Research Problems in Function Theory — Problem 4.11

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.Assume in addition $|z_j|\le1$ for every $j$. Let $\lambda_2$ denote planar Lebesgue measure and define the optimal lower-area function \[a_n=\inf_{z_1,\ldots,z_n\in\overline{\mathbb D}}\lambda_2\left(\left\{z:\prod_{j=1}^n|z-z_j|\le1\right\}\right). \]

Statement recorded in the supplied source.

Obtain lower bounds for $a_n$ in terms of $n$, and determine its sharp size and asymptotic behaviour. The source records that arbitrarily small areas are possible once the degree is sufficiently large, so the requested lower bound must depend on degree. [S2]

Short English statement

Find sharp lower bounds, depending on degree, for the area of a monic polynomial’s lemniscate when all zeros lie in the unit disc.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2304011 (AMR-022-4011). 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 4.11, its complete Update 4.11, and the relevant chapter notation. https://arxiv.org/pdf/1809.07200.