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

Myriam Ounaïes

TL;DR

The paper resolves Alexander's conjecture by proving that Cassels' inequality $\prod_{j\neq k}|1-\overline{z_j}z_k| \le \left(\frac{\rho^{2n}-1}{\rho^2-1}\right)^n$ holds for all $\rho>1$ when $|z_j|\le \rho$. The authors reduce the problem to an auxiliary additive inequality on the unit disk and employ a monotonicity argument to show the extremizers occur when $|z_j|=\rho$ and the $z_j$ form a regular $n$-gon on the circle of radius $\rho$. A key ingredient is an additive inequality involving Blaschke products and Cauchy-type integrals, which also yields a precise equality condition characterizing the extremizers as vertices of a regular polygon. The result sharpens prior Cassels bounds and completes the conjecture with a clear extremal structure, highlighting the role of regular polygons in achieving equality.

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.

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

TL;DR

The paper resolves Alexander's conjecture by proving that Cassels' inequality holds for all when . The authors reduce the problem to an auxiliary additive inequality on the unit disk and employ a monotonicity argument to show the extremizers occur when and the form a regular -gon on the circle of radius . A key ingredient is an additive inequality involving Blaschke products and Cauchy-type integrals, which also yields a precise equality condition characterizing the extremizers as vertices of a regular polygon. The result sharpens prior Cassels bounds and completes the conjecture with a clear extremal structure, highlighting the role of regular polygons in achieving equality.

Abstract

Let be complex numbers with , where . Cassels proved that, under an additional restriction on , the inequality holds. In a subsequent note, Alexander conjectured that this inequality is in fact valid without any restriction on . In this paper, we confirm Alexander's conjecture.
Paper Structure (5 sections, 4 theorems, 37 equations)

This paper contains 5 sections, 4 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. An auxiliary inequality
  4. Proof of the main theorem
  5. Proof of Theorem \ref{['thm:additive']}

Key Result

Theorem 1

Let $\rho>1$ and let $z_1,\dots,z_n$ be complex numbers in the disc $\{z:|z|\le \rho\}$. Suppose Then with equality if and only if the $z_j$ are the vertices of a regular $n$-gon inscribed in the unit circle.

Theorems & Definitions (6)

  • Theorem 1: Cassels
  • Theorem 2
  • Theorem 3
  • Corollary 4
  • proof : Proof of Theorem \ref{['thm:main']}
  • proof