Find a simpler proof of the coefficientwise nonnegativity inequality

Develop a simpler or more direct method for establishing the coefficientwise nonnegativity inequality in Theorem 4.3, which asserts that for n≥0, i≤⌊n/2⌋−1, and every r≥0, the sum of the products of binomial coefficients on the left-hand side of equation (4.3) is at least the corresponding sum on the right-hand side.

Background

The paper proves the required coefficientwise nonnegativity by interpreting both sides of the inequality as weighted counts of north-east lattice paths and then comparing the relevant path intersections through an injection and an involution. The authors emphasize that this argument involves a complicated family of lattice paths and does not yield a simple closed-form enumeration.

Although the inequality itself is established in the paper, the authors explicitly leave unresolved whether a simpler or more direct proof exists. This is a methodological open problem concerning the proof of the binomial-coefficient inequality underlying the preservation of log-concavity for gamma-polynomials.

References

We do not know of a simpler or more direct way of establishing this inequality, and our proof shows that the exact enumeration would not have a simple expression.

— Preservation of log-concavity on gamma polynomials  (2502.08948 - Ferroni et al., 13 Feb 2025) in Section 4, subsection “The proof of Theorem 4.2 via lattice paths” (proof of Theorem 4.3)