A Dittert-like inequality for homogeneous stable polynomials

We prove a coefficient inequality for every homogeneous real stable polynomial P(s, x1, …, xn) of degree n ≥ 1 with nonnegative coefficients. If P(1, 0, …, 0) = 1, ri = ∂xiP(1, 0, …, 0), and ∑i ri = n, the following inequality holds, where brackets denote coefficient extraction: [x1⋯xn] P(0, x1, …, xn) + 2 − ∏i ri − P(0, 1, …, 1) ≥ n!/nn. Equality holds exactly for P = (s + ∑i xi/n)n. The normalizations can be removed by rescaling P and the xi. We give applications to permanents, mixed discriminants, subpermanents, and stable combinatorial measures. In particular, the theorem gives an independent proof of Dittert’s conjecture for nonnegative matrices in every dimension.