Majorization of the Critical Points of a Polynomial by Its Zeros

Let z1, …, zn be the zeros of a polynomial f(z) and let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\zeta _{1},\cdots,\zeta_{n}$\end{document} be those of zf′(z). Suppose that for both polynomials the zeros are labelled in order of non-increasing modulus. We show that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum_{v=1}^{k}|\zeta_{v}|\leq \sum_{v=1}^{k}|z_{v}|,\qquad k = 1\cdots,n$$\end{document}, which means that the moduli of the zeros of f(z) weakly majorize those of zf′(z). This refines the Gauss-Lucas Theorem. Moreover, this weak majorization is preserved if we replace \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$|\zeta_{v}|\ {\rm by}\ \psi (|\zeta_{v}|)\ {\rm and}\ |z_{v}|\ {\rm by}\ \psi (|z_{v}|) $\end{document} forv = 1,…, n, where ψ o exp is any non-decreasing convex function on ℝ. Actually, we establish more general results which hold for a polynomial f and a certain multiplicative composition which may be interpreted as a Hadamard product of f with a polynomial from a certain class.