View source for Phragmén-Lindelöf theorem

A generalization of the [[Maximum-modulus principle|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 [[#References|[1]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p0726301.png" /> be a regular analytic function of a complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p0726302.png" /> in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p0726303.png" /> of the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p0726304.png" /> with boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p0726305.png" />. One says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p0726306.png" /> does not exceed a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p0726307.png" /> in modulus at a boundary point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p0726308.png" /> if

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p0726309.png" /></td> </tr></table>

that is, for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263010.png" /> there is a disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263011.png" /> (depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263013.png" />) with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263014.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263015.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263016.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263017.png" /> exceeds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263018.png" /> in modulus nowhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263019.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263020.png" /> everywhere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263021.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263022.png" /> does not exceed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263023.png" /> in modulus at any point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263024.png" /> not belonging to some set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263025.png" />. Suppose also that there is a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263026.png" /> with the following properties: a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263027.png" /> is regular in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263028.png" />; b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263029.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263030.png" />; c) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263031.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263032.png" />; and d) for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263033.png" /> the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263034.png" /> does not exceed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263035.png" /> in modulus at any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263036.png" />. Under these conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263037.png" /> everywhere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263038.png" />.

The Phragmén–Lindelöf theorem has received numerous applications, also often called Phragmén–Lindelöf theorems, and associated with a concrete form of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263041.png" /> (see [[#References|[1]]]–[[#References|[4]]], in particular the generalization given in [[#References|[4]]]). In applications <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263042.png" /> most often consists of the single point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263043.png" />. For example, suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263044.png" /> is regular in the angular domain

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263045.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

and does not exceed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263046.png" /> in modulus on the sides of the angle. Then the following alternative holds: Either

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263047.png" /></td> </tr></table>

everywhere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263048.png" />, or the maximum modulus

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263049.png" /></td> </tr></table>

increases faster than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263050.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263051.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263053.png" />. This theorem is obtained from propositions 1 and 2 for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263055.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263056.png" />.

The statements of 1 and 2 remain valid for a holomorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263057.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263058.png" />, given in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263059.png" /> of the complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263060.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263061.png" />. 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|subharmonic function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263062.png" /> defined in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263063.png" /> of a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263064.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263065.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263067.png" />, provided that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263068.png" /> is replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263069.png" /> and the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263070.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263071.png" />, is assumed to be logarithmically subharmonic (cf. [[Logarithmically-subharmonic function|Logarithmically-subharmonic function]]) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263072.png" /> (see [[#References|[5]]], [[#References|[6]]]). For example, suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263073.png" /> is a subharmonic function in the angular domain (*) and does not exceed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263074.png" /> in modulus on the sides of the angle. Then the following alternative holds: Either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263075.png" /> everywhere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263076.png" />, or the maximum

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263077.png" /></td> </tr></table>

increases faster than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263078.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263079.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263080.png" />. 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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263081.png" /></td> </tr></table>

in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263082.png" />. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263083.png" /> is a harmonic function in the cylinder <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263084.png" /> and on its lateral surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263085.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263086.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263087.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263088.png" />. Then either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263089.png" /> everywhere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263090.png" />, or the maximum

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263091.png" /></td> </tr></table>

increases, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263092.png" />, faster than

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263093.png" /></td> </tr></table>

for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263094.png" /> (see [[#References|[5]]]–[[#References|[8]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Phragmén,   E. Lindelöf,   "Sur une extension d'un principe classique de l'analyse"  ''Acta Math.'' , '''31'''  (1908)  pp. 381–406</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.C. Titchmarsh,   "The theory of functions" , Oxford Univ. Press  (1979)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S. Stoilov,   "The theory of functions of a complex variable" , '''1–2''' , Moscow  (1962)  (In Russian; translated from Rumanian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M.A. Evgrafov,   "Analytic functions" , Saunders , Philadelphia  (1966)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  I.I. Privalov,   "Subharmonic functions" , Moscow-Leningrad  (1937)  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  E.D. Solomentsev,   "Harmonic and subharmonic functions and their generalizations"  ''Itogi Nauk. Mat. Anal. Teor. Veryatnost. Regulirovanie. 1962''  (1964)  pp. 83–100  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  M.A. Evgrafov,   "Generalization of the Phragmén–Lindelöf theorems for analytic functions to solutions of various elliptic systems"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''27'''  (1963)  pp. 843–854  (In Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  E.M. Landis,   "Second-order equations of elliptic and parabolic type" , Moscow  (1971)  (In Russian)</TD></TR></table>



====Comments====
For Phragmén–Lindelöf type theorems for subharmonic functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263095.png" /> see [[#References|[a3]]].

Theorems of Phragmén–Lindelöf type are known also for parabolic equations. For instance, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263096.png" /> solves the [[Heat equation|heat equation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263097.png" /> in the half-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263098.png" /> and is continuous for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p07263099.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p072630100.png" /> implies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p072630101.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p072630102.png" /> in the strip <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p072630103.png" />, provided <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p072630104.png" /> satisfies the growth condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p072630105.png" /> for certain positive constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p072630106.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p072630107.png" /> uniformly with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072630/p072630108.png" />. Disregarding the growth condition above, it is possible to find unbounded solutions with bounded initial values. A well-known example is due to A.N. Tikhonov [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.N. Tikhonov,   "Uniqueness theorems for the heat equation"  ''Mat. Sb.'' , '''42'''  (1935)  pp. 199–216  (In Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.R. Cannon,   "The one-dimensional heat equation" , Addison-Wesley  (1984)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L.I. Ronkin,   "Inroduction to the theory of entire functions of several variables" , ''Transl. Math. Monogr.'' , '''44''' , Amer. Math. Soc.  (1974)  (Translated from Russian)</TD></TR></table>