Given f(z) is analytic in Domain D, is Arg|f(z)| harmonic?

Question

Given f(z) is analytic in Domain D, is Arg|f(z)| harmonic? If yes, in which domain?

Answer 1

Assuming you mean what you actually write, then yes, obviously: $|f(z)|$ is real and non-negative, so $\operatorname{Arg} |f(z)| = 0$. (Depending slightly on what you mean by $\operatorname{Arg} 0$.)

Answer 2

If you mean $ arg(f(z)) $, not necessarily the principal argument and if its continuous, then its definitely harmonic. But it may not always exist. You will need each connected component of $ D $ to be simply connected and $ f(z)\ne 0 \forall z\in D $ for the existence of the argument function.