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

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  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 
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 
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: (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 . 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 . Q.E.D. 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
The condition () can be relaxed to

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. See also 

 Hadamard three-lines theorem

References

 
  (Corr. ) (See chapter 5)
 
 

Mathematical principles
Theorems in complex analysis