Find Entire Functions with |f(z)|=1 for all |z|=1: Proving Vigorously

[iamqsqsqs]

Difficult with this problem.
Find all entire functions f such that |f(z)| = 1 for all z with |z| = 1.

Are there any vigorous proofs?

 

[DonAntonio]

Difficult with this problem.
Find all entire functions f such that |f(z)| = 1 for all z with |z| = 1.

Are there any vigorous proofs?

By the minimum modulus theorem, such a function must be constant...

DonAntonio

 

[micromass]

By the minimum modulus theorem, such a function must be constant...

DonAntonio

So the function f(z)=z is a constant then??

 

[DonAntonio]

So the function f(z)=z is a constant then??

Oops! I missed the "for all |z| = 1" in the OP...

DonAntonio