A rigidity theorem for special families of rational functions

We study the question of whether for a given nonconstant holomorphic function f there is a pair of domains U, V such that f is the only nonconstant holomorphic function with f(U) \subseteq V. We show existence of such a pair for several classes of rational functions, namely maps of degree 1 and 2 as well as arbitrary degree Blaschke products. We give explicit constructions of U and V, where possible. Consequences for the generalized Kobayashi and Caratheodory metrics are also presented.