Research Problems in Function Theory — Problem 6.70

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 $f(z)=z+\sum_{n=2}^{\infty}a_nz^n$. 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 $g=h=f$. Equip $H(\mathbb D)$, the vector space of holomorphic functions on $\mathbb D$, with uniform convergence on compact subsets. A support point of $S$ maximizes $\operatorname{Re}L$ over $S$ for a continuous complex-linear functional $L:H(\mathbb D)\to\mathbb C$ whose real part is not constant on $S$. This class-relative extreme-point convention and the distinct closed-convex-hull convention are stated separately in [S3].

Statement recorded in the supplied source.

Is every extreme point of $S$ a support point? Conversely, is every support point of $S$ an extreme point? [S2] Update 6.70 records counterexamples to the first implication by Hamilton and by Duren and Leung; it reports the converse as unresolved at the time of the update. [S2]

Short English statement

Do extreme points and support points coincide in the normalized univalent class?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306070 (AMR-022-6070). 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 6.70 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.