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Let $f:U\rightarrow\mathbb{C}$ be holomorphic on some open domain $U\subset\hat{\mathbb{C}}=\mathbb{C}\cup\{\infty\}$ and $f(z)\not=0$ for $z\in U$.
Is it true that $z\mapsto \log(|f(z)|)$ is harmonic on $U$ ?
I guess the answer is yes and if that is true, how can I see that without a long and nasty calculation?
Yes, this is true. If $f(z)$ is holomorphic and nonzero on $U$, then $1/f(z)$ and $f'(z)$ are also holomorphic, so $f'(z)/f(z)$ is holomorphic on $U$. Therefore its integral $\log(|f(z)|) + i\arg(f(z))$ is holomorphic on any simply connected subdomain of $U$, so in particular $\log(|f(z)|)$ is harmonic everywhere on $U$.
Locally (but not necessarily globally), $\log f(z)$ is an analytic function because $\log$ is an analytic function. The real part of $\log f(z)$ is $\log |f(z)|$, i.e. the polar representation of the complex number $w$ is $w = r e^{i\theta}$ where $r = |w|$, and $\log w = \log r + i \theta$ so $\text{Re}(\log w) = \log r = \log |w|$. The real part of an analytic function is harmonic.
