An inequality for coefficients of the real-rooted polynomials

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In this paper, we prove that if f(x)=∑k=0n(nk)akxk is a polynomial with real zeros only, then the sequence {ak}k=0n satisfies the following inequalities ak+12(1−1−ck)2/ak2≤(ak+12−akak+2)/(ak2−ak−1ak+1)≤ak+12(1+1−ck)2/ak2, where ck=akak+2/ak+12. This inequality is equivalent to the higher order Turán inequality. It 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≥55. We also find that for a positive and log-concave sequence {ak}k≥0, the inequality ak+2/ak≤(ak+12−akak+2)/(ak2−ak−1ak+1)≤ak+1/ak−1 is the sufficient condition for both the 2-log-concavity and the higher order Turán inequalities of {ak}k≥0. It is easy to verify that if ak2≥rak+1ak−1, where r≥2, then the sequence {ak}k≥0 satisfies this inequality.