Phragmén-Lindelöf theorem

First some notation. Let G be the extended boundary of G. That is, the boundary of G, plus optionally the point at infinity if in fact G is unboundedPlanetmathPlanetmath.


Let G be a simply connected region and let f:GC and φ:GC be analytic functionsMathworldPlanetmath. Furthermore suppose that φ never vanishes and is boundedPlanetmathPlanetmathPlanetmath on G. If M is a constant and G=AB such that

  1. 1.

    for every aA, lim¯za|f(z)|M, and

  2. 2.

    for every bB, and η>0, lim¯zb|f(z)||φ(z)|ηM,

then |f(z)|M for all zG.

This theorem is a generalizationPlanetmathPlanetmath of the maximal modulus principle, but instead of requiring that the function is bounded as we approach the boundary, we only need a restrictionPlanetmathPlanetmathPlanetmath on its growth to it to in fact be bounded in all of G.

If you let A=G (and φ1 perhaps), then you get almost exactly one version of the maximal modulus principle. In this case it turns out that G need not be simply connected since that is only needed to define zφ(z)η.

In fact the requirement that G be simply connected can be eased up a bit in this theorem since it is only needed locally. So the theorem is still true if for every point zG there exists an open neighbourhood N of z such that NG is simply connected.


  • 1 John B. Conway. . Springer-Verlag, New York, New York, 1978.
Title Phragmén-Lindelöf theorem
Canonical name PhragmenLindelofTheorem
Date of creation 2013-03-22 14:12:09
Last modified on 2013-03-22 14:12:09
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 9
Author jirka (4157)
Entry type Theorem
Classification msc 30C80
Synonym Phragmén-Lindelöf principle
Related topic MaximalModulusPrinciple