$L^r$ inequalities for the derivative of a polynomial

Abstract

Let $p(z)$ be a polynomial of degree $n$ having no zero in $|z|< k$, $k\leq 1$, then Govil [Proc. Nat. Acad. Sci., $\textbf{50}$, (1980), 50-52] proved

$\max\limits_{|z|=1}|p'(z)|\leq \dfrac{n}{1+k^{n}}\max\limits_{|z|=1}|p(z)|$,

provided $|p'(z)|$ and $|q'(z)|$ attain their maxima at the same point on the circle $|z|=1$, where

$\label{A}q(z)=z^{n}\overline{p\left(\frac{1}{\overline{z}}\right)}$.

In this paper, we not only obtain an integral mean inequality for the above inequality but also extend an improved version of it into $L^{r}$ norm.