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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?

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2 Answers 2

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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 (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.

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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.

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