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Phragmén-Lindelöf theorem
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(Theorem)
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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 1 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
- for every $a \in A$ , $\varlimsup_{z\rightarrow a} \lvert f(z) \rvert \leq M$ , and
- 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 force 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.
- 1
- John B. Conway. Functions of One Complex Variable I. Springer-Verlag, New York, New York, 1978.
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"Phragmén-Lindelöf theorem" is owned by jirka.
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Cross-references: neighbourhood, open, growth, restriction, function, maximal modulus principle, theorem, bounded, vanishes, analytic functions, region, simply connected, unbounded, infinity, point, plus, boundary, extended boundary
This is version 6 of Phragmén-Lindelöf theorem, born on 2004-02-26, modified 2005-03-07.
Object id is 5634, canonical name is PhragmenLindelofTheorem.
Accessed 3011 times total.
Classification:
| AMS MSC: | 30C80 (Functions of a complex variable :: Geometric function theory :: Maximum principle; Schwarz's lemma, Lindelöf principle, analogues and generalizations; subordination) |
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Pending Errata and Addenda
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