# Phragmén–Lindelöf principle 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 {displaystyle f} that prevents it from "blowing up" and allows the Phragmén–Lindelöf principle to be applied. To this end, we now include the condition that {displaystyle |f(z)|0} the auxiliary function {displaystyle h_{epsilon }} by {textstyle h_{epsilon }(z)=e^{-epsilon (e^{bz}+e^{-bz})}} . Moreover, for a given {displaystyle a>0} , we define {displaystyle S_{a}} to be the open rectangle in the complex plane enclosed within the vertices {displaystyle {apm ipi /2,-apm ipi /2}} . Now, fix {displaystyle epsilon >0} and consider the function {displaystyle fh_{epsilon }} . It can be shown that {displaystyle |f(z)h_{epsilon }(z)|to 0} as {displaystyle |Re (z)|to infty } . This allows us to find an {displaystyle x_{0}} such that {displaystyle |f(z)h_{epsilon }(z)|leq 1} whenever {displaystyle zin {overline {S}}} and {displaystyle |Re (z)|geq x_{0}} . Because {displaystyle S_{x_{0}}} is a bounded region, and {displaystyle |f(z)h_{epsilon }(z)|leq 1} for all {displaystyle zin partial S_{x_{0}}} , the maximum modulus principle implies that {displaystyle |f(z)h_{epsilon }(z)|leq 1} for all {displaystyle zin {overline {S_{x_{0}}}}} . Since {displaystyle |f(z)h_{epsilon }(z)|leq 1} whenever {displaystyle zin S} and {displaystyle |Re (z)|>x_{0}} , {displaystyle |f(z)h_{epsilon }(z)|leq 1} in fact holds for all {displaystyle zin S} . Finally, because {displaystyle fh_{epsilon }to f} as {displaystyle epsilon to 0} , we conclude that {displaystyle |f(z)|leq 1} for all {displaystyle zin S} . 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 {displaystyle F} be a function that is holomorphic in a sector {displaystyle S=left{z,{big |},alpha 0} , then (1) holds also for all {displaystyle zin S} .

Remarks The condition (2) can be relaxed to {displaystyle liminf _{rto infty }sup _{alpha

Si quieres conocer otros artículos parecidos a Phragmén–Lindelöf principle puedes visitar la categoría Mathematical principles.

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