Phragmén–Lindelöf principle

In mathematics, the Phragmén–Lindelöf principle is a 1908 extension by Lars Edvard Phragmén (1863–1937) and Ernst Leonard Lindelöf of the maximum modulus principle of complex analysis, to unbounded domains.

Background

In complex function theory it is known that if a function f is holomorphic in a bounded domain D, and is continuous on the boundary of D, then the maximum of |f| must be attained on the boundary of D. If, however, the region D is not bounded, then this is no longer true, as may be seen by examining the function $g(z) = \exp(\exp(z))$ in the strip $-\pi/2 < \mbox{Im} \{ z \} < \pi/2.$ The difficulty here is that the function g tends to infinity 'very' rapidly as z tends to infinity along the positive real axis.

The Phragmén–Lindelöf principle shows that in certain circumstances, and by limiting the rapidity with which f is allowed to tend to infinity, it is possible to prove that f is actually bounded in the unbounded domain.

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.

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

• (Corr. )
• (See chapter 5)