It's kind of trivial, if was in then by the maximum modulus principle would be constant on and hence on . So unless you were assuming was constant this is a contradiction, and hence must belong to the boundary of .
Let U be a connected open set, and its closure. Let f be a continuous function on , analytic and nonconstant on U. If is a maximum point for f, then lies on the boundary of
Now, the book says this comes from a direct application of the global maximum modulus principle, which states:
If U is a connected open set, and f is analytic on U that has a maximum point in U. Then f is constant on U.
I understand the proof of the principle, but how do I apply that to the first statement? Thanks.