AbstractThe ratio monotonicity of a polynomial is a stronger property than log-concavity.Let P (x) be a polynomial with nonnegative and nondecreasing coefficients. Weprove the ratio monotone property of P (x + 1), which leads to the log-concavity ofP (x + c) for any c ≥ 1 due to Llamas and Mart´ınez-Bernal. As a consequence, weobtain the ratio monotonicity of the Boros-Moll polynomials obtained by Chen andXia without resorting to the recurrence relations of the coefficients...