Document Type : Research Paper

Authors

• Uzma Mubeen Ahanger
• Wali Mohammad Shah
• Lubna Wali Shah

Department of Mathematics, Central University of Kashmir, Ganderbal-191201, Jammu and Kashmir, India

Abstract

Let ${\mathcal{R}}_n$ be the set of rational functions with prescribed poles. It is known that if $r\in {\mathcal{R}}_n,$ such that $r(z)\ne 0$ in $|z|<1,$ then
     $$\begin{array}{r}\underset{|z|=1}{sup}|r^{{}^{\prime }}(z)|\le \frac{|{\mathcal{B}}^{{}^{\prime }}(z)|}{2}\underset{|z|=1}{sup}|r(z)|\end{array}$$
     and in case $r(z)=0$ in $|z|\le 1,$ then
     $$\begin{array}{r}\underset{|z|=1}{sup}|r^{{}^{\prime }}(z)|\ge \frac{|{\mathcal{B}}^{{}^{\prime }}(z)|}{2}\underset{|z|=1}{sup}|r(z)|,\end{array}$$
where $\mathcal{B}(z)$ is the Blashke product. The main aim of this paper is to relax the condition that all poles of $r(z)$ lie outside the unit circle and instead assume their location anywhere off the unit circle in the complex plane $\mathbb{C}.$ The results so obtained besides the above inequalities generalize some other well-known estimates for the derivative of rational functions $r\in {\mathcal{R}}_n$ with prescribed poles and restricted zeros.

Keywords

• Inequalities
• Polynomials
• Rational functions
• Off the unit circle
• Poles
• Zeros