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Let $f \in O(D)$ for some domain $D$. Prove that if $|f(z)|$ is a constant, then $f(z) =$ const on $D$.

It seems to me it is the direct application of the following version of maximum modulus principle:

Let f be a function holomorphic on some connected open subset $D$ of the complex plane $\mathbb{C}$ and taking complex values. If $z_0$ is a point in $D$ such that

$|f(z_0)|\ge |f(z)|$ for all z in a neighborhood of $z_0$, then the function f is constant on D.

luka5z
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    The symbol $O(D)$ means ring of holomorphic functions on $D,$ I guess. Then as you have mentioned, it is a direct corollary of maximum modulus principle. – Krish Jan 18 '15 at 12:28
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    You can also derive that directly from the Cauchy-Riemann equations, without using the maximum modulus principle, compare http://math.stackexchange.com/questions/244342/if-the-absolute-value-of-an-analytic-function-f-is-a-constant-must-f-be-a-c. – Martin R Jan 18 '15 at 12:37

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