On a property of rational functions. II

Abstract

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