Problem. Find all meromorphic functions $f$ s.t. $|f(z)|=1$ wherever $|z|=1$.
I know this problem has been solved in this posting. However, I cannot show $f$ has only finite number of poles and zeros. Any hints?
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Sangchul Lee
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See for example https://math.stackexchange.com/q/3883269/42969 or https://math.stackexchange.com/q/783741/42969 – Martin R Apr 21 '23 at 14:06
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@MartinR Thanks!But I think these examples are just about functions in the disc.I want to know why meromorphic functions in the complex plane satisfying the condition above has finite poles and zeros. – faith in prime Apr 21 '23 at 14:26
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2A meromorphic function in the plane is in particular meromorphic in the disk, therefore those solutions can all be applied to your case. – Martin R Apr 21 '23 at 14:31
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Unless I'm mistaken, if two meromorphic functions are equal on the disk, they are also equal on $\mathbb C$. So if some set of meromorphic functions are the only ones that satisfy this condition on the disk, they are also the only ones satisfying it on $\mathbb C$. – eyeballfrog Apr 21 '23 at 14:39
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@MartinR Thank you very much! – faith in prime Apr 21 '23 at 14:55
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@eyeballfrog Oh,I see.Thank you! – faith in prime Apr 21 '23 at 14:55