A remark about positive polynomials

The following theorem is proved. {\bf Theorem.} {\it Let $P(x) = \sum_{k=0}^{2n} a_k x^k$ be a polynomial with positive coefficients. If the inequalities $\frac{a_{2k+1}^2}{a_{2k}a_{2k+ 2}} < \frac{1}{cos^2(\frac{\pi}{n+2})}$ hold for all $k=0, 1, ..., n-1,$ then $P(x)>0$ for every $x\in\mathbb{R}$ .} We show that the constant $\frac{1}{cos^2(\frac{\pi}{n+2})}$ in this theorem could not be increased. We also present some corollaries of this theorem.