A real-valued function $u\geq 0$ such that $\log u$ is subharmonic is called logarithmically subharmonic. One easy theorem is that the sum of logarithmically subharmonic functions is also logarithmically subharmonic.
(This can be proved by computation of the Laplacian). It follows that
logarithmically subharmonic functions form a convex cone. The pointwise maximum of several logarithmically subharmonic functions is logarithmically subharmonic. Moreover, the class is closed with respect to certain limits. Evidently, in dimension 2, this class contains all functions
of the form $|f|$, where $f$ is analytic, and one can show that this is the
minimal class containing all $|f|^\alpha, \alpha>0$ where
$f$ is analytic, and closed with respect to $L^1_{\mathrm{loc}}$ limits.