Annales Academię Scientiarum Fennicę

Mathematica

Volumen 37, 2012, 277-284

# A RIGIDITY THEOREM FOR SPECIAL FAMILIES OF RATIONAL FUNCTIONS

## Greg Markowsky

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.

Full document as PDF file

doi:10.5186/aasfm.2012.3717

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