A remark about positive polynomials

Abstract

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 $\frac{a_{2k+1}^2}{a_{2k}{a}_{2k+2}}<\frac{1}{co{s}^2(\frac{\pi }{n+2})}$ hold for all $k=0,1,...,n-1,$ then $P(x)>0$ for every $x\in \mathbb{R}$ .} We show that the constant $\frac{1}{co{s}^2(\frac{\pi }{n+2})}$ in this theorem could not be increased. We also present some corollaries of this theorem.

Submitted to the journal "Mathematical Inequalities and Applications" on September 29, 2008