An inequality for the norm of a polynomial factor

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 o f the form parallel toq parallel to (E) less than or equal to C-n parallel top parallel to (E), where parallel to.parallel to (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 t he general result to the cases of a disk and a segment.