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Let $f$ be a function which is continuous on the closed unit disc and analytic on the open disc. Assume that $|f(z)|=1$ whenever $|z|=1$. Show that the function $f$ can be extended meromorphically to the whole complex plane with at most a finite number of poles.

I think it feasible to construct such a meromorphic function but how, and is it possible to avoid such constructions?

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marked as duplicate by José Carlos Santos complex-analysis Jan 2 at 12:00

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  • $\begingroup$ We can show that $f$ is a finite Blaschke product. It can be extended meromorphically with at most finite number of poles. $\endgroup$ – Song Jan 2 at 11:40