Rudin's Real and complex analysis [17.3] asserts that if $\Omega$ is a region (connected open set in the plane), $f$ is holomorphic in $\Omega$, and $f$ is not identically $0$, then $\log |f|$ is subharmonic in $\Omega$.
However, the definition of subharmonic function [Definition 17.1] $u$ in Rudin's book includes that, if a closed ball $\bar{D}(a; r)$ is a subset of $\Omega$, then the integral $\int_{-\pi}^{\pi}u(a + re^{i\theta})$ is not $- \infty$.
However, I think the given conditions on $f$ are not sufficient to show this condition for the subharmonic function. Would anyone be able to help me?