Research Problems in Function Theory — Problem 4.8

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.

Let $n\ge1$ and let $f(z)=\prod_{j=1}^n(z-z_j)$ be a monic polynomial of degree $n$, with zeros repeated according to multiplicity. Its filled unit lemniscate is the compact set \[E_f=\{z\in\mathbb C:|f(z)|\le1\}. \] The monic normalization is the one used in the source's Problem 4.7 and its following lemniscate questions.

Statement recorded in the supplied source.

If $E_f$ is connected, is it true that \[\max_{z\in E_f}|f'(z)|\le\tfrac12n^2?\] The source's Update 4.8 explains that this historical bound is false and gives the sharp replacement \[\max_{z\in E_f}|f'(z)|\le 2^{1/n-1}n^2. \] The original bound and its corrected sharp version concern the same monic connected-lemniscate class. [S2]

Short English statement

Bound the derivative of a monic polynomial on its connected unit lemniscate. The originally proposed constant is false; the source records the stated sharp correction.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2304008 (AMR-022-4008). 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. Lemniscate notation preceding Problem 4.7, Problem 4.7, Problem 4.8 and its complete Update 4.8. https://arxiv.org/pdf/1809.07200.