Research Problems in Function Theory — Problem 5.50

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $\mathbb D=\{z\in\mathbb C:|z|<1\}$ and $\mathbb T=\partial\mathbb D$. Call an analytic function $f$ annular if there are Jordan curves $J_n\subset\mathbb D$ such that $J_n$ lies in the bounded component of $\mathbb C\setminus J_{n+1}$, for every $\varepsilon>0$ all sufficiently late $J_n$ lie in $\{1-\varepsilon<|z|<1\}$, and $\min_{J_n}|f|\to\infty$. For $a\in\mathbb C$, let \[Z_{\partial}(f-a)=\{\zeta\in\mathbb T:\exists\text{ distinct }z_j\in\mathbb D,\ z_j\to\zeta,\ f(z_j)=a\},\quad S(f)=\{a\in\mathbb C:Z_{\partial}(f-a)\ne\mathbb T\}.\] This is the boundary accumulation set of the zeros, not the zero set itself.

Statement recorded in the supplied source.

Does there exist an annular $f$ for which $S(f)$ is countably infinite, that is, $|S(f)|=\aleph_0$? The update’s countability result for singular values of a strongly annular function concerns a different specified set. [S2]

Short English statement

Can an annular analytic function have exactly countably infinitely many values whose preimages fail to accumulate at every boundary point?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2305050 (AMR-022-5050). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory (2018), arXiv:1809.07200. Problem 5.50, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.