Computational Methods and Function Theory 3 (2003), No. 1, 95--103
Copyright Heldermann Verlag 2003
Gerhard Schmeisser
schmeisser@mi.uni-erlangen.de, Mathematical Institute, University of Erlangen-Nuremberg, Bismarckstraße 1 1/2, 91054 Erlangen, Germany.
Let $z_1,\dots, z_n$ be the zeros of a polynomial $f(z)$ and let $\zeta_1,\dots, \zeta_n$ be those of $zf'(z)$. Suppose that for both polynomials the zeros are labelled in order of non-increasing modulus. We show that $$ \sum_{\nu=1}^k \abs{\zeta_\nu} \le \sum_{\nu=1}^k \abs{z_\nu}, \qquad k=1,\dots,n, $$ 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 $\abs{\zeta_\nu}$ by $\psi(\abs{\zeta_\nu})$ and $\abs{z_\nu}$ by $\psi(\abs{z_\nu})$ for $\nu=1,\dots,n$, where $\psi\circ\exp$ is any non-decreasing convex function on $\mathbb{R}$. 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.
Keywords: Majorization, zeros of polynomials, critical points, Gauss-Lucas Theorem, inequalities.
MSC 2000: 30C10, 30C15.
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