CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-28

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The logarithm of the modulus of a holomorphic function is plurisubharmonic

Statement

Let $Ω \subseteq C^{m}$ be a domain and let $f$ be holomorphic on $Ω$ , not identically zero on any connected component. Define

$u (z) = \log ∣ f (z) ∣,$

with the convention $u (z) = - \infty$ at the zeros of $f$ . Then $u$ is plurisubharmonic on $Ω$ .

Facts & Assumptions

Given: A holomorphic function $f$ on a domain $Ω \subseteq C^{m}$ , not identically zero on any connected component.

[L1]

Plurisubharmonicity is tested on affine complex lines (Plurisubharmonic functions).

[L2]

For a one-variable holomorphic function, the logarithm of the modulus is subharmonic with the value $- \infty$ at its zeros (The logarithm of the modulus of a holomorphic function is subharmonic).

Proof

technique · direct

1.1L2given

Fix an affine complex line in $Ω$ . The restriction of $f$ to that line is a one-variable holomorphic function, and by the componentwise hypothesis it is not identically zero on the connected component under consideration. Therefore [L2] makes the restriction of $u = \log ∣ f ∣$ subharmonic or identically $- \infty$ there.

2.1L1step 1.1∎

The function $u$ is upper semicontinuous because it is a logarithm of a continuous modulus away from the zero set and has value $- \infty$ on the zero set. Step 1.1 is exactly the line test from [L1], so $u$ is plurisubharmonic on $Ω$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Jiří Lebl, Tasty Bits of Several Complex Variables, §2.4 (standard reference, not scraped)
Harold P. Boas, Lecture Notes on Several Complex Variables, §3.2.4 (standard reference, not scraped)