Liouville Theorem says the range of an entire function can not be confined, unless that’s a constant function. In other words, the modulus of a non-const entire function can not be bounded.

In harmonic function version, that’s pretty similar, but more precise. The statement is, there should be neither upper bound nor lower bound exists for a non-const harmonic function defined on the whole complex plane. That is, if a harmonic function has max or min on the whole complex plane, then that guy should be constant.

The proof is easy. W.L.O.G, considering a positive harmonic function u on the whole plane. We can find its harmonic conjugate v, to form an entire function f = u + iv. Then, g = exp(-f) should also be entire, with modulus bounded by 1 as the real part of g is exp(-u) < 1 (as u > 0). Then by Liouville, g is const, from which we get f is const, which implies its real part, our harmonic function u, is also const. Then we done.