Research Problems in Function Theory — Problem 6.65

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 $f(z)=z+\sum_{n=2}^{\infty}a_nz^n$. For $M>1$, let $S^*(M)$ consist of the functions $f\in S$ with $|f(z)|\le M$ throughout $\mathbb D$ and with image starlike about $0$, meaning $tw\in f(\mathbb D)$ whenever $w\in f(\mathbb D)$ and $0\le t\le1$.

Statement recorded in the supplied source.

For every real $M$ with $e<M<5$, determine the exact value of $\sup\{|a_3|:f\in S^*(M)\}$. [S2]

Short English statement

Find the sharp third-coefficient bound for bounded starlike functions in the remaining parameter range between e and five.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306065 (AMR-022-6065). 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 6.65 and its complete update; Chapter 6 initial notation. https://arxiv.org/abs/1809.07200.