Liouville's Theorem states that if a function is bounded and holomorphic on the complex plane (i.e. bounded and entire), then it is a constant function.
What if we consider the following, slightly modified scenario:
Suppose a function $f$ is holomorphic and has constant modulus on a bounded domain $D$ (e.g. a small disk).
Can we use Liouville's Theorem to somehow conclude that $f$ is a constant function? (either on $D$ or on the whole of the complex plane?)
Best Answer
I don't see how you could use Liouville's theorem to prove that, but it does follow from Cauchy-Riemann's equations.
If you assume that $f$ is entire, use Cauchy-Riemann's equation on $|f|^2 = u^2 + v^2$ to show that both $u$ and $v$ must be constant on $D$. After that it follows from the uniqueness theorem that $f$ is constant everywhere.