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 $r(z)\neq 0$ in $ |z|<1,$ then
     \begin{align*}
     \sup_{|z|=1}|r^{'}(z)|\leq \frac{|\mathcal{B}^{'}(z)|}{2}\sup_{|z|=1}|r(z)|
     \end{align*}
     and in case $r(z)=0$ in $|z|\leq 1,$ then
     \begin{align*}
     \sup_{|z|=1}|r^{'}(z)|\geq \frac{|\mathcal{B}^{'}(z)|}{2}\sup_{|z|=1}|r(z)|,
     \end{align*}
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