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

Document Type : Research Paper


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


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
\ \ 0<q<\infty,\\
 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 $q>0$,
\left\|p'\right\|_{q}\leq \frac{n}{\|k+z\|_q}\|p\|_{q}.
In this paper, we shall present an interesting generalization and refinement of this result which include some previous results.