Inequalities Pertaining to Rational Functions with Prescribed Poles

Abstract

Let  $\mathbf{R_{n}}$ be the space of rational functions with prescribed poles. If $t_{1},t_{2},...,t_{n}$ are the zeros of $B(z)+\lambda$ and $s_{1},s_{2},...,s_{n}$ are zeros of $B(z)-\lambda,$ where $B(z)$ is the Blaschke product and  $\lambda\in T,$ then for $z\in T$ $$|r^{\prime}(z)|\le \frac {|B^{\prime}(z)|}{2}[(\max_{1\le k\le n}|r(t_{k})|)^{2}+(\max_{1\le k\le n}|r(s_{k})|)^{2}].$$  Let $r,s\in \mathbf{R_{n}}$ and assume $s$ has all its $n$ zeros in $ D^{-}\cup T$ and $|r(z)|\le |s(z)|$ for $z\in T,$ then for any $\alpha$ with $|\alpha|\le \frac {1}{2}$ and for $z\in T$ $$ |r^{\prime}(z)+\alpha B^{\prime}(z)r(z)|\le|s^{\prime}(z)+\alpha B^{\prime}(z)s(z)|.$$ In this paper, we consider a more general class of rational functions $rof \in \mathbf{R}_{m^{\star}n}, $ defined by $(rof)(z)=r(f(z))$, where $f(z)$ is a polynomial of  degree $m^{\star}$  and prove some generalizations of the above inequalities.