Research Problems in Function Theory — Problem 6.47

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 $\mathbb T=\partial\mathbb D$. Let $S$ consist of holomorphic injective functions $f(z)=z+\sum_{n\ge2}a_nz^n$ on $\mathbb D$; set $a_0=0$, $a_1=1$. A function in $S$ is starlike if its image is starlike about $0$, and convex if its image is a convex domain.Let $S'=\{f\in S:f'\text{ is injective on }\mathbb D\}$, and for $n\ge2$ let $A_n=\sup_{f\in S'}|a_n|$.

Statement recorded in the supplied source.

Determine the sharp constants $A_n$ for all $n\ge2$. In particular, determine $A_2$ and the best coefficient consequences of that bound for the higher $A_n$. [S2]

Short English statement

Find sharp coefficient bounds when both the normalized univalent function and its derivative are injective.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306047 (AMR-022-6047). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), arXiv:1809.07200, Chapter 6, Problem 6.47, complete update and preceding notation. https://arxiv.org/abs/1809.07200.