Research Problems in Function Theory — Problem 5.59

Definitions and mathematical statement

Fix a holomorphic function $g$ on $\mathbb D$. A function $f$ is subordinate to $g$, written $f\prec g$, if $f=g\circ\omega$ for some holomorphic $\omega:\mathbb D\to\mathbb D$ with $\omega(0)=0$. Let $S_g$ be the family of all such functions. Equip the vector space of holomorphic functions on $\mathbb D$ with uniform convergence on compact subsets, and let $K_g=\overline{\operatorname{co}}(S_g)$ be its closed convex hull, where convex coefficients are real and nonnegative. A member $h\in K_g$ is extreme if $h=tu+(1-t)v$, $u,v\in K_g$, $0<t<1$, implies $u=v=h$. Find general conditions on $g$ ensuring that every extreme point of $K_g$ is a rotation of the argument of $g$: \[ \operatorname{Ext}(K_g)\subseteq \{z\mapsto g(e^{it}z):0\le t<2\pi\}. \] Where possible, determine exactly which of these rotations are extreme. The source supplies several special functions for which the desired description is known, and suggests regular positive-coefficient hypotheses as a possible route. Neither univalence of $g$ nor boundedness of $g$ is part of the general setup; any extra assumption should appear in the proposed characterization.

Short English statement

When are all extreme points of the closed convex hull of a subordination family obtained simply by rotating the argument of its defining function?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2305059 (AMR-022-5059), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Hayman - Research Problems in Function Theory (2018), Problem 5.59. Original source indexed by AMR-022-5059. https://arxiv.org/abs/1809.07200.