Is absolute value of an analytic function a harmonic function? It is well known that if $f(x+i y) = u(x, y) + i v(x, y)$ is an analytic function of variable $z = x + iy$ then both $u$ and $v$ are harmonic functions.
Does $|f| = \sqrt{u^2 + v^2}$ have any special properties, in particular is $|f|$ harmonic?

It is generally not harmonic. For example let $f(z)=z$.
Then $\Delta |f|(z)=|z|^{-1},$ which is certainly not zero. 
However, $\log |f|$ is harmonic if $f$ is analytic.
