A remark on two famous theorems concerning polynomials (Q1091498)

In the study of polynomials of a single complex variable, geometric relationships between the zeros of a polynomial and those of its derivative are sought. Two examples are provided by two classical theorems. (A) Laguerre: Let \({\mathcal C}\) be the image of the unit disk under a linear transformation and let P(z) be a polynomial of degree n. If P(z) has no zeros in \({\mathcal C}\), then for any (\(\zeta\),z)\(\in {\mathcal C}\times {\mathcal C}\), \((\zeta -z)P'(z)+nP(z)\neq 0\). (B) Gauss-Lucas: The zeros of P'(z) lie in the convex hull of the zeros of P(z). Elementary arguments establish the equivalence of (A) and (B). \{Reviewer's remark: Corresponding results are implied for homogeneous polynomials on abstract spaces.\}