Document Type : Research Paper

Author

• Ahmad Zireh

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

Abstract

Let $f(z)$ be an analytic function on the unit disk $\{z\in \mathbb{C},|z|\le 1\}$, for each $q>0$, the $\Vert f{\Vert }_q$ is defined as follows
$$\begin{array}{r}\begin{array}{rl}& {\Vert f\Vert }_q:={\left\{\frac{1}{2\pi }{\int }_0^{2\pi }{|f(e^{i\theta })|}^{q}d\theta \right\}}^{1/q},0<q<\mathrm{\infty },\\ & {\Vert f\Vert }_{\mathrm{\infty }}:=\underset{|z|=1}{max}|f(z)|.\end{array}\end{array}$$
 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 $k\ge 1$, then for each $q>0$,
$$\begin{array}{r}{\Vert p^{\prime }\Vert }_q\le \frac{n}{\Vert k+z{\Vert }_q}\Vert p{\Vert }_q.\end{array}$$
In this paper, we shall present an interesting generalization and refinement of this result which include some previous results.

Keywords

• Derivative
• Polynomial
• $L^q$ Inequality
• Maximum modulus
• Restricted Zeros