Inequalities for the polar derivative of a polynomial

Let P(z) be a polynomial of degree n having all its zeros in |z|≤1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|z|\le 1$$\end{document}, then according to Turan (Compositio Mathematica 7:89–95, 2004) max|Z|=1|P′(z)|≥n2max|Z|=1|P(z)|.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \max \limits _{|Z|=1}|P'(z)|\ge \frac{n}{2}\max \limits _{|Z|=1}|P(z)|. \end{aligned}$$\end{document}In this paper, we shall use polar derivative and establish a generalisation and an extension of this result. Our results also generalize variety of other results.