Annales Academię Scientiarum Fennicę
Mathematica
Volumen 37, 2012, 277-284
Monash University, Department of Mathematical Sciences
Victoria 3800, Australia; gmarkowsky 'at' gmail.com
Abstract. 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 Carathéodory metrics are also presented.
2010 Mathematics Subject Classification: Primary 30E99.
Key words: Complex variables, rational functions, generalized Kobayashi metric, generalized Caratheodory metric.
Reference to this article: G. Markowsky: A rigidity theorem for special families of rational functions. Ann. Acad. Sci. Fenn. Math. 37 (2012), 277-284.
doi:10.5186/aasfm.2012.3717
Copyright © 2012 by Academia Scientiarum Fennica