Research Problems in Function Theory — Problem 6.66

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.

The exterior class $\Sigma$ consists of holomorphic injective functions on $\{z:|z|>1\}$ with Laurent expansion $f(z)=z+b_0+\sum_{j=1}^{\infty}b_jz^{-j}$. A function $f$ in a class $\mathcal A$ is an extreme point of that class if $f=tg+(1-t)h$, with $g,h\in\mathcal A$ and $0<t<1$, forces $g=h=f$. Let $\Sigma_0=\{g\in\Sigma:b_0=0\}$, and write $K_g=\mathbb C\setminus g(\{|z|>1\})$ for the compact omitted set. Measure means two-dimensional Lebesgue measure. This class-relative extreme-point convention and the distinct closed-convex-hull convention are stated separately in [S3].

Statement recorded in the supplied source.

Characterize all extreme points of $\Sigma_0$. Is $g\in\Sigma_0$ an extreme point if and only if the omitted set $K_g$ has area zero? The source records the zero-area condition as sufficient; the question asks whether it is also necessary. [S2] Update 6.66 attributes an affirmative answer to Hamilton. [S2]

Short English statement

Characterize extreme normalized exterior maps: must their omitted sets have zero area?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306066 (AMR-022-6066). 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 6.66 and its complete update; Chapter 6 initial notation. https://arxiv.org/abs/1809.07200.
• S3 V. Allu and A. Pandey, Support points of some classes of analytic and univalent functions (2022), Introduction and Preliminaries, p.3. https://arxiv.org/pdf/2209.10607.