# Phragmén-Lindelöf theorem

First some notation. Let $\partial_{\infty}G$ 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 $f\colon G\to{\mathbb{C}}$ and $\varphi\colon G\to{\mathbb{C}}$ be analytic functions. Furthermore suppose that $\varphi$ never vanishes and is bounded on $G$. If $M$ is a constant and $\partial_{\infty}G=A\cup B$ such that

1. 1.

for every $a\in A$, $\varlimsup_{z\rightarrow a}\lvert f(z)\rvert\leq M$, and

2. 2.

for every $b\in B$, and $\eta>0$, $\varlimsup_{z\rightarrow b}\lvert f(z)\rvert\lvert\varphi(z)\rvert^{\eta}\leq M$,

then $\lvert f(z)\rvert\leq M$ for all $z\in G$.

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 $A=\partial_{\infty}G$ (and $\varphi\equiv 1$ 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 $z\mapsto\varphi(z)^{\eta}$.

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 $z\in\partial_{\infty}G$ there exists an open neighbourhood $N$ of $z$ such that $N\cap G$ is simply connected.

## References

• 1 John B. Conway. . Springer-Verlag, New York, New York, 1978.
Title Phragmén-Lindelöf theorem PhragmenLindelofTheorem 2013-03-22 14:12:09 2013-03-22 14:12:09 jirka (4157) jirka (4157) 9 jirka (4157) Theorem msc 30C80 Phragmén-Lindelöf principle MaximalModulusPrinciple