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

Let $S(\varphi )=\{z:\mathrm{\mid }\arg (z)\mathrm{\mid }\ge \varphi \}$ be a sector on the complex plane $\text{CC}$. If $\varphi \ge \pi \mathrm{/}2$, then $S(\varphi )$ is a convex set and, according to the Gauss-Lucas theorem, if a polynomial $p(z)$ has all its zeros on $S(\varphi )$, 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 $\varphi <\pi \mathrm{/}2$, when the sector is not a convex set on the complex plane.