# plurisubharmonic function

###### Definition.

Let $f\colon G\subset{\mathbb{C}}^{n}\to{\mathbb{R}}$ 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 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.

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

## References

• 1 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
Title plurisubharmonic function PlurisubharmonicFunction 2013-03-22 14:29:09 2013-03-22 14:29:09 jirka (4157) jirka (4157) 9 jirka (4157) Definition msc 31C10 msc 32U05 plurisubharmonic plurisuperharmonic function pseudoconvex function