Let $P(z)$ be a polynomial of degree $n$ having no zero in $|z|<k$ where $k\geq 1,$ then for every real or complex number $\alpha$ with $|\alpha|\geq 1$ it is known \begin{ equation*} \underset{ |z|=1}{ \max}|D_\alpha P(z)|\leq n\left(\dfrac{ |\alpha|+k}{ 1+k}\right)\underset{ |z|=1}{ \max}|P(z)|, \end{ equation*} where $D_\alpha P(z)=nP(z)+(\alpha-z)P^{ \prime}(z)$ denote the polar derivative of the polynomial $P(z)$ of degree $n$ with respect to a point $\alpha\in\mathbb{ C}.$ In this paper, by a simple method, a refinement of above inequality and other related results are obtained.
Categories: math.CV