Research Problems in Function Theory — Problem 4.2

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 $p(z)=\sum_{k=0}^na_kz^k$ be a nonconstant complex polynomial of degree $n\ge1$ whose zeros all lie on $\mathbb T$. Put \[A(p)=\max_{0\le k\le n}|a_k|,\qquad M(p)=\max_{|z|=1}|p(z)|. \]

Statement recorded in the supplied source.

Is $M(p)\ge2A(p)$ for every such polynomial? [S2]

Short English statement

Does a polynomial with all zeros on the unit circle have maximum modulus there at least twice its largest coefficient?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2304002 (AMR-022-4002). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory, arXiv:1809.07200, 2018. Problem 4.2, its complete Update 4.2, and the relevant chapter notation. https://arxiv.org/pdf/1809.07200.