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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  $  f ( z) $
be a regular analytic function of a complex variable  $  z $
in a domain  $  D $
of the plane  $  \mathbf C $
with boundary  $  \Gamma $. 
One says that  $  f ( z) $
does not exceed a number  $  M $
in modulus at a boundary point  $  \zeta \in \Gamma $
if

$$ 
\lim\limits _ {z \rightarrow \zeta }  \sup _ {z \in D } \ 
| f ( z) |  \leq   M ,
$$

that is, for every  $  \epsilon > 0 $
there is a disc  $  \Delta $(
depending on  $  \zeta $
and  $  \epsilon $) 
with centre  $  \zeta $
such that  $  | f ( z) | < M + \epsilon $
for  $  z \in D \cap \Delta $. 
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  $  f ( z) $
exceeds  $  M $
in modulus nowhere on  $  \Gamma $, 
then  $  | f ( z) | \leq  M $
everywhere in  $  D $. 
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  $  f ( z) $
does not exceed  $  M $
in modulus at any point of  $  \Gamma $
not belonging to some set  $  E \subset  \Gamma $. 
Suppose also that there is a function  $  \omega ( z) $
with the following properties: a)  $  \omega ( z) $
is regular in  $  D $; 
b)  $  | \omega ( z) | < 1 $
in  $  D $; 
c)  $  \omega ( z) \neq 0 $
in  $  D $; 
and d) for every  $  \sigma > 0 $
the function  $  | \omega ( z) |  ^  \sigma  | f ( z) | $
does not exceed  $  M $
in modulus at any point  $  \zeta \in E $. 
Under these conditions  $  | f ( z) | \leq  M $
everywhere in  $  D $.

The Phragmén–Lindelöf theorem has received numerous applications, also often called Phragmén–Lindelöf theorems, and associated with a concrete form of  $  D $, 
$  E $
and  $  \omega ( z) $(
see [[#References|[1]]]–[[#References|[4]]], in particular the generalization given in [[#References|[4]]]). In applications  $  E $
most often consists of the single point  $  \infty $. 
For example, suppose that  $  f ( z) $
is regular in the angular domain

$$ \tag{* }
D  =  \left \{ {z  =  r e ^ {i \phi } } : {| \phi | < \lambda 
\frac \pi {2}
 ,\ 
\lambda > 0 , 0 < r < \infty } \right \}
$$

and does not exceed  $  M $
in modulus on the sides of the angle. Then the following alternative holds: Either

$$ 
| f ( z) |  \leq   M
$$

everywhere in  $  D $, 
or the maximum modulus

$$ 
M ( r)  =  \max  \{ {| f ( z) | } : {| z | = r , z \in D } \}
$$

increases faster than  $   \mathop{\rm exp} ( r  ^ {k} ) $
as  $  r \rightarrow \infty $
for any  $  k $, 
$  0 < k < 1 / \lambda $. 
This theorem is obtained from propositions 1 and 2 for  $  \zeta = \infty $, 
$  \omega ( z) =  \mathop{\rm exp} ( - z ^ {k  ^  \prime  } ) $, 
where  $  k < k  ^  \prime  < 1 / \lambda $.

The statements of 1 and 2 remain valid for a holomorphic function  $  f ( z) $, 
$  z = ( z _ {1} \dots z _ {n} ) $, 
given in a domain  $  D $
of the complex space  $  \mathbf C  ^ {n} $, 
$  n \geq  1 $. 
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]]  $  u ( P) $
defined in a domain  $  D $
of a Euclidean space  $  \mathbf R  ^ {n} $, 
$  n \geq  2 $, 
or  $  \mathbf C  ^ {n} $, 
$  n \geq  1 $, 
provided that  $  | f ( z) | $
is replaced by  $  u( P) $
and the function  $  \omega $, 
$  0 < \omega < 1 $, 
is assumed to be logarithmically subharmonic (cf. [[Logarithmically-subharmonic function|Logarithmically-subharmonic function]]) in  $  D $(
see [[#References|[5]]], [[#References|[6]]]). For example, suppose that  $  u ( z) $
is a subharmonic function in the angular domain (*) and does not exceed  $  M $
in modulus on the sides of the angle. Then the following alternative holds: Either  $  u ( z) \leq  M $
everywhere in  $  D $, 
or the maximum

$$ 
M ( r)  =  \max  \{ {u ( z) } : {| z | = r , z \in D } \}
$$

increases faster than  $  r  ^ {k} $
for every  $  k $, 
$  0 < k < 1 / \lambda $. 
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

$$ 
D  =  \{ {( \rho , \phi , t ) } : {0 \leq  \rho < a , 0 \leq  \phi
< 2 \pi , | t | < \infty } \}
$$

in  $  \mathbf R  ^ {3} $. 
Suppose that  $  u ( P) $
is a harmonic function in the cylinder  $  D $
and on its lateral surface  $  \Gamma $, 
with  $  | u ( P) | \leq  M $
and  $  | \partial  u / \partial  n | \leq  M $
on  $  \Gamma $. 
Then either  $  | u ( P) | \leq  M $
everywhere in  $  D $, 
or the maximum

$$ 
M ( t)  =  \max  \{ {u ( \rho , \phi , t ) } : {0 \leq  \rho < a , 0 \leq 
\phi < 2 \pi } \}
$$

increases, as  $  | t | \rightarrow \infty $, 
faster than

$$ 
c \cdot  \mathop{\rm exp}
\left (

\frac{\pi  | t | }{e ^ {2 ( a + \epsilon ) } }

\right ) ,
$$

for any  $  \epsilon , c > 0 $(
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  $  \mathbf R  ^ {n} $
see [[#References|[a3]]].

Theorems of Phragmén–Lindelöf type are known also for parabolic equations. For instance, if  $  u( x, t) $
solves the [[Heat equation|heat equation]]  $  \Delta u - u _ {t} = 0 $
in the half-space  $  t > 0 $
and is continuous for  $  t = 0 $, 
then  $  | u( x, 0) | \leq  M $
implies that  $  | u( x, t) | \leq  M $
for all  $  ( x  ,t) $
in the strip  $  \mathbf R  ^ {n} \times ( 0, T) $, 
provided  $  u $
satisfies the growth condition  $  | u( x  ,t) | \leq  \alpha   \mathop{\rm exp} [ k x  ^ {2} ] $
for certain positive constants  $  \alpha $, 
$  k $
uniformly with respect to  $  t \in ( 0, T) $. 
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>