Generalizations of Bernstein and Turán-Type Inequalities for the Polar Derivative of a Complex Polynomial

Abstract

The goal of this paper is to extend to the polar derivatives some previous inequalities between the $$L^\gamma $$-norms of the derivative and of the polynomial itself, in case when the zeros are outside or inside some closed disk. Apart from this, our results also derive polar derivative analogues of some classical Bernstein and Turan-type inequalities that relate the sup-norms of a polynomial to that of its derivative on the unit circle.