Arguments of zeros of highly log concave polynomials (Q1951349)

The paper is concerned with conditions on polynomials guaranteeing that all roots lie in the sectors of the form \(\left\{ z\in {\mathbb C} : \left| \arg z\right| >\theta \right\} \) for all \(\theta \) with \(\pi >\theta \geq {\pi \over 2}\). Given a real polynomial \(p=\sum\limits_{i=0}^{n}c_{i}x^{i}\) with non-negative coefficients and \(n\geq 6\), put \(\beta (p):=\inf \left\{ {c_{i}^{2} \over c_{i+1}c_{i-1}} : i=1,\dots,n-1\right\}\) (so \(\beta (p)\geq 1\) entails that \(p\) is log concave). In particular, the author proves that if \(\beta (p)>1.45\dots\), then all roots of \(p\) are in the left half plane and, moreover, there is a function \(\beta_{0}(\theta)\) (for \({\pi \over 2} \leq \theta \leq \pi \)) such that if \(\beta \left( p\right) \geq \beta _{0}\left( \theta \right) \), then all zeros of \(p\) have arguments in the sector \(\left| \arg z\right| \geq \theta \) with the smallest possible \(\theta \). He determines this function and its inverse. This extends an earlier result by J. I. Hutchinson and D. C. Kurtz saying that if \(\beta \geq 4\), then all roots are real.