Integral mean estimates for the polar derivative of a polynomial

TL;DR

This paper refines and generalizes integral mean estimates for the polar derivative of polynomials with zeros confined within a disk, extending previous results to broader classes of polynomials.

Contribution

It provides a refined and generalized inequality for the polar derivative of polynomials with zeros in a disk, extending prior bounds to more general polynomial classes.

Findings

Extended inequalities for the polar derivative of polynomials

Generalized results to polynomials with specific zero distributions

Improved bounds for integral mean estimates

Abstract

Let $ P(z) $ be a polynomial of degree $ n $ having all zeros in $|z|\leq k$ where $k\leq 1,$ then it was proved by Dewan \textit{et al} that for every real or complex number $\alpha$ with $|\alpha|\geq k$ and each $r\geq 0$ $$ n(|\alpha|-k)\left\{\int\limits_{0}^{2\pi}\left|P\left(e^{i\theta}\right)\right|^r d\theta\right\}^{\frac{1}{r}}\leq\left\{\int\limits_{0}^{2\pi}\left|1+ke^{i\theta}\right|^r d\theta\right\}^{\frac{1}{r}}\underset{|z|=1}{Max}|D_\alpha P(z)|. $$ \indent In this paper, we shall present a refinement and generalization of above result and also extend it to the class of polynomials $P(z)=a_nz^n+\sum_{\nu=\mu}^{n}a_{n-\nu}z^{n-\nu},$ $1\leq\mu\leq n,$ having all its zeros in $|z|\leq k$ where $k\leq 1$ and thereby obtain certain generalizations of above and many other known results.