Alexander’s conjecture on an inequality of Cassels

Alexander’s conjecture concerns extending a product inequality of Cassels to all $\rho>1$. Myriam Ounaïes claims that the conjecture holds in full, with equality only for a regular $n$-gon on the boundary circle.

Known results

• Cassels proved the inequality under the restriction $\cos(\pi/n)\leq\rho^2/(\rho^4-\rho^2+1)$.
• Alexander observed that this restriction could be weakened to $\cos(\pi/n)\leq2\rho^2/(\rho^4+1)$ and conjectured validity for every $\rho>1$.
• Dubickas proved a related elementary-symmetric-function conjecture in degrees $1$ through $4$, which the paper says would imply Alexander’s conjecture.

September 2026 proof claim

Ounaïes’s preprint, also published in Complex Analysis and Operator Theory, claims the unrestricted inequality

$$\prod_{j\ne k}|1-\overline{z_j}z_k|\leq\left(\frac{\rho^{2n}-1}{\rho^2-1}\right)^n$$

\nfor $|z_j|\leq\rho$, with equality exactly at the vertices of a regular $n$-gon on $|z|=\rho$. The claim is unverified from the supplied evidence.

Current status (as of September 2026): Alexander’s conjecture is claimed solved by Ounaïes, but the proof and its exact publication scope remain independently unverified.