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arXiv · 2601.10411

A proof of Alexander's conjecture on an inequality of Cassels

Myriam Ounaïes
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Abstract

Let $z_1,\dots,z_n$ be complex numbers with $|z_j|\le ρ$, where $ρ>1$. Cassels proved that, under an additional restriction on $ρ$, the inequality \[ \prod_{j\ne k}\bigl|1-\overline{z_j}z_k\bigr| \le \left(\frac{ρ^{2n}-1}{ρ^2-1}\right)^{\!n} \] holds. In a subsequent note, Alexander conjectured that this inequality is in fact valid without any restriction on $ρ$. In this paper, we confirm Alexander's conjecture.

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Myriam Ounaïes. 2026-01-22. A proof of Alexander's conjecture on an inequality of Cassels. https://arxiv.org/abs/2601.10411

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