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

Abstract

We prove sharp distortion theorems for rational functions without poles or zeros in arbitrary 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. This has some interesting consequences for locally sourceless, locally irrotational plane flows from theoretical hydrodynamics and in control theory, which is illustrated by examples.

Keywords:

• Unit interpolation
• Poincaré metric
• Hyperbolic distance
• Simultaneous stabilization
• Distortion theorem
• Rational functions
• Locally sourceless
• Locally irrotational plane flows

Keywords:

• Classification Categories: 30C99
• Secondary 93B60
• 93D99

Acknowledgement

The author wishes to thank the unknown referee(s) for many valuable comments and Prof. Vincent Blondel for discussions on the subject.