Analogue of Gauss-Lucas theorem for non convex set on the complex plane

Abstract

Let $S(\phi)= \{z:\;|\arg(z)|\geq \phi\}$ be a sector on the complex plane $\CC$. If $\phi\geq \pi/2$, then $S(\phi)$ is a convex set and, according to the Gauss-Lucas theorem, if a polynomial $p(z)$ has all its zeros on $S(\phi)$, then the same is true for the zeros of all its derivatives. In this paper is proved that if the polynomial $p(z)$ is with real and non negative coefficients, then the same is true also for $\phi < \pi/2$, when the sector is not a convex set on the complex plane.