Inequalities for complex rational functions - Bidkham, M.; Khojastehnezhad, E. (Ukrains’kyi Matematychnyi Zhurnal, 2021)
UDC 517.5For 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.