A remark about positive polynomials

TL;DR

This paper proves a theorem providing conditions under which a polynomial with positive coefficients is positive everywhere, establishing the optimality of a specific constant and presenting related corollaries.

Contribution

The paper introduces a sharp inequality condition ensuring polynomial positivity and demonstrates the constant's optimality, extending understanding of positive polynomials.

Findings

The constant os^2(rac{\u03c0}{n+2}) is proven to be optimal.

Theorem provides a sufficient condition for polynomial positivity.

Corollaries extend the theorem's applicability.

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}{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.