Research Problems in Function Theory — Problem 6.73

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 $f(z)=z+\sum_{n=2}^{\infty}a_nz^n$.

Statement recorded in the supplied source.

Is it true for every $f\in S$ that \[\limsup_{n\to\infty}\left||a_{n+1}|-|a_n|\right|\le1?\] [S2] Update 6.73 reports Hamilton's stronger eventual bound $\left||a_{n+1}|-|a_n|\right|\le1$. [S2]

Short English statement

Are successive coefficient magnitudes of every normalized univalent function eventually separated by at most one in the limiting sense?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306073 (AMR-022-6073). 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 6.73 and its complete update; Chapter 6 initial notation. https://arxiv.org/abs/1809.07200.