In complex analysis, the Phragmén–Lindelöf principle (or method), first formulated by Lars Edvard Phragmén (1863–1937) and Ernst Leonard Lindelöf (1870–1946) in 1908, is a technique which employs an auxiliary, parameterized function to prove the boundedness of a holomorphic function
when an additional (usually mild) condition constraining the growth of
It is a generalization of the maximum modulus principle, which is only applicable to bounded domains.
In the theory of complex functions, it is known that the modulus (absolute value) of a holomorphic (complex differentiable) function in the interior of a bounded region is bounded by its modulus on the boundary of the region.
The maximum modulus principle is generally used to conclude that a holomorphic function is bounded in a region after showing that it is bounded on its boundary.
However, the maximum modulus principle cannot be applied to an unbounded region of the complex plane.
As a concrete example, let us examine the behavior of the holomorphic function
The difficulty here stems from the extremely fast growth of
is guaranteed to not be "too fast," as specified by an appropriate growth condition, the Phragmén–Lindelöf principle can be applied to show that boundedness of
is in fact bounded in the whole region, effectively extending the maximum modulus principle to unbounded regions.
In a typical Phragmén–Lindelöf argument, we introduce a certain multiplicative factor
; and (ii): the asymptotic behavior of
This allows us to apply the maximum modulus principle to first conclude that
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.
To continue the example above, we can impose a growth condition on a holomorphic function
that prevents it from "blowing up" and allows the Phragmén–Lindelöf principle to be applied.
To this end, we now include the condition that for some real constants
, and suppose there exist real constants
Note that this conclusion fails when
, precisely as the motivating counterexample in the previous section demonstrates.
The proof of this statement employs a typical Phragmén–Lindelöf argument:[2] Proof: (Sketch) We fix
to be the open rectangle in the complex plane enclosed within the vertices
Hence, the maximum modulus principle implies that
A particularly useful statement proved using the Phragmén–Lindelöf principle bounds holomorphic functions on a sector of the complex plane if it is bounded on its boundary.
This statement can be used to give a complex analytic proof of the Hardy's uncertainty principle, which states that a function and its Fourier transform cannot both decay faster than exponentially.
be a function that is holomorphic in a sector of central angle
The condition (2) can be relaxed to with the same conclusion.
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.
Carlson's theorem is an application of the principle to functions bounded on the imaginary axis.