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 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.