holomorphically convex

Let $G\subset{\mathbb{C}}^{n}$ 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

$\hat{K}_{G}:=\{z\in G\mid\lvert f(z)\rvert\leq\sup_{w\in K}\lvert f(w)\rvert% \text{ for all }f\in{\mathcal{O}}(G)\}.$

The domain $G$ is called holomorphically convex if for every $K\subset G$ compact in $G$ , $\hat{K}_{G}$ 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 $n=1$ $\hat{K}_{G}=K$ 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