Phragmén–Lindelöf principle


In complex analysis, the Phragmén–Lindelöf principle, first formulated by Lars Edvard Phragmén and Ernst Leonard Lindelöf in 1908, is a technique which employs an auxiliary, parameterized function to prove the boundedness of a holomorphic function on an unbounded domain when an additional condition constraining the growth of on is given. It is a generalization of the maximum modulus principle, which is only applicable to bounded domains.

Background

In the theory of complex functions, it is known that the modulus of a holomorphic function in the interior of a bounded region is bounded by its modulus on the boundary of the region. More precisely, if a non-constant function is holomorphic in a bounded region and continuous on its closure, then for all. This is known as the maximum modulus principle. 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 in the unbounded strip
Although, so that is bounded on boundary, grows rapidly without bound when along the positive real axis. The difficulty here stems from the extremely fast growth of along the positive real axis. If the growth rate 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 on the region's boundary implies that is in fact bounded in the whole region, effectively extending the maximum modulus principle to unbounded regions.

Outline of the technique

Suppose we are given a holomorphic function and an unbounded region, and we want to show that on. In a typical Phragmén–Lindelöf argument, we introduce a certain multiplicative factor satisfying to "subdue" the growth of. In particular, is chosen such that : is holomorphic for all and on the boundary of an appropriate bounded subregion ; and : the asymptotic behavior of allows us to establish that for . This allows us to apply the maximum modulus principle to first conclude that on and then extend the conclusion to all. Finally, we let so that for every in order to conclude that on.
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.

Example of application

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, for all. It can then be shown that for all implies that in fact holds for all. Thus, we have the following proposition:
Proposition. Let
Let be holomorphic on and continuous on, and suppose there exist real constants such that
for all and for all. Then ' for all .
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:
Proof: ' We fix and define for each the auxiliary function by. Moreover, for a given, we define to be the open rectangle in the complex plane enclosed within the vertices. Now, fix and consider the function. It can be shown that as. This allows us to find an such that whenever and. Because is a bounded region, and for all, the maximum modulus principle implies that for all. Since whenever and, in fact holds for all. Finally, because as, we conclude that for all. ∎

Phragmén–Lindelöf principle for a sector in the complex plane

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.
Proposition. Let be a function that is holomorphic in a sector
of central angle, and continuous on its boundary. If
for, and
for all, where and, then holds also for all.

Remarks

with the same conclusion.

Special cases

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.
Carlson's theorem is an application of the principle to functions bounded on the imaginary axis.