TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-28

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Basic stability operations for plurisubharmonic functions

Statement

Let $Ω \subseteq C^{m}$ be a domain.

If $u_{1}, \dots, u_{N}$ are plurisubharmonic on $Ω$ and $α_{1}, \dots, α_{N} \geq 0$ , then $α_{1} u_{1} + \dots + α_{N} u_{N}$ is plurisubharmonic, with the zero-coefficient terms omitted.
If $u_{1}, \dots, u_{N}$ are plurisubharmonic on $Ω$ , then $u (z) = \max {u_{1} (z), \dots, u_{N} (z)}$ is plurisubharmonic.
If $u : Ω \to R$ is plurisubharmonic and $ϕ : R \to R$ is convex and nondecreasing, then $ϕ \circ u$ is plurisubharmonic on $Ω$ .

Facts & Assumptions

Given: Plurisubharmonic functions on a domain $Ω \subseteq C^{m}$ ; for part 3, a real-valued plurisubharmonic function $u$ on $Ω$ and a convex nondecreasing function $ϕ : R \to R$ .

[L1]

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

[L2]

In one complex variable, nonnegative finite sums and finite maxima preserve subharmonicity (Positive linear combinations and finite maxima preserve subharmonicity).

Proof

technique · direct

1.1

Restrict every function in parts 1 and 2 to an affine complex line in $Ω$ . By [L1], each restriction is subharmonic or identically $- \infty$ on every connected component of the line domain, and [L2] shows that nonnegative finite sums and finite maxima preserve subharmonicity there. Another use of [L1] therefore gives parts 1 and 2.

L1L2given

2.1

For part 3, fix an affine complex line. Its restriction $v$ of $u$ is a real-valued subharmonic function by [L1]. The cited one-variable source result for convex nondecreasing compositions of subharmonic functions makes $ϕ \circ v$ subharmonic on each connected component. Because upper semicontinuity is preserved by nondecreasing composition, [L1] shows that $ϕ \circ u$ is plurisubharmonic.

L1step 1.1∎

Depends on

Used by

Hartogs pseudoconvexity yields a continuous plurisubharmonic exhaustion Theorem

Dependency tree · two levels

6 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 and Exercise 18 (standard reference, not scraped)