Maximum modulus principle on a bounded domain

If a function is continuous on the closure of a bounded plane domain and holomorphic inside, the compact closure supplies a point of maximum modulus. Unless the function is constant, the maximum modulus principle places every maximum on the boundary.

Let $f$ be bounded and holomorphic on a half-plane and continuous on its closure. A bound $|f|\le M$ on the boundary line propagates throughout the half-plane. Apply the bounded-domain principle to $f(z)z^{-1/n}$ on expanding truncated half-discs, then let $n\to \infty$.