Ratio Monotonicity of Polynomials Derived from Nondecreasing Sequences
William Y. C. Chen, Arthur L. B. Yang, and Elaine L. F. Zhou
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 ≥ 1 due to Llamas and Martínez-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.
AMS Classification: 05A20, 33F10
Keywords: log-concavity, ratio monotonicity, Boros-Moll polynomials.