Majorization of the critical points of a polynomial by its zeros
Let \(z_{1},\ldots,z_n\) be the zeros of a polynomial \(f(z)\) and let \(\zeta_{1},\ldots,\zeta_n\) be those of \(zf'(z)\). Suppose that for both polynomials the zeros are labelled in order of non-increasing modulus. The author shows that \[ \sum\limits^k_{\nu=1}| \zeta_\nu| \leq\sum\limits^k_{\nu=1}| z_\nu| ,\qquad k=1,\ldots,n, \] which means that the moduli of the zeros of \(f(z)\) weakly majorize those of \(zf'(z)\). This refines the Gauss-Lucas Theorem, which states that the convex hull of the zeros of a polynomial \(f\) contains all the critical points of \(f\). Moreover, this weak majorization is preserved if one replaces \(| \zeta_\nu| \) by \(\psi(| \zeta_\nu| )\) and \(| z_\nu| \) by \(\psi(| z_\nu| )\) for \(\nu=1,\ldots,n,\) where \(\psi \circ \exp \) is any non-decreasing convex function on \(\mathbb{R}\). Actually, more general results are established 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.
scientific article; zbMATH DE number 2112928