An inequality for coefficients of the real-rooted polynomials

JOURNAL OF NUMBER THEORY

卷:225

Article

Guo, Jeremy J. F.¹ 

[1] Tianjin Univ, Sch Math, Tianjin 300072, Peoples R China

关键词: Real-rooted polynomials;    Higher order Turan inequalities;    2-log-concavity;    Partition function;   

DOI  :  10.1016/j.jnt.2021.02.011

来源: Elsevier

【 摘 要 】

In this paper, we prove that if f (x) = Sigma(n)(k=0) ((n)(k)) a(k)x(k) polynomial with real zeros only, then the sequence {a(k)}(k=0)(n) satisfies the following inequalities a(k+1)(2) (1 - root 1 - c(k))(2)/a(k)(2) <= (a(k+1)(a) - a(k)a(k)+2)/(a(k)(2) - a(k-1)a(k+1)) <= a(k+1)(2) (1 + root 1 - c(k))(2)/a(k)(2), where c(k) = a(k)a(k+2)/a(k+1)(2). This inequality is equivalent to the higher order Turan inequality. It holds for the coefficients of the Riemann xi-function, the ultraspherical, Laguerre and Hermite polynomials, and the partition function. Moreover, as a corollary, for the partition function p(n), we prove that p(n)2 - p(n - 1)p(n + 1) is increasing for n >= 55. We also find that for a positive and log-concave sequence {a(k)}(k >= 0), the inequality a(k+2)/a(k) <= (a(k+1)(2) - a(k)a(k+2))/(a(k)(2) - a(k-1) a(k+1)) <= a(k+1)/a(k-1) is the sufficient condition for both the 2-log concavity and the higher order Turan inequalities of {a(k)}(k >= 0). It is easy to verify that if a(2)(k) >= ra(k+1)a(k-1), where r >= 2, then the sequence {a(k)}(k >= 0) satisfies this inequality. (C) 2021 Elsevier Inc. All rights reserved.

【 授权许可】

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