Sharp distortion theorems for rational functions without poles or zeros in simply connected domains

We prove distortion theorems for rational functions without poles or zeros in simply connected domains. The distance between the values u(z 1) and u(z 2) assumed by such a rational function is limited by its degree and by the hyperbolic distance between z 1 and z 2. For discs and halfplanes, the case of equality in these estimates is fully explored. These results have appUcations in control theory which are illustrated by examples.