# Phragmén-Lindelöf theorem

A generalization of the maximum-modulus principle for analytic functions to the case of functions that are given a priori as unbounded; it was first given in its simplest form by E. Phragmén and E. Lindelöf . Let be a regular analytic function of a complex variable in a domain of the plane with boundary . One says that does not exceed a number in modulus at a boundary point if that is, for every there is a disc (depending on and ) with centre such that for . The main content of the result of Phragmén and Lindelöf, in a somewhat modernized form, consists in the following two propositions, which are successive extensions of the maximum-modulus principle.

1) If the regular analytic function exceeds in modulus nowhere on , then everywhere in . This proposition is sometimes called the Phragmén–Lindelöf principle. It extends the maximum-modulus principle to functions about the behaviour of which on the boundary only partial information is available.

2) Suppose that the regular analytic function does not exceed in modulus at any point of not belonging to some set . Suppose also that there is a function with the following properties: a) is regular in ; b) in ; c) in ; and d) for every the function does not exceed in modulus at any point . Under these conditions everywhere in .

The Phragmén–Lindelöf theorem has received numerous applications, also often called Phragmén–Lindelöf theorems, and associated with a concrete form of , and (see , in particular the generalization given in ). In applications most often consists of the single point . For example, suppose that is regular in the angular domain (*)

and does not exceed in modulus on the sides of the angle. Then the following alternative holds: Either everywhere in , or the maximum modulus increases faster than as for any , . This theorem is obtained from propositions 1 and 2 for , , where .

The statements of 1 and 2 remain valid for a holomorphic function , , given in a domain of the complex space , . Many papers have been devoted to obtaining results of the type of the Phragmén–Lindelöf theorem for the solutions of partial differential equations and systems of equations of elliptic type. Propositions 1 and 2 remain true for a subharmonic function defined in a domain of a Euclidean space , , or , , provided that is replaced by and the function , , is assumed to be logarithmically subharmonic (cf. Logarithmically-subharmonic function) in (see , ). For example, suppose that is a subharmonic function in the angular domain (*) and does not exceed in modulus on the sides of the angle. Then the following alternative holds: Either everywhere in , or the maximum increases faster than for every , . There are also similar results for solutions of other elliptic equations. They may be called "weak" theorems of Phragmén–Lindelöf type, in the sense that, on account of their weak restriction only on the function itself on the boundary, one obtains a fairly weak assertion about its growth.

In other results, which may be called "strong" theorems of Phragmén–Lindelöf type, on account of the restriction on the function itself and its normal derivative on the boundary, one obtains a stronger assertion about its growth. An example is the following statement for the cylindrical domain in . Suppose that is a harmonic function in the cylinder and on its lateral surface , with and on . Then either everywhere in , or the maximum increases, as , faster than for any (see ).

How to Cite This Entry:
Phragmén-Lindelöf theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Phragm%C3%A9n-Lindel%C3%B6f_theorem&oldid=23452
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article