An Inequality for the Norm of a Polynomial Factor

Let p z be a monic polynomial of degree n with complex coef cients and let q z be its monic factor We prove an asymptotically sharp inequality of the form kqkE C n kpkE where k kE denotes the sup norm on a compact set E in the plane The best constant CE 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 Introduction Let p z be a monic polynomial of degree n with complex coe cients Suppose that p z has a monic factor q z so that p z q z r z where r z is also a monic polynomial De ne the uniform sup norm on a compact set E in the complex plane C by kfkE sup z E jf z j We study the inequalities of the following form kqkE C kpkE deg p n where the main problem is to nd the best the smallest constant CE such that is valid for any monic polynomial p z and any monic factor q z In the case E D where D fz jzj g the inequality was considered in a series of papers by Mignotte Granville and Glesser who obtained a number of improvements on the upper bound for CD D W Boyd made the nal step here by proving that kqkD kpkD Received by the editors November and in revised form November Mathematics Subject Classi cation Primary C C Secondary C A