L$^q$ inequalities for the ${s^{th}}$ derivative of a polynomial

Document Type: Research Paper

Author

Department of Mathematics, Shahrood University of Technology, Shahrood, Iran

Abstract

Let $f(z)$ be an analytic function on the unit disk $\{z\in\mathbb{C},\ |z|\leq 1\}$, for each $q>0$, the $\|f\|_{q}$ is defined as follows
\begin{align*}
\begin{split}
&\left\|f\right\|_q:=\left\{\frac{1}{2\pi}\int_0^{2\pi}\left|f(e^{i\theta})\right|^qd\theta\right\}^{1/q},\
\ \ 0<q<\infty,\\
&\left\|f\right\|_{\infty}:=\max_{|z|=1}\left|f(z)\right|.
\end{split}
\end{align*}
 Govil and Rahman [{\it Functions of exponential type not vanishing in a half-plane and related polynomials}, { Trans. Amer. Math. Soc.} {137} (1969) 501--517] proved that if $p(z)$ is a polynomial of degree $n$, which does not vanish in $|z|<k$, where $k\geq 1$, then for each $q>0$,
\begin{align*}
\left\|p'\right\|_{q}\leq \frac{n}{\|k+z\|_q}\|p\|_{q}.
\end{align*}
In this paper, we shall present an interesting generalization and refinement of this result which include some previous results.

Keywords