Modulus powers of holomorphic functions are subharmonic ($\Delta |f{|}^p=p^2|f{|}^{p-2}|f^{\prime }{|}^2\ge 0(f\ne 0,p>0)$)

For every holomorphic function and $p>0$, $|f{|}^p$ is subharmonic. Away from its zeros the displayed Laplacian calculation proves this. At the zeros, use the smooth approximations $(|f{|}^2+\epsilon {)}^{p/2}$, whose Laplacians are nonnegative, and let $\epsilon \downarrow 0$ in their mean inequalities. This permits Poisson integral comparison for fractional modulus powers even when a global analytic fractional power does not exist.