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

TL;DR

This paper provides a new, independent proof of an inequality related to the polar derivative of polynomials with restricted zeros, avoiding reliance on Laguerre's theorem, and clarifies a recent result in polynomial inequalities.

Contribution

It offers a correct, independent proof of a recent inequality involving the polar derivative of polynomials with restricted zeros, expanding understanding in polynomial inequality theory.

Findings

Proof of the inequality is independent of Laguerre's theorem.

Clarifies and validates a recent inequality in polynomial analysis.

Enhances theoretical understanding of polar derivatives 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].