dorsal/arxiv
View SchemaA proof of Alexander's conjecture on an inequality of Cassels
| Authors | Myriam Ounaïes |
|---|---|
| Categories | |
| ArXiv ID | 2601.10411vv2 |
| URL | https://arxiv.org/abs/2601.10411 |
| License | http://arxiv.org/licenses/nonexclusive-distrib/1.0/ |
Abstract
Let $z_1,\dots,z_n$ be complex numbers with $|z_j|\le \rho$, where $\rho>1$. Cassels proved that, under an additional restriction on $\rho$, the inequality \[ \prod_{j\ne k}\bigl|1-\overline{z_j}z_k\bigr| \le \left(\frac{\rho^{2n}-1}{\rho^2-1}\right)^{\!n} \] holds. In a subsequent note, Alexander conjectured that this inequality is in fact valid without any restriction on $\rho$. In this paper, we confirm Alexander's conjecture.
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"abstract": "Let $z_1,\\dots,z_n$ be complex numbers with $|z_j|\\le \\rho$, where $\\rho\u003e1$. Cassels proved that, under an additional restriction on $\\rho$, the inequality \\[ \\prod_{j\\ne k}\\bigl|1-\\overline{z_j}z_k\\bigr| \\le \\left(\\frac{\\rho^{2n}-1}{\\rho^2-1}\\right)^{\\!n} \\] holds. In a subsequent note, Alexander conjectured that this inequality is in fact valid without any restriction on $\\rho$. In this paper, we confirm Alexander\u0027s conjecture.",
"arxiv_id": "2601.10411",
"authors": [
"Myriam Ouna\u00efes"
],
"categories": [
"math.CV"
],
"license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
"title": "A proof of Alexander\u0027s conjecture on an inequality of Cassels",
"url": "https://arxiv.org/abs/2601.10411",
"version": "v2"
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