An inequality for the norm of a polynomial factor

Abstract

Let p ( z ) p(z) be a monic polynomial of degree n n , with complex coefficients, and let q ( z ) q(z) be its monic factor. We prove an asymptotically sharp inequality of the form ‖ q ‖ E ≤ C n ‖ p ‖ E \|q\|_{E} \le C^n \, \|p\|_E , where ‖ ⋅ ‖ E \|\cdot \|_E denotes the sup norm on a compact set E E in the plane. The best constant C E 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.