 The statement is

To proof this one needs to substitute Taylor Series expansions up to order 3 into the discrete Laplace operator.Similar to continuous subharmonic functions one can define "subharmonic functions" for finite-difference approximations formula_64

One can define a general stencil of "positive type" via

If formula_67 is (discrete) subharmonic then the following" mean value property" holds
where the approximation is evaluated on points of the grid, and the stencil is assumed to be of positive type.A similar mean value property also holds for the continuous case.