PhragménLindelöf theorem
First some notation. Let ${\partial}_{\mathrm{\infty}}G$ be the extended boundary of $G$. That is, the boundary of $G$, plus optionally the point at infinity if in fact $G$ is unbounded^{}.
Theorem.
Let $G$ be a simply connected region and let $f\mathrm{:}G\mathrm{\to}\mathrm{C}$ and $\phi \mathrm{:}G\mathrm{\to}\mathrm{C}$ be analytic functions^{}. Furthermore suppose that $\phi $ never vanishes and is bounded^{} on $G$. If $M$ is a constant and ${\mathrm{\partial}}_{\mathrm{\infty}}\mathit{}G\mathrm{=}A\mathrm{\cup}B$ such that

1.
for every $a\in A$, ${\overline{\mathrm{lim}}}_{z\to a}f(z)\le M$, and

2.
for every $b\in B$, and $\eta >0$, ${\overline{\mathrm{lim}}}_{z\to b}f(z){\phi (z)}^{\eta}\le M$,
then $\mathrm{}f\mathit{}\mathrm{(}z\mathrm{)}\mathrm{}\mathrm{\le}M$ for all $z\mathrm{\in}G$.
This theorem is a generalization^{} of the maximal modulus principle, but instead of requiring that the function is bounded as we approach the boundary, we only need a restriction^{} on its growth to it to in fact be bounded in all of $G$.
If you let $A={\partial}_{\mathrm{\infty}}G$ (and $\phi \equiv 1$ perhaps), then you get almost exactly one version of the maximal modulus principle. In this case it turns out that $G$ need not be simply connected since that is only needed to define $z\mapsto \phi {(z)}^{\eta}$.
In fact the requirement that $G$ be simply connected can be eased up a bit in this theorem since it is only needed locally. So the theorem is still true if for every point $z\in {\partial}_{\mathrm{\infty}}G$ there exists an open neighbourhood $N$ of $z$ such that $N\cap G$ is simply connected.
References
 1 John B. Conway. . SpringerVerlag, New York, New York, 1978.
Title  PhragménLindelöf theorem 

Canonical name  PhragmenLindelofTheorem 
Date of creation  20130322 14:12:09 
Last modified on  20130322 14:12:09 
Owner  jirka (4157) 
Last modified by  jirka (4157) 
Numerical id  9 
Author  jirka (4157) 
Entry type  Theorem 
Classification  msc 30C80 
Synonym  PhragménLindelöf principle 
Related topic  MaximalModulusPrinciple 