On the zeros and critical points of polynomials with nonnegative coefficients: a nonconvex analogue of the Gauss-Lucas theorem
The Gauss-Lucas theorem claims that the convex hull of the roots of a polynomial contains all the roots of its derivative. The paper extends this result for polynomials with real nonnegative coefficients. If \(S(\varphi)\) is the set af all those complex numbers \(z\), for which \(|\arg (z) |\geq \varphi\), it is shown that whenever a polynomial with nonnegative coefficients has all its roots in \(S(\varphi)\), then all the roots of its derivative lie in the same set.
scientific article; zbMATH DE number 6798752