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This and This problems discuses the characterization of Analytic functions which maps unit circle on to itself.

I would like to know the characterization of functions which map unit circle into itself with finitely many poles inside the open unit disk.
Is there any such characterization?

Arctic Char
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  • See also http://math.stackexchange.com/questions/1240087/meromorphic-functions-on-the-unit-disk. – lhf Dec 21 '15 at 10:36
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    How could a function with a pole on the circle map the circle to itself? – zhw. Dec 21 '15 at 22:11

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