Phragmén-Lindelöf principle

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

Reference

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