Research Problems in Function Theory — Problem 8.8

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 $\mathbb T=\partial\mathbb D$. For $p>0$, the Bergman space is $A^p(\mathbb D)=\{f\text{ analytic}:\|f\|_{A^p}=(\int_{\mathbb D}|f|^p\,dA)^{1/p}<\infty\}$, where $dA=dx\,dy$ is ordinary area measure. For a relatively closed $F\subset\mathbb D$, define $\|f\|_F=\sup_{z\in F}|f(z)|\in[0,\infty]$, with $\|f\|_\varnothing=0$, and its $A^2$ hull \[\widehat F_{A^2}=\{z\in\mathbb D:\ |f(z)|\le\|f\|_F\text{ for every }f\in A^2(\mathbb D)\}.\]

Statement recorded in the supplied source.

Give a geometric description of $\widehat F_{A^2}$ for every relatively closed $F\subset\mathbb D$. [S2]

Short English statement

Describe geometrically the hull of a relatively closed disk set defined by all Bergman functions.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2308008 (AMR-022-8008). 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 8.8, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.