Give an example of an analytic function

Question

I am trying to give an example for the following statement:

"Does there exist an analytic function $f=u+iv$ in $D=\{z: |z|<1\}$ such that $|f(z)|=x$ for $z=x+iy \in D$? Prove your response."

I'm unsure of what the question is asking, and I can't seem to find any example that makes this work. Can anyone help me out?

Answer 1

To make the question less trivial, it should be rephrased as

Does there exist holomorphic function $f:\mathbb D\to\mathbb C$ such that $|f(z)|=|\Re z|$?

Suppose such function exists.

Let $z=ia$ where $-1<a<1$. Then $|f(z)|=\Re ia =0$, hence $f=0$ on the segment of imaginary axis enclosed in the unit circle. By identity theorem, $f=0$ on $\mathbb D$.

However, when $z=\frac12$, $|f|=0\ne \Re z=\frac12$. Therefore such function does not exist.