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