On the polar derivative of a polynomial

TL;DR

This paper refines existing inequalities involving the polar derivative of polynomials, providing tighter bounds and related results using a simple methodological approach.

Contribution

It introduces a simplified method to improve bounds on the polar derivative of polynomials with no zeros inside a certain disk.

Findings

Refined inequality for the polar derivative of polynomials

Extended results related to polynomial bounds

Simplified proof technique for existing inequalities

Abstract

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.