Entire function confined to a half-plane is constant

If an entire function $g$ satisfies $\mathrm{Re}(cg)\ge b$ for constants $c\ne 0$ and real $b$, then $e^{-cg}$ is entire and has modulus at most $e^{-b}$. The Liouville theorem makes it constant, and differentiating gives $g^{\prime }=0$ 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.