Volume: 40 • Number: 4 • Pages: 299-316

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.