Phragmén-Lindelöf principle

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 function theory it is known that if a function "f" is holomorphic in a bounded domain "D", and is continuous on the 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 $g(z)\; =\; exp(exp(z))$ in the strip 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 and superharmonic functions.

Phragmén-Lindelöf principle for a sector in the complex plane

Let $F(z)$ be a function that is holomorphic in the sector

:

and continuous on its boundary. If

:$|F(z)|\; leq\; 1$ on the boundary and

:$|F(z)|\; leq\; Ce^\{c|z$ in the sector for some constants $c$ and $C$, then for all points $z\; in\; S$ we have

:$|F(z)|\; leq\; 1$.

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.

References

* cite journal
author = Pharagmén, Lars Edvard and Lindelöf, Ernst
title = Sur une extension d'un principe classique de l'analyse et sur quelques propriétés des fonctions monogénes dans le voisinage d'un point singulier
journal = Acta Math.
volume = 31
year = 1908
issue = 1
pages = 381–406
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 edition
publisher = Oxford University Press
YEAR = 1976
isbn = 0198533497 (See chapter 5)
*
* 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