The discussion revolves around finding entire functions \( f \) such that \( |f(z)| = 1 \) for all \( |z| = 1 \). Participants are seeking rigorous proofs related to this problem, exploring the implications of the minimum modulus theorem.
Participants do not reach a consensus, as there is confusion about the implications of the minimum modulus theorem and the nature of the function \( f(z) \).
There is a potential misunderstanding regarding the application of the minimum modulus theorem and its implications for the nature of entire functions under the specified conditions.
iamqsqsqs said: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 said:By the minimum modulus theorem, such a function must be constant...
DonAntonio
micromass said:So the function f(z)=z is a constant then??