Phragmén–Lindelöf principle

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 (i.e, ) on an unbounded domain when an additional (usually mild) 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

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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. More precisely, if a non-constant function   is holomorphic in a bounded region[1]   and continuous on its closure  , then   for all  . This is known as the maximum modulus principle. (In fact, since   is compact and   is continuous, there actually exists some   such that  .) 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

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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 (i):   is holomorphic for all   and   on the boundary   of an appropriate bounded subregion  ; and (ii): the asymptotic behavior of   allows us to establish that   for   (i.e., the unbounded part of   outside the closure of the bounded subregion). 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

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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:[2]

Proof: (Sketch) 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  . Because one can show that   for all  , it follows that   for  . Moreover, one can show for   that   uniformly as  . This allows us to find an   such that   whenever   and  . Now consider the bounded rectangular region  . We have established that   for all  . Hence, the maximum modulus principle implies that   for all  . Since   also holds whenever   and  , we have in fact shown that   holds for all  . Finally, because   as  , we conclude that   for all  . Q.E.D.

Phragmén–Lindelöf principle for a sector in the complex plane

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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.[3]

Proposition. Let   be a function that is holomorphic in a sector

 

of central angle  , and continuous on its boundary. If

  (1)

for  , and

  (2)

for all  , where   and  , then   holds also for all  .

Remarks

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The condition (2) can be relaxed to

  (3)

with the same conclusion.

Special cases

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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.

See also

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References

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  1. ^ The term region is not uniformly employed in the literature; here, a region is taken to mean a nonempty connected open subset of the complex plane.
  2. ^ Rudin, Walter (1987). Real and Complex Analysis. New York: McGraw-Hill. pp. 257–259. ISBN 0070542341.
  3. ^ Tao, Terence (2009-02-18). "Hardy's Uncertainty Principle". Updates on my research and expository papers, discussion of open problems, and other maths-related topics. By Terence Tao.