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

TL;DR

This paper proves Alexander's conjecture that a certain inequality involving complex numbers and their magnitudes holds universally without restrictions on the parameter, extending Cassels' earlier result.

Contribution

The paper confirms Alexander's conjecture, establishing the inequality's validity for all complex numbers with magnitude greater than one, removing previous restrictions.

Findings

Confirmed Alexander's conjecture for all ho > 1

Extended Cassels' inequality without restrictions

Provides a complete proof of the conjecture

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.