(\Omega)$ such that no more than three consecutive derivatives tend to infinity along any given sequence?

— On Simply Connected Domains Supporting an Unbounded Analytic Function with Bounded Derivative  (2609.20607 - MacMahon, 17 Sep 2026) in Section 6, Open Questions

Characterize those domains $\Lambda \subset \Bbb{R}n$ for which, given any unbounded harmonic function $u$ on $\Lambda$, there exists a sequence ${x_n}{n \in \Bbb{N} \subset \Lambda$ along which $u(x_j)$ $\max{1 \leq i \leq n} |\partial_i u(x_j)|$, $\max_{1 \leq i \leq n} |\partial2_i u(x_j)|$, $\cdots$, $\max_{1 \leq i \leq n} |\partial{(k)}_i u(x_j)| \rightarrow \infty$ as $j \rightarrow \infty$.

— On Simply Connected Domains Supporting an Unbounded Analytic Function with Bounded Derivative  (2609.20607 - MacMahon, 17 Sep 2026) in Section 6, Open Questions