Problem 1756 (difficulty: 5/10)

(a) Prove that for all \(\displaystyle f\in\C[z]\) one can find \(\displaystyle g \in \C[z]\) with the property that \(\displaystyle g\) has no roots inside the unit disc and \(\displaystyle |g(z)|=|f(z)|\) for \(\displaystyle |z|=1\).

(b) Prove the same for meromorphic functions on \(\displaystyle \C\). For all meromorphic \(\displaystyle f\) one can find a meromorphic \(\displaystyle g\) which has no poles or zeros inside the unit disc and which satisfies \(\displaystyle |g|=|f|\) on the unit circle.

Give me another random problem!