Inequalities for the polar derivative of a complex polynomial

Let P(z):= ?n v=0 avzv be a univariate complex coefficient polynomial of degree n. Then as generalization well-known classical inequality Tur?n [25], it was shown by Govil [7] that if P(z) has all its zeros in |z| ? k, k 1, then max |z|=1 |P?(z)| n/1 + kn |P(z)|, whereas, , 0 < again [6] who gave an extension the Erd?s-Lax [13], obtaining provided and |Q?(z)| attain maximum at same point on = where Q(z) ?znP (1/z). In this paper, we obtain several generalizations refinements above inequalities related results while taking into account placement extremal coefficients underlying polynomial. Moreover, some concrete numerical examples are presented, showing situations, bounds obtained our can considerably sharper than ones previously known.