Phragmén–Lindelöf principle

In complex analysis, the Phragmén–Lindelöf principle (or method) is a 1908 extension by Lars Edvard Phragmén (1863–1937) and Ernst Leonard Lindelöf of the maximum modulus principle to unbounded 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 $f:\mathbb {C} \to \mathbb {C}$ is holomorphic in a bounded region^[1] $\Omega$ and continuous on its closure ${\overline {\Omega }}=\Omega \cup \partial \Omega$ , then $|f(z_{0})|<\sup _{z\in \partial \Omega }|f(z)|$ for all $z_{0}\in \Omega$ . This is known as the maximum modulus principle. (In fact, since $\overline {\Omega }$ is compact and $|f|$ is continuous, there actually exists some $w_{0}\in \partial \Omega$ such that $|f(w_{0})|=\sup _{z\in \partial \Omega }|f(z)|$ .) 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 the boundary of that region.

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 function $g(z)=\exp(\exp(z))$ in the unbounded strip $S={\Big \{}z:\Im (z)\in {\big (}-{\frac {\pi }{2}},{\frac {\pi }{2}}{\big )}{\Big \}}$ . Although $|g(x\pm \pi i/2)|=1$ , so that $|g|$ is bounded on $\partial S$ , $|g|$ grows rapidly without bound when $|z|\to\infty$ along the positive real axis. The difficulty here stems from the extremely fast growth of $|g|$ along the positive real axis. If the growth rate of $|g|$ 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 $|g|$ on the region's boundary implies that $|g|$ is in fact bounded in the whole region, effectively extending the maximum modulus principle to unbounded regions.

Suppose we are given a holomorphic function $g$ and an unbounded region $S$ . In a typical Phragmén–Lindelöf argument, we introduce a multiplicative factor $h_{\epsilon }$ to "subdue" the growth of $g$ , such that $|gh_{\epsilon }|$ is bounded on the boundary of a bounded subregion of $S$ and we can apply the maximum modulus principle to $gh_{\epsilon }$ . We then argue that, by passing to the limit, the subregion could be expanded to encompass all of $S$ , allowing boundedness to be established for $|gh_{\epsilon }|$ for all $z\in S$ . Finally, we let $\epsilon\to 0$ so that $gh_{\epsilon }\to g$ in order to conclude that $|g|$ must also be bounded in $S$ .

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 $g$ that prevents it from "blowing up" and allows the Phragmén–Lindelöf principle to be applied. Let the function $g:\mathbb {C} \to \mathbb {C}$ be holomorphic on $S$ and continuous on ${\overline {S}}$ , where $S={\Big \{}z:\Im (z)\in {\big (}-{\frac {\pi }{2}},{\frac {\pi }{2}}{\big )}{\Big \}}$ . If we now include the condition that $|g(z)|<\exp {\big (}A\exp(c\cdot |\Re (z)|){\big )}$ for some real constants $c<1$ and $A<\infty$ for all $z\in S$ , then it can be shown that $|g(z)|\leq 1$ for all $z\in \partial S$ implies that $|g(z)|\leq 1$ actually holds for all $z\in S$ . Note that this conclusion fails when $c=1$ , precisely as the motivating counterexample in the previous section demonstrates.

The proof of this theorem employs a typical Phragmén–Lindelöf argument:^[2]

Sketch of Proof: We first choose $b\in (c,1)$ and define for each $\epsilon >0$ the auxiliary function $h_{\epsilon }$ by $h_{\epsilon }(z)=\exp(-\epsilon (e^{bz}+e^{-bz}))$ . We consider the function $gh_{\epsilon }$ . The growth properties of $gh_{\epsilon }$ allow us to find an $x_{0}$ such that $|gh_{\epsilon }|\leq 1$ whenever $z\in {\overline {S}}$ and $|\Re (z)|\geq x_{0}$ . In particular, for such an $x_{0}$ , $|gh_{\epsilon }|\leq 1$ holds for all $z\in \partial S_{x_{0}}$ , where $S_{x_{0}}$ is the open rectangle in the complex plane defined by the vertices $\{\varepsilon _{1}x_{0}+i\varepsilon _{2}(\pi /2):\varepsilon _{1},\varepsilon _{2}\in \{-1,1\}\}$ . Because $S_{x_{0}}$ is a bounded region, the maximum modulus principle is applicable and implies that $|gh_{\epsilon }|\leq 1$ for all $z\in S_{x_{0}}$ . But since $x_{0}$ can be made arbitrarily large, this result actually extends to all of $S$ , so that $|gh_{\epsilon }|\leq 1$ , for all $\epsilon >0$ and $z\in S$ . Finally, letting $\epsilon\to 0$ , so that $h_{\epsilon }(z)\to 1$ for every $z$ , allows us to conclude that $|g(z)|\leq 1$ for all $z\in S$ . $\blacksquare$

Phragmén–Lindelöf principle for a sector in the complex plane

Let F(z) be a function that is holomorphic in a sector

S=\left\{z\,{\big |}\,\alpha <\arg z<\beta \right\}

of angle π/λ = β–α, and continuous on its boundary. If

$|F(z)|\leq 1$

(1)

for z on the boundary of S, and

$|F(z)|\leq e^{{C|z|^{\rho }}}$

(2)

for all z in S, where 0≤ρ<λ and C>0, then (1) holds also for all z in S.

Remarks

The condition (2) can be relaxed to

$\liminf _{{r\to \infty }}\sup _{{\alpha <\theta <\beta }}{\frac {\log |F(re^{{i\theta }})|}{r^{\rho }}}=0\quad {\text{for some}}\quad 0\leq \rho <\lambda ~,$

(3)

with the same conclusion.

Phragmén–Lindelöf principle for strips

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.

Other special cases

Carlson's theorem is an application of the principle to functions bounded on the imaginary axis.

Applications

The principle is used to prove Hardy's uncertainty principle, which states that a function and its Fourier transform cannot both decay faster than exponentially.

References

↑ The term region is not uniformly employed in the literature; here, a region is taken to mean a nonempty connected open subset of the complex plane.
↑ Rudin, Walter (1987). Real and Complex Analysis (PDF). New York: McGraw-Hill. pp. 257–259. ISBN 0070542341.

Phragmén, Lars Edvard; Lindelöf, Ernst (1908). "Sur une extension d'un principe classique de l'analyse et sur quelques propriétés des fonctions monogènes dans le voisinage d'un point singulier". Acta Math. 31 (1): 381–406. ISSN 0001-5962. doi:10.1007/BF02415450.
Riesz, Marcel (1920). "Sur le principe de Phragmén-Lindelöf". Proceedings of the Cambridge Philosophical Society. 20. (Corr. "Sur le principe de Phragmén-Lindelöf". 21. 1921. )
Titchmarsh, Edward Charles (1976). The Theory of Functions (Second ed.). Oxford University Press. ISBN 0-19-853349-7. (See chapter 5)
E.D. Solomentsev (2001) [1994], "Phragmén–Lindelöf theorem", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Stein, Elias M. and Shakarchi, Rami (2003). Complex analysis. Princeton Lectures in Analysis, II. Princeton, NJ: Princeton University Press. ISBN 0-691-11385-8.

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