# 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 {style d'affichage f} that prevents it from "blowing up" and allows the Phragmén–Lindelöf principle to be applied. À cette fin, we now include the condition that {style d'affichage |F(z)|0} the auxiliary function {style d'affichage h_{epsilon }} par {style de texte h_{epsilon }(z)=e^{-epsilon (e ^{bz}+e ^{-bz})}} . En outre, for a given {displaystyle a>0} , we define {style d'affichage S_{un}} to be the open rectangle in the complex plane enclosed within the vertices {style d'affichage {apm ipi /2,-apm ipi /2}} . À présent, fix {displaystyle epsilon >0} and consider the function {displaystyle fh_{epsilon }} . It can be shown that {style d'affichage |F(z)h_{epsilon }(z)|à 0} comme {style d'affichage |Re (z)|to infty } . This allows us to find an {style d'affichage x_{0}} tel que {style d'affichage |F(z)h_{epsilon }(z)|leq 1} chaque fois que {displaystyle zin {surligner {S}}} et {style d'affichage |Re (z)|geq x_{0}} . Car {style d'affichage S_{X_{0}}} is a bounded region, et {style d'affichage |F(z)h_{epsilon }(z)|leq 1} pour tous {displaystyle zin partial S_{X_{0}}} , the maximum modulus principle implies that {style d'affichage |F(z)h_{epsilon }(z)|leq 1} pour tous {displaystyle zin {surligner {S_{X_{0}}}}} . Depuis {style d'affichage |F(z)h_{epsilon }(z)|leq 1} chaque fois que {displaystyle zin S} et {style d'affichage |Re (z)|>x_{0}} , {style d'affichage |F(z)h_{epsilon }(z)|leq 1} in fact holds for all {displaystyle zin S} . Pour terminer, car {displaystyle fh_{epsilon }to f} comme {displaystyle epsilon to 0} , nous concluons que {style d'affichage |F(z)|leq 1} pour tous {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] Proposition. Laisser {style d'affichage F} be a function that is holomorphic in a sector {displaystyle S=left{z,{gros |},alpha 0} , alors (1) holds also for all {displaystyle zin S} .

Remarks The condition (2) can be relaxed to {style d'affichage limite _{rto infty }souper _{alpha

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