Inequalities for complex rational functions

Abstract

UDC 517.5

For the rational function $r(z)=p(z)/H(z)$ having all its zeros in $|z|\leq 1,$ it is known that
\begin{equation*}
|r'(z)|\geq\dfrac{1}{2}|B'(z)||r(z)|\quad \text{for}\quad |z|=1,
\end{equation*}
where $H(z)=\prod_{j=1}^n(z - c_j),$ $|c_j|>1,$ $n$ is a positive integer, $B(z)=H^*(z)/H(z),$ and $H^*(z)=z^n\overline{H(1/\overline{z})}.$
In this paper, we improve the above mentioned inequality for the rational function $r(z)$ with all zeros in $|z|\leq 1$ and a zero of order $s$ at the origin.
Our main results refine and generalize some known rational inequalities.