Mean value type inequalities for quasinearly subharmonic functions

The mean value type inequality is characteristic for upper semi-continuous functions to be subharmonic. Quasinearly subharmonic functions generalize subharmonic functions. The necessary and sufficient conditions under which subsets of balls are “big enough” for the characterization of non-negative, quasinearly subharmonic functions by mean value inequalities are found. Similar result is obtained also for generalized mean value inequalities where, instead of balls, we consider arbitrary bounded sets, which have non-void interiors and instead of the volume of ball some functions depending on the diameter of considering set.