Perron family


Let G be a region, G the extended boundary of G and S(G) the set of subharmonic functionsMathworldPlanetmath on G, then if f:G is a continuous functionMathworldPlanetmath then the set

𝒫(f,G):={φ:φS(G) and lim supzaφ(z)f(a) for all aG},

is called the Perron family.

One thing to note is the 𝒫(f,G) is never empty. This is because f is continuous on G it attains a maximum, say |f|<M, then the function φ(z):=-M is in 𝒫(f,G).


Let G be a region and f:G be a continuous function then the function u:G


is called the Perron function associated with f.

Here is the reason for all these definitions.


Let GC be a region and suppose f:GR is a continuous function. If u:GR is the Perron function associated with f, then u is a harmonic functionMathworldPlanetmath.

Compare this with Rado’s theorem ( which works with harmonic functions with range in 2, but also gives a much stronger statement.


  • 1 John B. Conway. . Springer-Verlag, New York, New York, 1978.
Title Perron family
Canonical name PerronFamily
Date of creation 2013-03-22 14:19:42
Last modified on 2013-03-22 14:19:42
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 6
Author jirka (4157)
Entry type Definition
Classification msc 31B05
Related topic RadosTheorem
Defines Perron function