Research Problems in Function Theory — Problem 6.83

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 $\mathbb T=\partial\mathbb D$. The class $S$ consists of holomorphic injective functions $f:\mathbb D\to\mathbb C$ normalized by $f(0)=0$ and $f'(0)=1$, so $f(z)=z+\sum_{n=2}^{\infty}a_nz^n$. Let $(z_n)_{n\ge1}$ be a sequence of points in $\mathbb D$. Agreement is required at every listed point; repetitions impose the same condition again, rather than an additional zero multiplicity.

Statement recorded in the supplied source.

Characterize exactly those sequences $(z_n)$ for which there exist distinct functions $f,g\in S$ with $f(z_n)=g(z_n)$ for every $n$. [S2] The source gives the Blaschke summability condition $\sum_n(1-|z_n|)<\infty$ as necessary for a sequence of distinct points. When repetitions occur this condition concerns the distinct agreement points. Update 6.83 records Overholt's partial solution. [S2]

Short English statement

Which sets of sample points can fail to distinguish two normalized univalent functions?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306083 (AMR-022-6083). 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 6.83 and its complete update; Chapter 6 initial notation. https://arxiv.org/abs/1809.07200.