An inequality for the norm of a polynomial factor

Abstract

Let $p(z)$ be a monic polynomial of degree $n$, with complex coefficients, and let $q(z)$ be its monic factor. We prove an asymptotically sharp inequality of the form $\Vert q{\Vert }_E\le C^n\Vert p{\Vert }_E$, where $\Vert \cdot {\Vert }_E$ denotes the sup norm on a compact set $E$ in the plane. The best constant $C_E$ in this inequality is found by potential theoretic methods. We also consider applications of the general result to the cases of a disk and a segment.

9 pages, 2 figures, AMS-LaTeX, to appear in Proc. Amer. Math. Soc