plurisubharmonic function

Definition.

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

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

Definition.

A continuous plurisubharmonic function is said to be a pseudoconvex function.

Note that since plurisubharmonic is a long word, many authors abbreviate with psh, plsh, or plush.

Title	plurisubharmonic function
Canonical name	PlurisubharmonicFunction
Date of creation	2013-03-22 14:29:09
Last modified on	2013-03-22 14:29:09
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	9
Author	jirka (4157)
Entry type	Definition
Classification	msc 31C10
Classification	msc 32U05
Synonym	plurisubharmonic
Defines	plurisuperharmonic function
Defines	pseudoconvex function