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

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 ۰ in |z|<۱, then     \begin{align*}     \sup_{|z|=۱}|r^{'}(z)|\leq \frac{|\mathcal{B}^{'}(z)|}{۲}\sup_{|z|=۱}|r(z)|     \end{align*}     and in case r(z)=۰ in |z|\leq ۱, then     \begin{align*}     \sup_{|z|=۱}|r^{'}(z)|\geq \frac{|\mathcal{B}^{'}(z)|}{۲}\sup_{|z|=۱}|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.

کلیدواژه ها:

Inequalities ، polynomials ، Rational functions ، Off the unit circle ، poles ، Zeros