Research Problems in Function Theory — Problem 7.46

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 $D\subset\mathbb C$ be an unbounded domain and let $f$ be holomorphic in $D$ with a continuous extension to its Euclidean closure. The condition $f(z)=o(|z|)$ at infinity means \[\lim_{R\to\infty}\sup_{z\in D,\ |z|\ge R}\frac{|f(z)|}{|z|}=0.\]

Statement recorded in the supplied source.

If $|f(z)|\le1$ for every finite $z\in\partial D$ and $f(z)=o(|z|)$ at infinity, must $|f(z)|\le1$ for every $z\in D$, for an arbitrary unbounded plane domain? [S2] The source compares the growth assumption with the exterior-disc case. Update7.46 reports Fuchs's affirmative result, followed by a more general result of Gehring, Hayman and Hinkkanen. [S2]

Short English statement

Does a boundary bound of one and uniform sublinear growth give the same bound throughout every unbounded plane domain?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307046 (AMR-022-7046). CC BY4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Chapter7, Problem7.46 and its complete update. https://arxiv.org/abs/1809.07200.