On an inequality concerning the polar derivative of a polynomial with restricted zeros

Let $D_{\alpha }P(z)=nP(z)+(\alpha -z)P^{\mathrm{\prime }}(z)$ denote the polar derivative of a polynomial $P(z)$ of degree $n$ with respect to a point $\alpha \in \mathbb{C}\mathrm{.}$ In this paper, we present a correct proof, independent of Laguerre’s theorem, of an inequality concerning the polar derivative of a polynomial with restricted zeros recently formulated by K. K. Dewan, Naresh Singh, Abdullah Mir, [Extensions of some polynomial inequalities to the polar derivative, \emph{J. Math. Anal. Appl.,} $\mathbf{\text{352}}$ (2009) 807-815].