Research Problems in Function Theory — Problem 9.7

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.

For a closed $E\subset\mathbb C$, let $A(E)=\{g\in C(E):g\text{ is analytic on }E^{\circ}\}$. Call $E$ weak Arakelian if for every $g\in A(E)$ there is an entire function $F$ such that, for every sequence $(z_j)\subset E$, \[|F(z_j)|\to\infty\quad\Longleftrightarrow\quad |g(z_j)|\to\infty.\] The roles of the prescribed function and the entire function are separated here by the names $g$ and $F$.

Statement recorded in the supplied source.

Give necessary and sufficient geometric conditions for a closed planar set $E$ to be weak Arakelian. [S2]

Short English statement

Characterize closed sets on which each continuous interior-analytic function has the same diverging-value sequences as some entire function.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2309007 (AMR-022-9007). 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 9.7, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.