An inequality for coefficients of the real-rooted polynomials

In this paper, we prove that if f ( x ) = ∑ k 0 n a is polynomial with real zeros only, then the sequence { } satisfies following inequalities + 1 2 − c / ≤ , where . This inequality equivalent to higher order Turán inequality. It holds for coefficients of Riemann ξ -function, ultraspherical, Laguerre and Hermite polynomials, partition function. Moreover, as corollary, function p increasing ≥ 55 We also find positive log-concave sufficient condition both 2-log-concavity easy verify r