Ratio Monotonicity of Polynomials Derived from Nondecreasing Sequences

TL;DR

This paper proves that polynomials with nondecreasing nonnegative coefficients exhibit ratio monotonicity when shifted by 1, which implies their log-concavity for any shift c ≥ 1, extending to Boros-Moll polynomials.

Contribution

It establishes the ratio monotonicity of P(x+1) for such polynomials, providing a new proof for Boros-Moll polynomials without using recurrence relations.

Findings

Proves ratio monotonicity of P(x+1) for polynomials with nondecreasing coefficients.

Shows that P(x+c) is log-concave for c ≥ 1.

Extends ratio monotonicity results to Boros-Moll polynomials.

Abstract

The ratio monotonicity of a polynomial is a stronger property than log-concavity. Let P(x) be a polynomial with nonnegative and nondecreasing coefficients. We prove the ratio monotone property of P(x+1), which leads to the log-concavity of P(x+c) for any $c\geq 1$ due to Llamas and Mart\'{\i}nez-Bernal. As a consequence, we obtain the ratio monotonicity of the Boros-Moll polynomials obtained by Chen and Xia without resorting to the recurrence relations of the coefficients.