# holomorphically convex

Let $G\subset {\u2102}^{n}$ be a domain, or alternatively for a more general definition let $G$ be an $n$ dimensional complex analytic manifold.
Further let $\mathcal{O}(G)$ stand for the set of holomorphic functions^{} on $G$.

###### Definition.

Let $K\subset G$ be a compact set. We define the holomorphically of $K$ as

$${\widehat{K}}_{G}:=\{z\in G\mid |f(z)|\le \underset{w\in K}{sup}|f(w)|\text{for all}f\in \mathcal{O}(G)\}.$$ |

The domain $G$ is called holomorphically convex if for every $K\subset G$ compact in $G$, ${\widehat{K}}_{G}$ is also compact in $G$. Sometimes this is just abbreviated as holomorph-convex.

Note that when $n=1$, any domain $G$ is holomorphically convex since when $n=1$ ${\widehat{K}}_{G}=K$ for all compact $K\subset G$. Also note that this is the same as being a domain of holomorphy.

## References

- 1 Lars Hörmander. , North-Holland Publishing Company, New York, New York, 1973.
- 2 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.

Title | holomorphically convex |
---|---|

Canonical name | HolomorphicallyConvex |

Date of creation | 2013-03-22 15:04:33 |

Last modified on | 2013-03-22 15:04:33 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 8 |

Author | jirka (4157) |

Entry type | Definition |

Classification | msc 32E05 |

Synonym | holomorph-convex |

Related topic | PolynomiallyConvexHull |

Related topic | SteinManifold |

Defines | holomorphically convex hull |