The Caratheodory and Kobayashi metrics and applications in complex   analysis
  The Caratheodory and Kobayashi metrics have proved to be important tools in the function theory of several complex variables. But they are less familiar in the context of one complex variable. Our purpose here is to gather in one place the basic ideas about these important invariant metrics for domains in the plane and to provide some illuminating examples and applications. 
Steven G. Krantz