Subharmonic Function

Let  be an open set and  a real-valued continuous function on . Suppose that for each closed disk  and every real-valued harmonic function  defined on a neighborhood of  which satisfies  on , it holds that on the open disk . Then  is said to be subharmonic on  (Krantz 1999, p. 99).

1. If  are subharmonic on , then so is .

2. If  is subharmonic on  and  is a constant, than  is subharmonic on .

3. If  are subharmonic on , then  is also subharmonic on .

REFERENCES:

Krantz, S. G. "The Dirichlet Problem and Subharmonic Functions." §7.7 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 97-101, 1999.