An Inequality for Coefficients of the Real-rooted Polynomials

Abstract

In this paper, we prove that if $f(x)={\sum }_{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)a_k{x}^k$ is a polynomial with real zeros only, then the sequence $\{a_k{\}}_{k=0}^n$ satisfies the following inequalities $a_{k+1}^2(1-\sqrt{1-c_k}{)}^2/a_k^2\le (a_{k+1}^2-a_k{a}_{k+2})/(a_k^2-a_{k-1}{a}_{k+1})\le a_{k+1}^2(1+\sqrt{1-c_k}{)}^2/a_k^2$, where $c_k=a_k{a}_{k+2}/a_{k+1}^2$. This inequality 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 $n\ge 55$. We also find that for a positive and log-concave sequence $\{a_k{\}}_{k\ge 0}$, the inequality $a_{k+2}/a_k\le (a_{k+1}^2-a_k{a}_{k+2})/(a_k^2-a_{k-1}{a}_{k+1})\le a_{k+1}/a_{k-1}$ is the sufficient condition for both the $2$-log-concavity and the higher order Tur{á}n inequalities of $\{a_k{\}}_{k\ge 0}$. It is easy to verify that if $a_k^2\ge r{a}_{k+1}{a}_{k-1}$, where $r\ge 2$, then the sequence $\{a_k{\}}_{k\ge 0}$ satisfies this inequality.