It is known that the logarithm of the modulus of an *analytic* function $f: D \subset \mathbb C \rightarrow \mathbb C$ ($D$ is a domain) is subharmonic. I have two questions:


(1) Are there some weaker conditions than analyticity that ensure the same result?


(2) Is there any characterization of functions $f$ such that $\log |f|$ is subharmonic?



$\log |f|$ is subharmonic
Conditions for a log-subharmonic function $f$ on $D\in\mathbb{R}^n$ are described by Mochizuki in [A Class of Subharmonic Functions and Integral Inequalities](https://www.jstage.jst.go.jp/article/iis/10/2/10_2_153/_pdf) (2004).


The conditions are phrased in terms of an inequality for the volume average $A\_p$ of $|f|^p$ and the surface average $M$ of $|f|$, in the form $A\_p\leq M^p$ for any closed ball in $D$.  

If $n\geq 2$ the condition $A\_{1+2/n}\leq M^{1+2/n}$ is sufficient for $\log|f|$ to be subharmonic. If $n=2$ this condition is also necessary, if $n>2$ it is not.


