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Báo cáo toán học: "Ratio Monotonicity of Polynomials Derived from Nondecreasing Sequences"

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@nankai.edu.cn 2yang@nankai.edu.cn 3zhoulf@cfc.nankai.edu.cn 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 _ 1.1 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 1.2 and al a. - - 2 am-i a m -1 _ am- m 1. 1.3 Given a polynomial P x a0 a1x amxm with positive .