Mean value type inequalities for quasinearly subharmonic functions

TL;DR

This paper establishes conditions under which mean value inequalities hold for quasinearly subharmonic functions, extending classical results to more general sets and functions.

Contribution

It provides necessary and sufficient conditions for mean value inequalities to apply to quasinearly subharmonic functions over various sets.

Findings

Characterization of big enough subsets for mean value inequalities

Extension to generalized mean value inequalities with arbitrary sets

Conditions involving functions of the radius for mean inequalities

Abstract

The mean value inequality is characteristic for upper semicontinuous functions to be subharmonic. Quasinearly subharmonic functions generalize subharmonic functions. We find the necessary and sufficient conditions under which subsets of balls are big enough for the catheterization of nonnegative, quasinearly subharmonic functions by mean value inequalities. 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 some functions depending on the radius of this ball.