Bounded half-plane maximum principle
= Bounded half-plane maximum principle
Let $f$ be bounded and holomorphic on a half-plane and continuous on its closure. A bound $|f|\leq M$ 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 $n\to\infty$.