Research Problems in Function Theory — Problem 6.75

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 $\mathbb T=\partial\mathbb D$. For $n\in\mathbb N$, let $\mathcal P_n$ be the holomorphic injective polynomials $p(z)=z+\sum_{m=2}^n a_mz^m$ on $\mathbb D$ (degree at most $n$). Set $a_1=1$ and $A_m(n)=\max_{p\in\mathcal P_n}|a_m|$ for $1\le m\le n$.

Statement recorded in the supplied source.

For every fixed $n$, is there an index $n_0\in\{1,\ldots,n\}$ with \[A_1(n)<\cdots<A_{n_0}(n)>A_{n_0+1}(n)>\cdots>A_n(n),\] allowing either side to be empty? Also determine the least $n_1\in\{1,\ldots,n\}$ for which $A_m(n)\le1$ whenever $n_1\le m\le n$. [S2]

Short English statement

Are the sharp polynomial coefficient bounds strictly unimodal, and where does their final tail first stay at or below one?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306075 (AMR-022-6075). 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 6.75 and its complete update; Chapter 6 initial notation. https://arxiv.org/abs/1809.07200.