On the polar derivative of a polynomial

Abstract

Let $P(z)$ be a polynomial of degree $n$ having nozeros in $|z|<k$ where $k\geq 1$. Then it is known that for everyreal or complex number $\alpha$ with $|\alpha|\geq 1$,$$\max_{|z|=1}|D_\alpha P(z)|\leqn\left(\dfrac{|\alpha|+k}{1+k}\right)\max_{|z|=1}|P(z)|,$$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 $\alphaın\Bbb{C}$. In this paper, by a simplemethod, a refinement of the above inequality and other relatedresults are obtained.