On Bernstein-type inequalities for polynomials involving the polar derivative

Abstract

. In this paper, we establish some upper bound estimates for the polar derivative of a polynomial not vanishing in a disk | z | < k , k (cid:2) 1 with a zero of multiplicity s , 0 (cid:3) s (cid:3) n − 1 at the origin. The obtained results enable us to derive polar derivative analogues of some well known Bernstein-type inequalities as special cases.