 Laplacian identity for the squared modulus of a holomorphic function

Write a holomorphic function as $f=u+iv$. The Cauchy-Riemann equations make $u,v$ harmonic functions and give $u_x=v_y$, $u_y=-v_x$. The product rule therefore gives $\Delta (u^2+v^2)=2(|\nabla u{|}^2+|\nabla v{|}^2)=4|f^{\prime }{|}^2$. Thus its squared modulus is subharmonic and is harmonic only where $f^{\prime }$ vanishes.