A Note On Generalized $L_p$ Inequalities for the polar derivative of a polynomial

TL;DR

This paper establishes new $L_p$ inequalities for the polar derivative of polynomials with zeros in a disk, generalizing and refining previous results in polynomial inequalities.

Contribution

It introduces generalized $L_p$ inequalities for the polar derivative of polynomials, extending known bounds to broader conditions.

Findings

Refined bounds for polar derivatives with zeros in a disk

Generalized inequalities applicable to a wider class of polynomials

Improved understanding of polynomial derivative behavior in complex analysis

Abstract

Let \( P(z) \) be a polynomial of degree \( n \) and $\alpha \in \mathbb{C}$. The polar derivative of \( P(z) \), denoted by \( D_\alpha P(z) \) and is defined by $D_\alpha P(z): = nP(z) + (\alpha -z)P'(z)$. The polar derivative \( D_\alpha P(z) \) is a polynomial of degree at most \( n - 1 \) and it generalizes the ordinary derivative \( P'(z) \). In this paper, we establish some \( L_p \) inequalities for the polar derivative of a polynomial with all its zeros located within a prescribed disk. Our results refine and generalize previously known findings.