Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài:Ratio Monotonicity of Polynomials Derived from Nondecreasing Sequences. | Ratio Monotonicity of Polynomials Derived from Nondecreasing Sequences William Y. C. Chen1 Arthur L. B. Yang2 Elaine L. F. Zhou3 Center for Combinatorics LPMC-TJKLC Nankai University Tianjin 300071 P. R. China 1chen@ 2yang@ 3zhoulf@ Submitted July 28 2010 Accepted Nov 26 2010 Published Dec 10 2010 Mathematics Subject Classification 05A20 33F10 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 Martinez-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. Keywords log-concavity ratio monotonicity Boros-Moll polynomials. 1 Introduction This paper is concerned with the ratio monotone property of polynomials derived from nonnegative and nondecreasing sequences. A sequence ak 0 k m of positive real numbers is said to be unimodal if there exists an integer r 0 such that a0 ar-1 ar ar 1 am and it is said to be spiral if am ao am-i ai a _ where mm stands for the largest integer not exceeding mm. We say that a sequence ak 0 k m is log-concave if for any 1 k m 1 ak ak iak-i 0 THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 N37 1 or equivalently a0 a1 am-1 a1 _ a2 am It is easy to see that either log-concavity or the spiral property implies unimodality while a log-concave sequence is not necessarily spiral and vice versa. A stronger property which implies both log-concavity and the spiral property was introduced by Chen and Xia 6 and is called the ratio monotonicity. A sequence of positive real numbers ak 0 k m is said to be ratio monotone if am am-1 a0 a1 ai a - m-1 a . 1 and al a. - - 2 am-i a m -1 _ am- m 1. Given a polynomial P x a0 a1x amxm with positive .