Research Problems in Function Theory — Problem 5.75

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 $\mathbb T=\partial\mathbb D$. If $f$ is analytic, the length of its parametrized radial image at $\zeta\in\mathbb T$ is \[\operatorname{length}(r\mapsto f(r\zeta),\ 0\le r<1)=\int_0^1|f^{\prime}(r\zeta)|\,dr\in[0,\infty].\]

Statement recorded in the supplied source.

Does there exist an analytic $f$ on $\mathbb D$ with $\sup_{\mathbb D}|f|<\infty$ for which this length is infinite for every $\zeta\in\mathbb T$? [S2]

Short English statement

Can a bounded analytic disk function have an image of infinite length along every radius?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2305075 (AMR-022-5075). 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 5.75, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.