# 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 {Anzeigestil 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 {Anzeigestil |f(z)|0} the auxiliary function {Anzeigestil h_{Epsilon }} durch {textstyle h_{Epsilon }(z)=e^{-Epsilon (e^{bz}+e^{-bz})}} . Darüber hinaus, für ein gegebenes {displaystyle a>0} , we define {Anzeigestil S_{a}} to be the open rectangle in the complex plane enclosed within the vertices {Anzeigestil {apm ipi /2,-apm ipi /2}} . Jetzt, fix {displaystyle epsilon >0} and consider the function {displaystyle fh_{Epsilon }} . It can be shown that {Anzeigestil |f(z)h_{Epsilon }(z)|zu 0} wie {Anzeigestil |Re (z)|to infty } . This allows us to find an {Anzeigestil x_{0}} so dass {Anzeigestil |f(z)h_{Epsilon }(z)|leq 1} wann immer {displaystyle zin {überstreichen {S}}} und {Anzeigestil |Re (z)|geq x_{0}} . Da {Anzeigestil S_{x_{0}}} is a bounded region, und {Anzeigestil |f(z)h_{Epsilon }(z)|leq 1} für alle {displaystyle zin partial S_{x_{0}}} , the maximum modulus principle implies that {Anzeigestil |f(z)h_{Epsilon }(z)|leq 1} für alle {displaystyle zin {überstreichen {S_{x_{0}}}}} . Seit {Anzeigestil |f(z)h_{Epsilon }(z)|leq 1} wann immer {displaystyle zin S} und {Anzeigestil |Re (z)|>x_{0}} , {Anzeigestil |f(z)h_{Epsilon }(z)|leq 1} in fact holds for all {displaystyle zin S} . Endlich, Weil {displaystyle fh_{Epsilon }to f} wie {displaystyle epsilon to 0} , Wir schließen daraus {Anzeigestil |f(z)|leq 1} für alle {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.[3] Vorschlag. Lassen {Anzeigestil F} be a function that is holomorphic in a sector {displaystyle S=left{z,{groß |},Alpha 0} , dann (1) holds also for all {displaystyle zin S} .

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

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