Subharmonicity of the logarithmic modulus

Statement

Let $f$ be a holomorphic function on a planar domain $U$ , not identically zero. Extending $\log|f|$ by $-\infty$ at zeros, the function $\log|f|$ is subharmonic. Moreover, in the sense of distributions,

\Delta\log|f|=2\pi\sum_{a\in U}\operatorname{ord}_a(f)\,\delta_a.

Away from the zeros

Where $f\ne0$ , a local holomorphic logarithm exists and $\log|f|$ is the real part of that logarithm, hence harmonic. All positive Laplacian mass is concentrated at the zeros.

Multiplicity

If $f(z)=(z-a)^m g(z)$ with $g(a)\ne0$ , then the contribution at $a$ is $2\pi m\delta_a$ . Thus the Riesz measure of $\log|f|$ is exactly the zero-counting measure with multiplicity.

References

Thomas Ransford, Potential Theory in the Complex Plane, Cambridge University Press, 1995. DOI record.