One-radius results for supermedian functions on R-d, d <= 2

Hansen W, Nikolov N. One-radius results for supermedian functions on R-d, d &lt;= 2. Mathematische Annalen. 2010;348(3):565-575.A classical result states that every lower bounded superharmonic function on R-2 is constant. In this paper the following (stronger) one-circle version is proven. If f : R-2 --> (-infinity, infinity] is lower semicontinuous, lim inf(|x| --> infinity) f (x)/ln |x| >= 0, and, for every x --> R-2, 1/(2 pi) integral(2 pi)(0) f (x + r (x)e(it)) dt (0,infinity) is continuous, sup(x)is an element of(R2) (r (x) - |x|) < infinity, and inf(x is an element of R2) (r (x) - |x|) = -infinity, then f is constant. Moreover, it is shown that, assuming r <= c|.| + M on R-d, d <= 2, and taking averages on {y is an element of R-d : | y - x| <= r (x)}, such a result of Liouville type holds for supermedian functions if and only if c <= c(0), where c(0) = 1, if d = 2, whereas 2.50 < c(0) < 2.51, if d = 1