plurisubharmonic function

Definition.

Let $f:G\subset {ℂ}^n\to ℝ$ be an upper semi-continuous function. $f$ is called plurisubharmonicif 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 likewe have a superharmonic function, but again it just means that $-f$ isplurisubharmonic, 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 abbreviatewith psh, plsh, or plush.