Research Problems in Function Theory — Problem 7.4

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.

For $n\geq1$ let $z_1=1$ and let $z_2,\ldots,z_n$ be arbitrary complex numbers, with no modulus restriction. Put $S_k=\sum_{j=1}^{n}z_j^k$ and define \[c_*=\sup\left\{c\in\mathbb R:\max_{1\leq k\leq n}|S_k|>c \text{ for every }n\geq1\text{ and every such }(z_1,\ldots,z_n)\right\}.\] Equivalently, $c_*$ is the infimum of these maxima over all sizes and tuples; the source records that $c=1/3$ is valid.

Statement recorded in the supplied source.

Determine $c_*$, the sharp universal constant in this power-sum bound, including whether the strict inequality remains valid at the sharp endpoint. [S2]

Short English statement

Find the sharp uniform lower bound for a power sum of a complex tuple containing 1.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307004 (AMR-022-7004). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Problem 7.4, complete problem and update. https://arxiv.org/abs/1809.07200.