  For a real polynomial $p = \sum_{i=0}^{n} c_ix^i$ with no negative
coefficients and $n\geq 6$, let $\beta (p) = \inf_{i=1}^{n-1}
c_i^2/c_{i+1}c_{i-1}$ (so $\beta (p) \geq 1$ entails that $p$ is log concave).
If $\beta(p) > 1.45...$, then all roots of $p$ are in the left half plane, and
moreover, there is a function $\beta_0 (\theta)$ (for $\pi/2 \leq \theta \leq
\pi$) \st $\beta \geq \beta_0(\theta)$ entails all roots of $p$ have arguments
in the sector $| \arg z| \geq \theta$ with the smallest possible $\theta$; we
determine exactly what this function (and its inverse) is (it turns out to be
piecewise smooth, and quite tractible). This is a one-parameter extension of
Kurtz's theorem (which asserts that $\beta \geq 4$ entails all roots are real).
We also prove a version of Kurtz's theorem with real (not necessarily
nonnegative) coefficients.
