The latter case happens if and only if there exists no Green function, or equivalently there exists no non-constant subharmonic function bounded from above.
Then -h extends to be a subharmonic function on D, and there exists a harmonic function k on D and a constant b [greater than or equal to] 0, such that
Keywords: Holomorphic function, subharmonic function, Hausdorff measure, exceptional sets.
For the definition and properties of subharmonic functions, see e.
The authors develop the foundations of potential theory on the Berkovich projective line, including the definition of a measure-valued Laplacian operator, capacity theory, and a theory of harmonic and subharmonic functions
Lyubarskii, Frames in the Bargmann space of entire functions and subharmonic functions
HAYMAN, Subharmonic Functions
, Volume 2, Academic Press, 1989.
Specific topics explored include zeros of functions in weighted and Bergman spaces, Blaschke-type conditions for analytic and subharmonic functions
, interpolation sequences for the Bernstein algebra, differentiability of functions of contractions, and free interpolation in the Nevanlinna class.
On subharmonic functions
and differential geometry in the large, Comment.
For a description of subharmonic functions
and their properties, see for example .
max], as the maximum of subharmonic functions
, is again subharmonic in C.
Other topics include Blaschke sets for Bergman spaces, a representation formula for reproducing subharmonic functions
in the unit disc, and the trigonometric obstacle problem and weak factorization.