On a property of rational functions. II

Abstract:

It is shown that if ${r_n}(z)$ is a rational function of degree $n$ such that ${r_n}(0) = 1,{\lim _{|z| \to \infty }}|{r_n}(z)| = 0$ and all its poles lie in $|{\zeta _1}| \leqq |z| \leqq 1$ then ${\max _{|z| = \rho < |{\zeta _1}|}}|{r_n}(z)| \geqq 1/(1 - {\rho ^n})$.