On discrete form of the mean value inequality for subharmonic functions

Abstract

We obtain a sufficient condition for the subharmonicity of a function u(x,y) = u(z), z ∈ G ⊂ ℝ2, in which the mean value inequality has discrete form. Namely, it is assumed that for each point ζ ∈ G there are a circle of arbitrarily small radius centered at ζ and a set of nodes lying on this circle for which the value u(ζ) does not exceed the arithmetic mean of the function values in the nodes of this set. A necessary and sufficient condition for the location of the nodes of the set is established when executed, the function u(z) satisfying at each point of G such discrete form of mean value inequality, and, additionally, some condition of summability and continuity in the directions is subharmonic in the domain G.