Preservation of log-concavity on gamma polynomials

Every symmetric polynomial h(x) with center of symmetry n/2 can be expressed as a linear combination in the basis xi(1 + x)n−2i. The γ- polynomial of h(x), which we denote γh(x), records the coefficients of this linear combination. Two decades ago, Br¨and´en [Electron. J. Combin. 11 (2004/06), Research Paper 9] and Gal [Discrete Comput. Geom. 34 (2005), pp. 269–284] independently showed that if γh(x) has nonpositive real roots only, then so does h(x). More recently, Br¨and´en, Ferroni, and Jochemko [Preservation of inequalities under Hadamard products, (2024) Preprint, arXiv:2408.12386] proved using Lorentzian polynomials that if γh(x) is ultra log-concave, then so is h(x), and they raised the question of whether a similar statement can be proved for the usual notion of log-concavity. The purpose of this article is to show that the answer to the question of Br¨and´en, Ferroni, and Jochemko is affirmative. One of the crucial ingredients of the proof is an inequality involving binomial numbers that we establish via a path-counting argument.