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

N. A. Rather; Danish Rashid Bhat; Tanveer Bhat

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arXiv · 2505.06539

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

N. A. Rather·Danish Rashid Bhat·Tanveer Bhat

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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.

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N. A. Rather, Danish Rashid Bhat, Tanveer Bhat. 2025-05-10. A Note On Generalized $L_p$ Inequalities for the polar derivative of a polynomial. https://arxiv.org/abs/2505.06539

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