A proof of Alexander's conjecture on an inequality of Cassels
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908780502974464 |
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| author | Ounaïes, Myriam |
| author_facet | Ounaïes, Myriam |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_10411 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A proof of Alexander's conjecture on an inequality of Cassels Ounaïes, Myriam Complex Variables 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. |
| title | A proof of Alexander's conjecture on an inequality of Cassels |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2601.10411 |