Research Problems in Function Theory — Problem 6.90

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 $\mathbb T=\partial\mathbb D$. A compact set $K$ has positive logarithmic capacity if it supports a probability measure $\mu$ with finite logarithmic energy $\iint\log(1/|\zeta-\eta|)\,d\mu(\zeta)d\mu(\eta)$. For a general set use its inner logarithmic capacity, the supremum of capacities of its compact subsets. Boundary values here are finite angular limits: limits within each fixed Stolz angle at a point of $\mathbb T$. A boundary uniqueness set for univalent functions is a set on which agreement of these limits for two holomorphic injective disc functions forces their equality throughout the disc.

Statement recorded in the supplied source.

Must every $E\subset\mathbb T$ of positive logarithmic capacity be a boundary uniqueness set for holomorphic injective functions on $\mathbb D$? [S2] Update 6.90 reports Hamilton's negative answer. The earlier Carleson comparison concerns finite Dirichlet integral, not the Hardy class $H^1$. [S2]

Short English statement

Does agreement on a boundary set of positive logarithmic capacity determine a univalent function uniquely?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306090 (AMR-022-6090). 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 6, Problem 6.90 and its complete update; Chapter 6 initial notation. https://arxiv.org/abs/1809.07200.