Upper bound for the central two-step Laplace–Pólya ratio J_n(2)/J_n(0)

Prove that for every integer n ≥ 4, the Laplace–Pólya integral satisfies J_n(2)/J_n(0) ≤ n (n^2 − 2) / (n + 2)^3, where J_n(r) = (1/π) ∫_{−∞}^{∞} sinc^n(t) · cos(r t) dt with sinc(x) = sin(x)/x for x ≠ 0 and sinc(0) = 1.

Background

The paper studies the Laplace–Pólya integral J_n(r) and provides lower and upper bounds for two-step ratios, including central values related to volumes of diagonal sections of the n-dimensional cube and to Eulerian numbers. Corollary 1.2 gives bounds on J_n(2)/J_n(0), and the authors propose a stronger upper bound as a conjecture.

For even n, the proposed inequality follows from results of Lesieur and Nicolas via analytic methods, but the authors aim for a combinatorial proof. Establishing this bound uniformly for all n ≥ 4 would sharpen known estimates and align combinatorial techniques with analytic results.