论文标题
实用多项式系数的不平等
An Inequality for Coefficients of the Real-rooted Polynomials
论文作者
论文摘要
在本文中，我们证明，如果$ f（x）= \ sum_ {k = 0}^n {n \ select k} a_kx^k $仅具有真实零的多项式，那么序列$ \ {a_k \} _ {k = 0}^n $满足以下量不满意美元a_ {k+1}^2（1+ \ sqrt {1-c_k}）^2/a_k^2 $，其中$ c_k = a_ka_ {k+2}/a__ {k+1}^2 $。这种不等式适用于Riemann $ξ$ - 功能，超强，拉瓜和赫米特多项式的系数以及分区函数。此外，作为推论，对于分区功能$ p（n）$，我们证明$ p（n）^2-p（n-1）p（n+1）$在$ n \ geq 55 $中增加。我们还发现，对于积极和log-concave序列$ \ {a_k \} _ {k \ geq 0} $，不等式$ a_ {k+2}/a_k \ leq（a_ {k+1} \ leq a_ {k+1}/a_ {k-1} $是$ 2 $ -log-concavity和高级tur {á} n不平等的高级条件的足够条件。很容易验证，如果$ a_k^2 \ geq ra_ {k+1} a_ {k-1} $，其中$ r \ geq 2 $，则序列$ \ {a_k \} _ {k \ geq 0} $满足这种不平等。
In this paper, we prove that if $f(x)=\sum_{k=0}^n{n\choose k}a_kx^k$ is a polynomial with real zeros only, then the sequence $\{a_k\}_{k=0}^n$ satisfies the following inequalities $a_{k+1}^2(1-\sqrt{1-c_k})^2/a_k^2 \leq(a_{k+1}^2-a_ka_{k+2})/(a_k^2-a_{k-1}a_{k+1}) \leq a_{k+1}^2(1+\sqrt{1-c_k})^2/a_k^2$, where $c_k=a_ka_{k+2}/a_{k+1}^2$. This inequality holds for the coefficients of the Riemann $ξ$-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 $n\geq 55$. We also find that for a positive and log-concave sequence $\{a_k\}_{k\geq 0}$, the inequality $a_{k+2}/a_k\leq (a_{k+1}^2-a_ka_{k+2})/(a_k^2-a_{k-1}a_{k+1}) \leq a_{k+1}/a_{k-1}$ is the sufficient condition for both the $2$-log-concavity and the higher order Tur{á}n inequalities of $\{a_k\}_{k\geq 0}$. It is easy to verify that if $a_k^2\geq ra_{k+1}a_{k-1}$, where $r\geq 2$, then the sequence $\{a_k\}_{k\geq 0}$ satisfies this inequality.
