= Entire function confined to a half-plane is constant
If an <entire function> $g$ satisfies $\operatorname{Re}(cg)\ge b$ for constants $c\ne0$ and real $b$, then $e^{-cg}$ is entire and has modulus at most $e^{-b}$. The <Liouville theorem> makes it constant, and differentiating gives $g'=0$ since the exponential never vanishes. The conclusion also applies after rotating and translating any containing half-plane; separate bounds on the real and imaginary parts are unnecessary.
Back to article page
