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 $\xi$-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 $n\geq 55$. 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{ \'a}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.
Categories: math.CO