On the Zeros and Critical Points of Polynomials with Nonnegative Coefficients: A Nonconvex Analogue of the Gauss–Lucas Theorem

Abstract

In this work, we present a nonconvex analogue of the classical Gauss–Lucas theorem stating that the critical points of a polynomial with complex coefficients are in the convex hull of its zeros. We show that if the polynomial p(z) of degree n has nonnegative coefficients and zeros in the sector $\{z \in \mathcal C: |\arg (z)| \ge \varphi \}$, for some $\varphi \in [0,\pi ]$, then the critical points of p(z) are also in that sector. Clearly, when $\varphi \in [\pi /2,\pi ]$, our result follows from the classical Gauss–Lucas theorem. But when $\varphi \in [0,\pi /2)$, we obtain a nonconvex analogue.