LIOUVILLES THEOREM AND THE RESTRICTED MEAN-VALUE PROPERTY IN THE PLANE

Let U be a domain in R-2 such that U-c is polar and let r be a real fu nction on U such that 0 < r less than or equal to \.\ + M. A positive numerical function f on U is called r-supermedian if, for every x is a n element of U, the average of f on the disk of center r and radius r( x) is at most f(x) The purpose of this note is to give a short proof f or the fact that every 1.s.c. r-median function is constant. (C) Elsev ier, Paris.