Research Problems in Function Theory — Problem 6.35

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.For $f\in S$, let $O_f=\mathbb D\setminus f(\mathbb D)$. Area means two-dimensional Lebesgue measure. A finite omitted configuration is a list $w_1,\ldots,w_m\in\mathbb D$, with $m\ge1$.

Statement recorded in the supplied source.

Characterize the subsets $O\subset\mathbb D$ that equal $O_f$ for some $f\in S$; determine $\sup_{f\in S}\operatorname{area}(O_f)$ and the maximizing maps; and give necessary and sufficient conditions on any finite configuration for the existence of $f\in S$ with all $w_j\in O_f$. Include the two-point and three-point cases explicitly in that classification. [S2]

Short English statement

Classify the omitted parts of the unit disc for normalized univalent maps, maximize their area, and decide which finite point configurations can be omitted together.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306035 (AMR-022-6035). 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 6.35, complete update and preceding notation. https://arxiv.org/abs/1809.07200.