Research Problems in Function Theory — Problem 6.55

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.

Write $\mathbb D=\{z\in\mathbb C:|z|<1\}$ and $\mathbb T=\partial\mathbb D$. Let $S$ consist of holomorphic injective functions $f(z)=z+\sum_{n\ge2}a_nz^n$ on $\mathbb D$; set $a_0=0$, $a_1=1$. A function in $S$ is starlike if its image is starlike about $0$, and convex if its image is a convex domain.Suppose $f$ is bounded and starlike. For $E\subset\mathbb T$ at whose points the radial limits exist, write $f(E)=\{\lim_{r\uparrow1}f(r\xi):\xi\in E\}$. For a compact $K\subset\mathbb T$, define its logarithmic capacity by $\operatorname{cap}K=\exp(-\inf_\mu\iint\log(1/|z-w|)\,d\mu(z)d\mu(w))$, where $\mu$ ranges over Borel probability measures on $K$; extend capacity to arbitrary sets by outer capacity. Let $\mathcal H^1$ be one-dimensional Hausdorff measure, and $\mathcal H^{\log}$ the Hausdorff measure with gauge $h(t)=1/\log(1/t)$ near $t=0$, with $h(0)=0$.

Statement recorded in the supplied source.

Does $\operatorname{cap}E=0$ imply $\mathcal H^1(f(E))=0$ for every such $f$? If not known in that generality, does the conclusion at least hold whenever $\mathcal H^{\log}(E)=0$? [S2]

Short English statement

Must a bounded starlike boundary map send a capacity-zero set, or at least a logarithmic-measure-zero set, to a set of zero length?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306055 (AMR-022-6055). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), arXiv:1809.07200, Chapter 6, Problem 6.55, complete update and preceding notation. https://arxiv.org/abs/1809.07200.