Integral mean estimates for the polar derivative of a polynomial

Abstract

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

8 pages