On discrete form of the mean value inequality for subharmonic functions
Dmitrii Sergeevich Telyakovskii · Journal of Physics Conference Series · 2019
We obtain a sufficient condition for the subharmonicity of a function u ( x,y ) = u ( z ), z ∈ G ⊂ 2 , 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 .