Research Problems in Function Theory — Problem 6.71

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 $g=h=f$. A normalized function $f$ is close-to-convex if there is a convex univalent function $g$ on $\mathbb D$ and a real $\eta$ such that $\operatorname{Re}(e^{i\eta}f'/g')>0$ on $\mathbb D$. A bound $I_f(R)=O((1-R)^{-1})$ means there are constants $C_f,R_f$ with $I_f(R)\le C_f/(1-R)$ for $R_f<R<1$; the constant may depend on $f$. This class-relative extreme-point convention and the distinct closed-convex-hull convention are stated separately in [S3].

Statement recorded in the supplied source.

For \[I_f(R)=\int_0^{2\pi}\left|\frac{f'(Re^{it})}{f(Re^{it})}\right|^2\,dt,\] does $I_f(R)=O((1-R)^{-1})$ as $R\to1^-$ hold for every extreme point $f$ of $S$? Does the same estimate hold for every close-to-convex $f\in S$? The general recorded estimate has the additional factor $\log(1/(1-R))$. [S2]

Short English statement

Can the extra logarithmic factor in this integral estimate be removed for extreme univalent functions and for close-to-convex functions?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306071 (AMR-022-6071). 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 6.71 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.