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

Let $z_1,\dots ,z_n$ be complex numbers with $\mathrm{\mid }{z}_j\mathrm{\mid }\le \rho$, where $\rho >1$. Cassels proved that, under an additional restriction on $\rho$, the inequality $$\prod _{j\mathrm{\ne }k}\mid 1-\overline{z_j}{z}_k\mid \le {\left(\frac{{\rho }^{2n}-1}{{\rho }^2-1}\right)}^{\text{\hspace{0.17em}⁣}n}$$ 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.