For definition of subharmonic functions on Riemannian manifolds, the defining inequalities between the subharmonic function $f$ and the harmonic function $f_1$ should be the reverse, in accordance with the definition given above 8and in accordance with the heuristical content of the name SUBharmonic).
I'm not sure what he means precisely, possibly due to notational confusion. RayTalk 19:23, 10 November 2010 (UTC)
I think you confused and in your counterexample. If you interpret , (as one is supposed to do), the inequality really holds. Vigfus (talk) 20:23, 3 August 2012 (UTC)
Maximum Principle(with modulus) for harmonic functions.
Let's suppose that is a harmonic function non-constant,and is an open simply connected set.I want to prove that there is not with .We observe this equality is equivalent to .Since is subharmonic,we find a contradiction with the maximum principle for subharmonic functions.