On an Inequality of Paul Turan Concerning Polynomials-II

Abstract

Let $P(z)$ be a polynomial of degree $n$ and for any complex number $\alpha$, let $D_{\alpha}P(z)=nP(z)+(\alpha-z)P'(z)$ denote the polar derivative of the polynomial $P(z)$ with respect to $\alpha$. In this paper, we obtain inequalities for the polar derivative of a polynomial having all zeros inside a circle. Our results shall generalize and sharpen some well-known results of Turan, Govil, Dewan et al. and others.