Best constant in the Turan-Atkinson power-sum inequality (Problem 7.4)
Statement
Let $z_1=1$ and let $z_2,\dots,z_n$ be arbitrary complex numbers, and set $S_k=\sum_{i=1}^{n}z_i^k$. Atkinson proved that $\max_{1\le k\le n}|S_k|>c$ for an absolute constant $c>0$. What is the best (largest) admissible value of $c$? Equivalently, writing $R_n=\inf_{z_2,\dots,z_n\in\mathbb{C}}\max_{1\le k\le n}|S_k|$, determine the best universal constant $c=\inf_{n\ge2}R_n$ (and, if it exists, the limit $\lim_{n\to\infty}R_n$). The normalization $z_1=1$ is a genuine constraint: since any $z_i$ with $|z_i|>1$ only enlarges $\max_k|S_k|$, one may equivalently assume $|z_i|\le1$ for all $i$ with $z_1=1$ of maximal modulus.
Acceptance. ADVANCES: certified high-precision values of $R_n$ (with the extremal configuration and rigorous two-sided bounds) for $n$ beyond the reach of prior work, tabulating the extremizers. ADVANCES: a rigorous improvement of Biro's bounds $\tfrac12+q\le R_n\le\tfrac56$; a proof that $\lim_n R_n$ exists. FULLY RESOLVES: the exact best constant $c=\inf_{n}R_n$ (equivalently $\lim_n R_n$ if it exists) with matching lower- and upper-bound proofs. Numerical estimates without certified bounds do NOT qualify.
Background
From Turan's power-sum theory; Problem 7.4 in W. K. Hayman & E. F. Lingham, Research Problems in Function Theory (Fiftieth Anniversary Edition, Springer 2019), source of record arXiv:1809.07200, whose Update 7.4 states 'No progress on this problem has been reported to us.' Frontier: Atkinson (1961, 'On sums of powers of complex numbers') proved $R_n>1/6$ in his normalization (Hayman-Lingham print $c=1/3$; both refer to the same 1961 theorem). A. Biro, 'An improved estimate in a power sum problem of Turan', Indagationes Math. (N.S.) 11(3) (2000) 343-358, sharpened this to $\tfrac12+q\le R_n\le\tfrac56$ for some $q>0$ and all large $n$; the exact best constant, and even whether $\lim_n R_n$ exists, remain undetermined. CAUTION - do not conflate with two neighbouring problems: (a) the pure power-sum problem with all $|z_k|=1$ (studied by J. Andersson and others), where $\inf\max$ grows like $\sqrt n$ and where major recent progress exists; and (b) Turan's separate theorem for $|z_k|\ge1$, which gives $\inf\max_{1\le k\le n}|S_k|=1$ exactly. Problem 7.4 is the $z_1=1$, otherwise-unconstrained case with an absolute-constant answer. For each $n$, $R_n$ is a concrete continuous min-max over $\mathbb{C}^{n-1}$ amenable to global optimization with certified verification. Vetted open as of 2026-07-06 (high confidence; Biro's gap unmoved since 2000; no 2020-2026 resolution found).
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Hayman & Lingham, Research Problems in Function Theory (New Edition) - Problem 7.4 (Turan-Atkinson) | arxiv |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.