Research Problems in Function Theory — Problem 2.54

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 set $S\subseteq\mathbb C$, a function is bounded on $S$ if there is a finite $C$ with $|f(z)|\le C$ for all $z\in S$, and bounded away from zero there if some $c>0$ satisfies $|f(z)|\ge c$ for all $z\in S$. A transcendental entire function is entire and is not a polynomial.

Statement recorded in the supplied source.

Let $E\subseteq\mathbb C$ be closed. Suppose that one transcendental entire function is bounded on $E$ and another is bounded away from zero on $\mathbb C\setminus E$. Must there be a single transcendental entire function $h$ and constants $C<\infty$, $c>0$ satisfying \[|h(z)|\le C\ (z\in E),\qquad |h(z)|\ge c\ (z\in\mathbb C\setminus E)?\] [S2] Update 2.54 records no progress on the simultaneous-existence question. [S2]

Short English statement

Can the two separate entire-function existence properties of a closed set and its complement always be realised by one function?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2302054 (AMR-022-2054). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Chapter 2, Problem 2.54 and Update 2.54. https://arxiv.org/abs/1809.07200.