Hartogs functions


Let Gn be an open set and let G be the smallest class of functionsMathworldPlanetmath on G to {-} that contains all of the functions zlog|f(z)| where f is holomorphic on G and such that G is closed with respect to the following conditions:

  • If φ1,φ2G, then φ1+φ2G.

  • If φG then aφ1G for all a0.

  • If {φk}G and φ1φ2, then limkφkG.

  • If {φk}G and the sequence is uniformly bounded above on compact sets, then supkφkG.

  • If φG and φ^(w):=lim supwzφ(w), then φ^G

  • If φ|UU for all UG where U is relatively compact (the closure of U is compact), then φG.

These functions are called the Hartogs functions.

It is known that if n=1 then the upper semi-continuous Hartogs functions are precisely the subharmonic functions on G.

Theorem (H. Bremerman).

All plurisubharmonic functionsMathworldPlanetmath are Hartogs functions if G is a domain of holomorphy.


  • 1 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
Title Hartogs functions
Canonical name HartogsFunctions
Date of creation 2013-03-22 14:29:27
Last modified on 2013-03-22 14:29:27
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 7
Author jirka (4157)
Entry type Definition
Classification msc 32U05