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In [[mathematics]], the '''Phragmén–Lindelöf principle''' is a 1908 extension by [[Lars Edvard Phragmén]] (1863–1937) and [[Ernst Leonard Lindelöf]] of the [[maximum modulus principle]] of [[complex analysis]], to unbounded domains.

==Background==

In [[complex analysis|complex function theory]] it is known that if a function ''f'' is [[holomorphic]] in a '''[[bounded]]'''  domain ''D'', and is [[continuous function|continuous]] on the [[Boundary (topology)|boundary]] of ''D'', then the maximum of |''f''| must be attained on the boundary of ''D''. If, however, the region ''D'' is not bounded, then this is no longer true, as may be seen by examining the function <math>g(z) = \exp(\exp(z))</math> in the strip <math>-\pi/2 < \mbox{Im} \{ z \} < \pi/2.</math> The difficulty here is that the function ''g'' tends to infinity 'very' rapidly as ''z'' tends to infinity along the positive real axis.

The Phragmén–Lindelöf principle shows that in certain circumstances, and by limiting the rapidity with which ''f'' is allowed to tend to infinity, then it is possible to prove that ''f'' is actually bounded in the [[unbounded]] domain.

In the literature of complex analysis, there are many examples of the Phragmén–Lindelöf principle applied to unbounded regions of differing types, and also a version of this principle may be applied in a similar fashion to [[subharmonic function|subharmonic]] and superharmonic functions.

== Phragmén–Lindelöf principle for a sector in the complex plane ==
Let ''F(z)'' be a function that is [[holomorphic]] in a [[Circular sector|sector]] 

:<math> S = \left\{ z \, \big| \, \alpha < \arg z < \beta \right\} </math>

of angle ''&pi;''/''&lambda;'' = ''&beta;''&ndash;''&alpha;'', and continuous on its boundary. If 

:{{NumBlk|:|<math>|F(z)| \leq 1</math>|{{EquationRef|1}}}}

for ''z'' on the boundary of ''S'', and

:{{NumBlk|:|<math>|F(z)| \leq e^{C |z|^\rho}</math>|{{EquationRef|2}}}}
 
for all ''z'' in ''S'', where 0≤''&rho;''<''&lambda;'' and ''C''>0, then ({{EquationNote|1}}) holds also for all ''z'' in ''S''.

===Remarks===
* The condition ({{EquationNote|2}}) can be relaxed to 

:{{NumBlk|:|<math>\liminf_{r \to \infty} \sup_{\alpha < \theta < \beta} \frac{\log|F(re^{i\theta})|}{r^\rho} = 0 \quad \text{for some} \quad 0 \leq \rho < \lambda~,</math>|{{EquationRef|3}}}}

with the same conclusion.

== Phragmén–Lindelöf principle for strips ==
In practice the point 0 is often transformed into the point ∞ of the [[Riemann sphere]]. This gives a version of the principle that applies to strips, for example bounded by two lines of constant [[real part]] in the complex plane. This special case is sometimes known as [[Lindelöf's theorem]].

==Other special cases==
* [[Carlson's theorem]] is an application of the principle to functions bounded on the imaginary axis.

==Applications==

The principle is used to prove [[Hardy's uncertainty principle]].

==References==

* {{cite journal
|   author = Phragmén, Lars Edvard and Lindelöf, Ernst
|    title = Sur une extension d'un principe classique de l'analyse et sur quelques propriétés des fonctions monogènes dans le voisinage d'un point singulier
|  journal = Acta Math.
|   volume = 31
|     year = 1908
|    issue = 1
|    pages = 381&ndash;406
|     issn = 0001-5962
|    doi = 10.1007/BF02415450
}}
* {{cite journal
|     last = Riesz
|    first = Marcel
|authorlink= Marcel Riesz
|    title = Sur le principe de Phragmén-Lindelöf
|  journal = Proceedings of the Cambridge Philosophical Society
|   volume = 20
|     year = 1920
}} (Corr. {{cite journal | title = Sur le principe de Phragmén-Lindelöf | volume = 21 | year = 1921 }})
* {{cite book
|     last = Titchmarsh
|    first = Edward Charles
|authorlink= Edward Charles Titchmarsh
|    title = The Theory of Functions
|  edition = Second
|publisher = Oxford University Press
|     year = 1976
|     isbn = 0-19-853349-7
}} (See chapter 5)
* {{springer|author=E.D. Solomentsev|title=Phragmén–Lindelöf theorem|id=p/p072630}}
* {{cite book
|   author = Stein, Elias M. and Shakarchi, Rami
|    title = Complex analysis
|   series = Princeton Lectures in Analysis, II
|publisher = Princeton University Press
| location = Princeton, NJ
|     year = 2003
|     isbn = 0-691-11385-8
}}

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[[Category:Complex analysis]]
[[Category:Mathematical principles]]
[[Category:Theorems in complex analysis]]