Some generalizations involving the polar derivative for an inequality of Paul Turán.

Let p(z) be a polynomial of degree n and for a complex number α, let D_αp(z) np(z) + (α - z)p'(z) denote the polar derivative of the polynomial p(z) with respect to α. In this paper, we obtain inequalities for the polar derivative of a polynomial having all its zeros in /z/ ≤ K. Our results generalize and sharpen a famous inequality of Turán and some other known results in this direction. [ABSTRACT FROM AUTHOR]