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?)