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

Abstract

Let $D_{\alpha }P(z)=nP(z)+(\alpha -z)P^{\prime }(z)$ denote the polar derivative of a polynomial $P(z)$ of degree $n$ with respect to a point $\alpha \in \mathbb{C}.$ 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.,} \textbf{352} (2009) 807-815].

In this paper, we point out an error in the proof of a result due to K. K. Dewan, Naresh Singh, Abdullah Mir, [Extensions of some polynomial inequalities to the polar derivative, \emph{J. Math. Anal. Appl.,} \textbf{352} (2009) 807-815], and give its correct proof