Let Ω be a bounded open set in R
n. An immediate consequence of the maximum principle is that if s is a function continuous on and subharmonic on Ω, then
Of course (1) is no longer true if Ω is not bounded. For example in C ∼ R
2 consider the functions
However, if we restrict the growth of s, then (1) may still hold even if the open set Ω is no longer bounded and such is the theme of Phragmèn-Lindelöf type theorems. If we assume even more, namely, that s is upper-bounded, then we can again infer (1) for unbounded open sets Ω. We shall return to this point later.
In the present note, we wish to prove (1) for an arbitrary subharmonic function s on an open subset Ω of R
. In particular, we do not assume that s is bounded or even of restricted growth. Rather, we impose restrictions on the (possibly unbounded) set Ω.