# plurisubharmonic function

###### Definition.

Let $f:G\subset {\u2102}^{n}\to \mathbb{R}$ be an upper semi-continuous function^{}. $f$ is called plurisubharmonic
if for every complex line $\{a+bz\mid z\in \u2102\}$
the function $z\mapsto f(a+bz)$ is a subharmonic function on the set
$\{z\in \u2102\mid a+bz\in G\}$.

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.

## References

- 1 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.

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 |