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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?)

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    A constant modulus on the closure of a domain gives that the function is constant.2012-12-05

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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.

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    Yes. (But of course then you only conclude that $f$ is constant on $D$.)2012-12-05