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

TL;DR

This work studies $L_p$-type inequalities for the polar derivative $D_{\alpha}P$ of a degree-$n$ polynomial whose zeros lie in a disk $|z|\le k$ with $k\le 1$. By leveraging a key lemma involving a reversal polynomial $Q(z)$ and a subordination argument, the authors derive an integral bound: $$n(|\alpha|-k^{\mu})\left(\int_{0}^{2\pi}\left|\frac{P(e^{i\theta})+\frac{m\beta}{k^{n-\mu}}}{|D_{\alpha}P(e^{i\theta})| - \frac{nm}{k^{n-\mu}}}\right|^p d\theta\right)^{1/p} \leq \left(\int_{0}^{2\pi}|1+k^{\mu}e^{i\theta}|^p d\theta\right)^{1/p},$$ where $m=\min_{|z|=k}|P(z)|$, $|\alpha|\ge k^{\mu}$, and $|\beta|\le 1$. The results generalize known $L_p$-type inequalities, extend to lacunary polynomials, and recover the $p\to\infty$ case as well as the $\mu=1$ specialization. The proofs combine Gauss-Lucas, subordination, and Hölder-type arguments. Overall, the paper advances the theory of polynomial inequalities for polar derivatives under zero-location constraints, with potential implications for complex approximation and analysis.

Abstract

Let \( P(z) \) be a polynomial of degree \( n \) and $α\in \mathbb{C}$. The polar derivative of \( P(z) \), denoted by \( D_αP(z) \) and is defined by $D_αP(z): = nP(z) + (α-z)P'(z)$. The polar derivative \( D_αP(z) \) 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.

Theorems & Definitions (12)

• Theorem 1
• Remark 1
• Corollary 1
• Corollary 2
• Theorem 2
• Corollary 3
• Theorem 3
• Remark 2
• Lemma 1
• proof : Proof of Theorem \ref{['th2']}