Plurisubharmonic function

Definition

Let $f\colon G\subset {\mathbb {C} }^{n}\to {\mathbb {C} }$ be an upper semi-continuous function. $f$ is called plurisubharmonic if for every complex line $\{a+bz\mid z\in {\mathbb {C} }\}$ the function $z\mapsto f(a+bz)$ is a subharmonic function on the set $\{z\in {\mathbb {C} }\mid a+bz\in G\}$.

Similarly we could also definie a plurisuperharmonic function like a superharmonic function, but again it just means that $-f$ is plurisubharmonic and so this extra term is not very useful.

Alterantive definition

If $f$ is a plurisubharmonic function and further $f$ is continuous, then $f$ is called a pseudoconvex function.

Note: Some authors abbreviate plurisubharmonic with psh, plsh or plush.

Bibliography

1. Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.

plurisubharmonic function at PlanetMath.