Inequalities for the rational functions with no Poles on the unit circle

Ahanger, Uzma Mubeen; Shah, Wali Mohammad; Shah, Lubna Wali

doi:10.22075/ijnaa.2022.25770.3124

Inequalities for the rational functions with no Poles on the unit circle

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

10.22075/ijnaa.2022.25770.3124

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