A remark about positive polynomials

Details A remark about positive polynomials A remark about positive polynomials

2009-10-24

arxiv.org/abs/0910.4673v1

Mathematics

Classical Analysis and ODEs

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

Scientific paper

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.

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