Analogue of Gauss-Lucas theorem for non convex set on the complex plane
Let $S(ϕ)= \{z:\;|\arg(z)|\geq ϕ\}$ be a sector on the complex plane $\CC$. If $ϕ\geq π/2$, then $S(ϕ)$ is a convex set and, according to the Gauss-Lucas theorem, if a polynomial $p(z)$ has all its zeros on $S(ϕ)$, 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 $ϕ< π/2$, when the sector is not a convex set on the complex plane.
math.CV↗