Phragmén–Lindelöf principle

An extension by Phragmén and Lindelöf of the maximum modulus principle to unbounded domains. Let $F(z)$ be a function that is holomorphic in the sector $S=\{z:-\pi /4<argz<\pi /4\}$ and continuous on its boundary. If $|F(z)|\leq 1$ on the boundary and $|F(z)|\leq Ce^{c|z|}$ in the sector for some constants $c$ and $C$, then for all $z\in S,|F(z)|\leq 1$.

Reference

Stein and Shakarchi. Complex Analysis. Princeton: Princeton University Press, 2003.