Research Problems in Function Theory — Problem 4.24

Definitions and mathematical statement

Let $\mathbb D=\{z\in\mathbb C:|z|<1\}$ and let $P(z)=\sum_{k=0}^{n}a_kz^k$ be a nonconstant polynomial of degree $n\geq1$. Call $P$ self-inversive if $a_0a_n\ne0$ and there is a complex number $\eta$ with $|\eta|=1$ such that \[ P(z)=\eta z^n\overline{P(1/\overline z)}. \] The right-hand side is understood as the polynomial obtained after expanding the conjugated expression. Equivalently, its zero multiset is invariant under reflection $\zeta\mapsto1/\overline\zeta$ in the unit circle, with multiplicities preserved.

Put $A=\max_{0\leq k\leq n}|a_k|$. Must there be a point $w_0\in \mathbb C$ such that \[ \{w:|w-w_0|<A\}\subseteq P(\mathbb D)? \] The center is free and may depend on $P$. The assertion is about the ordinary image set: each point of the disk needs at least one preimage in $\mathbb D$, but no common univalent inverse branch is required. Since $P$ is nonconstant, its image is an open domain. The radius is tied to the largest coefficient and scales correctly when $P$ is multiplied by a nonzero complex number. The question ranges over every degree and all self-inversive polynomials, including those with repeated zeros or zeros lying on the unit circle.

Short English statement

Does the image of the unit disk under every self-inversive polynomial contain a disk as large as its largest coefficient?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2304024 (AMR-022-4024), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Hayman - Research Problems in Function Theory (2018), Problem 4.24. Original source indexed by AMR-022-4024. https://arxiv.org/abs/1809.07200.