F depends only on |z| then f must be constant?

[zardiac]

Homework Statement

Hi!
I am stuck on an exercise in the complex analysis course. The problem is:
"Show that the only analytic functions f that depends on only |z| must be the constant functions."

Homework Equations

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The Attempt at a Solution

I am not sure if I undertand the question, is it f(z)=|z|, or maybe f(z)=g(|z|)?

 

[Office_Shredder]

It's the lattet. Note this is equivalent to saying that f is constant on each circle of radius r

 

[zardiac]

Thanks a lot!
I'll try to solve this now, I'll be back if I get stuck. ;)