Polynomials with nonnegative coefficients whose zeros have modulus one

@article{Evans1991PolynomialsWN,
title={Polynomials with nonnegative coefficients whose zeros have modulus one},
author={R. J. Evans and J. R. Greene},
journal={Siam Journal on Mathematical Analysis},
year={1991},
volume={22},
pages={1173-1182}
}

Define $p(z) = \prod _{j = 0}^{n - 1} (z - e^{i(\theta + \alpha j)} )$ for $\alpha > 0$ and $\theta \geq 0$ with ${\pi / 2} - (n - 1){\alpha / 2} \leq \theta \leq \pi - (n - 1){\alpha / 2}$. It is proved that if $0 < \alpha < {\pi / n}$, then the $2n + 1$ coefficients of $p(z)$ are all positive. It is also proved that if for some point $\theta $, all coefficients of $p(z)$ are nonnegative, then each coefficient is an increasing function of $\theta $ in a neighborhood of this point. A similarâ€¦Â CONTINUE READING