On the polar derivative of a polynomial

Abstract

Let $P(z)$ be a polynomial of degree $n$ having nozeros in $|z|<k$ where $k\ge 1$. Then it is known that for everyreal or complex number $\alpha$ with $|\alpha |\ge 1$,$$\underset{|z|=1}{max}|D_{\alpha }P(z)|\text{leqn}\left({\displaystyle \frac{|\alpha |+k}{1+k}}\right)\underset{|z|=1}{max}|P(z)|,$$where $D_{\alpha }P(z)=nP(z)+(\alpha -z)P^{\prime }(z)$ denotes thepolar derivative of the polynomial $P(z)$ of degree $n$ withrespect to a point $\alpha ın\mathbb{C}$. In this paper, by a simplemethod, a refinement of the above inequality and other relatedresults are obtained.