A ratio of symmetric polynomials

Let H_i(x) = h_ℓ(1, …, 1, x, …, x) be the complete homogeneous symmetric polynomial of degree ℓ in n variables, with i of them equal to x and the rest equal to 1. Rachid Ait-Haddou conjectured on MathOverflow (question 467261, 2024) that Ψ_{i,n} = H_i² / (H_{i+1} H_{i-1}) is strictly increasing on (1, ∞) for 2 ≤ i ≤ n - 2. The note proves it for every ℓ ≥ 1 and every 1 ≤ i ≤ n - 1. Treating i as a real variable t, the logarithmic derivative of Ψ_{i,n} is a positive multiple of a second difference in t of P_{ℓ-1}/P_ℓ, where the P_ℓ form a Meixner family; that ratio is convex because the zeros interlace.

Preprint v1, 9 October 2026, not peer reviewed. Theorem 1 and the lemmas of its proof are also formally verified in Lean 4 (lean/, standard axioms only). The exposition has not yet been independently reviewed. The author used an AI tool (Claude, Anthropic) in this work, as described in the paper’s acknowledgements, and is responsible for its content.