Inequalities concerning polynomial and its derivative

Abstract

In this paper, some sharp inequalities for ordinary derivative $P'(z)$ and polar derivative $D_\alpha P(z)=nP(z)+(\alpha-z)P'(z)$ are obtained by including some of the coefficients and modulus of each individual zero of a polynomial $P(z)$ of degree $n$ not vanishing in the region $|z|>k$, $k\geq 1$. Our results also improve the bounds of Tur\'an's and Aziz's inequalities.