We find necessary and sufficient conditions under which subsets of balls are big enough for the characterization of nonnegative,
quasinearly subharmonic functions by mean value inequalities. A similar result is obtained also for generalized mean value inequalities where,
instead of balls, we consider arbitrary bounded sets which have nonvoid
interiors and instead of the volume of ball we use functions depending
on the radius of this ball.
Keywords: Subharmonic, nearly subharmonic, quasinearly subharmonic, mean value
