Namespaces
Variants
Actions

Difference between revisions of "Lindelöf principle"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (moved Lindelöf principle to Lindelof principle: ascii title)
m (tex encoded by computer)
 
(One intermediate revision by the same user not shown)
Line 1: Line 1:
A fundamental qualitative variational principle (cf. [[Variational principles (in complex function theory)|Variational principles (in complex function theory)]]) in the theory of [[Conformal mapping|conformal mapping]], discovered by E. Lindelöf [[#References|[1]]]. Suppose that two simply-connected domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l0589701.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l0589702.png" /> in the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l0589703.png" />-plane are such that their boundaries <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l0589704.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l0589705.png" />, respectively, consist of finitely many Jordans arcs, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l0589706.png" /> is contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l0589707.png" />, and suppose that the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l0589708.png" />. Suppose also that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l0589709.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897010.png" /> are functions that realise a conformal mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897012.png" />, respectively, onto the unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897013.png" /> and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897015.png" />. The Lindelöf principle states that under these conditions: 1) the inverse image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897016.png" /> of the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897018.png" />, under the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897019.png" /> lies inside the inverse image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897020.png" /> of the same domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897021.png" /> under the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897022.png" />, and their boundaries <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897024.png" /> can be in contact only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897025.png" />; 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897026.png" />, equality being possible only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897027.png" />; and 3) if there is a common point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897028.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897030.png" />, then
+
<!--
 +
l0589701.png
 +
$#A+1 = 77 n = 0
 +
$#C+1 = 77 : ~/encyclopedia/old_files/data/L058/L.0508970 Lindel\AGof principle
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
<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/l/l058/l058970/l05897031.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
equality being possible only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897032.png" />. In other words, if the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897033.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897034.png" /> is pushed inward, then: a) all level curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897035.png" />, that is, the inverse images of the circles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897036.png" />, are contracted; b) the stretching at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897037.png" /> increases; and c) the stretching at fixed points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897038.png" /> of the boundary decreases.
+
A fundamental qualitative variational principle (cf. [[Variational principles (in complex function theory)|Variational principles (in complex function theory)]]) in the theory of [[Conformal mapping|conformal mapping]], discovered by E. Lindelöf [[#References|[1]]]. Suppose that two simply-connected domains  $  D $
 +
and  $  \widetilde{D}  $
 +
in the complex  $  z $-
 +
plane are such that their boundaries  $  \Gamma $
 +
and  $  \widetilde \Gamma  $,
 +
respectively, consist of finitely many Jordans arcs,  $  \widetilde{D}  $
 +
is contained in  $  D $,  
 +
and suppose that the point  $  z _ {0} \in \widetilde{D}  \subset  D $.
 +
Suppose also that  $  w = f ( z) $
 +
and  $  w = \widetilde{f}  ( z) $
 +
are functions that realise a conformal mapping of  $  D $
 +
and  $  \widetilde{D}  $,
 +
respectively, onto the unit disc  $  \Delta = \{ {w } : {| w | < 1 } \} $
 +
and that  $  f ( z _ {0} ) = 0 $,  
 +
$  \widetilde{f}  ( z _ {0} ) = 0 $.  
 +
The Lindelöf principle states that under these conditions: 1) the inverse image  $  \widetilde{D}  _  \rho  $
 +
of the domain  $  | w | < \rho $,
 +
0 < \rho < 1 $,
 +
under the mapping  $  w = \widetilde{f}  ( z) $
 +
lies inside the inverse image  $  D _  \rho  $
 +
of the same domain  $  | w | < \rho $
 +
under the mapping  $  w = f ( z) $,
 +
and their boundaries  $  \widetilde \Gamma  _  \rho  $
 +
and  $  \Gamma _  \rho  $
 +
can be in contact only if  $  \widetilde{D}  = D $;
 +
2)  $  | \widetilde{f}  {} ^ { \prime } ( z _ {0} ) | \geq  | f ^ { \prime } ( z _ {0} ) | $,
 +
equality being possible only if  $  \widetilde{D}  = D $;  
 +
and 3) if there is a common point  $  z _ {1} $
 +
of $  \widetilde \Gamma  $
 +
and  $  \Gamma $,
 +
then
  
From the information given in the Lindelöf principle it also follows that the length of the image of an arc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897039.png" /> of the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897040.png" /> subject to indentation along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897041.png" /> never exceeds the length of the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897042.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897043.png" />), and equality holds only in case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897044.png" />. This consequence of the Lindelöf principle is also known as Montel's principle.
+
$$
 +
| \widetilde{f}  {} ^ { \prime } ( z _ {1} ) |  \leq  | f ^ { \prime } ( z _ {1} ) | ,
 +
$$
  
In the more general situation when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897046.png" /> are finitely-connected domains bounded by finitely many Jordan curves situated in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897047.png" />-plane and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897048.png" />-plane, respectively, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897049.png" /> is a meromorphic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897050.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897051.png" />, the Lindelöf principle consists of the following. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897052.png" /> is a point in the image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897053.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897056.png" /> is the set of points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897057.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897059.png" /> is the multiplicity of the zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897060.png" /> of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897062.png" /> is the [[Green function|Green function]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897063.png" /> with pole <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897064.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897065.png" /> is the Green function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897066.png" /> with pole <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897067.png" />, then the inequality
+
equality being possible only if  $  \widetilde{D}  = D $.  
 +
In other words, if the boundary  $  \Gamma $
 +
of $  D $
 +
is pushed inward, then: a) all level curves  $  \Gamma _  \rho  $,  
 +
that is, the inverse images of the circles  $  | w | = \rho $,
 +
are contracted; b) the stretching at the point  $  z _ {0} $
 +
increases; and c) the stretching at fixed points  $  z _ {1} $
 +
of the boundary decreases.
  
<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/l/l058/l058970/l05897068.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
From the information given in the Lindelöf principle it also follows that the length of the image of an arc  $  \gamma $
 +
of the boundary  $  \Gamma $
 +
subject to indentation along  $  \widetilde \gamma  $
 +
never exceeds the length of the image of  $  \widetilde \gamma  $(
 +
$  \textrm{ length }  f ( \gamma ) \leq  \textrm{ length }  \widetilde{f}  ( \widetilde \gamma  ) $),
 +
and equality holds only in case  $  \widetilde \Gamma  = \Gamma $.  
 +
This consequence of the Lindelöf principle is also known as Montel's principle.
  
holds for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897069.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897070.png" />. If equality holds in (1) for at least one point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897071.png" />, then it holds everywhere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897072.png" />. In particular, the inequality
+
In the more general situation when  $  \widetilde{D}  $
 +
and  $  D $
 +
are finitely-connected domains bounded by finitely many Jordan curves situated in the  $  z $-
 +
plane and  $  w $-
 +
plane, respectively, and  $  w = f ( z) $
 +
is a meromorphic function in  $  \widetilde{D}  $
 +
with values in  $  D $,  
 +
the Lindelöf principle consists of the following. If $  w _ {0} $
 +
is a point in the image  $  f ( \widetilde{D}  ) $
 +
of  $  \widetilde{D}  $,
 +
$  \{ z _  \nu  \} $,
 +
$  \nu = 0 , 1 \dots $
 +
is the set of points of  $  \widetilde{D}  $
 +
for which  $  f ( z _  \nu  ) = w _ {0} $,
 +
$  m _  \nu  $
 +
is the multiplicity of the zero  $  z _  \nu  $
 +
of the function  $  f ( z) - w _ {0} $,
 +
$  \widetilde{g}  ( z , z _  \nu  ) $
 +
is the [[Green function|Green function]] of  $  \widetilde{D}  $
 +
with pole  $  z _  \nu  $,
 +
and  $  g ( w , w _ {0} ) $
 +
is the Green function of  $  D $
 +
with pole  $  w _ {0} $,  
 +
then the inequality
  
<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/l/l058/l058970/l05897073.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{1 }
 +
g ( w , w _ {0} )  \geq  \sum _ {\nu = 0 } ^  \infty 
 +
m _  \nu  \widetilde{g}  ( z , z _  \nu  )
 +
$$
  
which follows from (1), was obtained by Lindelöf in [[#References|[1]]]. It implies that the image of the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897074.png" /> always lies inside the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897075.png" />.
+
holds for all  $  z \in \widetilde{D}  $,
 +
$  w= f ( z) $.
 +
If equality holds in (1) for at least one point  $  z \in \widetilde{D}  $,  
 +
then it holds everywhere in $  \widetilde{D}  $.  
 +
In particular, the inequality
  
The Lindelöf principle in the general form (1) can be applied to arbitrary domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897076.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058970/l05897077.png" />, but the conclusion concerning equality is not generally true here. The Lindelöf principle makes it possible to obtain many quantitative estimates for the variation of a conformal mapping under variation of the domain (see [[#References|[3]]]). It is closely connected with the [[Subordination principle|subordination principle]], and it can also be regarded as a generalization of the [[Schwarz lemma|Schwarz lemma]].
+
$$ \tag{2 }
 +
g ( w , w _ {0} )  \geq  \widetilde{g}  ( z , z _ {0} ) ,
 +
$$
 +
 
 +
which follows from (1), was obtained by Lindelöf in [[#References|[1]]]. It implies that the image of the domain  $  \{ {z } : {\widetilde{g}  ( z , z _ {0} ) > \lambda } \} $
 +
always lies inside the domain  $  \{ {w } : {g ( w , w _ {0} ) > \lambda } \} $.
 +
 
 +
The Lindelöf principle in the general form (1) can be applied to arbitrary domains $  \widetilde{D}  $
 +
and $  D $,  
 +
but the conclusion concerning equality is not generally true here. The Lindelöf principle makes it possible to obtain many quantitative estimates for the variation of a conformal mapping under variation of the domain (see [[#References|[3]]]). It is closely connected with the [[Subordination principle|subordination principle]], and it can also be regarded as a generalization of the [[Schwarz lemma|Schwarz lemma]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Lindelöf,  "Mémoire sur certaines inegualités dans la théorie des fonctions monogènes et sur quelques propriétes nouvelles de ces fonctions dans la voisinage d'un point singulier essentiel"  ''Acta. Soc. Sci. Fennica'' , '''35''' :  7  (1909)  pp. 1–35</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.M. Goluzin,  "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.A. Lavrent'ev,  B.V. Shabat,  "Methoden der komplexen Funktionentheorie" , Deutsch. Verlag Wissenschaft.  (1967)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</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">[5]</TD> <TD valign="top">  M.A. Lavrent'ev,  "Variational methods for boundary value problems for systems of elliptic equations" , Noordhoff  (1963)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Lindelöf,  "Mémoire sur certaines inegualités dans la théorie des fonctions monogènes et sur quelques propriétes nouvelles de ces fonctions dans la voisinage d'un point singulier essentiel"  ''Acta. Soc. Sci. Fennica'' , '''35''' :  7  (1909)  pp. 1–35</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.M. Goluzin,  "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.A. Lavrent'ev,  B.V. Shabat,  "Methoden der komplexen Funktionentheorie" , Deutsch. Verlag Wissenschaft.  (1967)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</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">[5]</TD> <TD valign="top">  M.A. Lavrent'ev,  "Variational methods for boundary value problems for systems of elliptic equations" , Noordhoff  (1963)  (Translated from Russian)</TD></TR></table>

Latest revision as of 22:16, 5 June 2020


A fundamental qualitative variational principle (cf. Variational principles (in complex function theory)) in the theory of conformal mapping, discovered by E. Lindelöf [1]. Suppose that two simply-connected domains $ D $ and $ \widetilde{D} $ in the complex $ z $- plane are such that their boundaries $ \Gamma $ and $ \widetilde \Gamma $, respectively, consist of finitely many Jordans arcs, $ \widetilde{D} $ is contained in $ D $, and suppose that the point $ z _ {0} \in \widetilde{D} \subset D $. Suppose also that $ w = f ( z) $ and $ w = \widetilde{f} ( z) $ are functions that realise a conformal mapping of $ D $ and $ \widetilde{D} $, respectively, onto the unit disc $ \Delta = \{ {w } : {| w | < 1 } \} $ and that $ f ( z _ {0} ) = 0 $, $ \widetilde{f} ( z _ {0} ) = 0 $. The Lindelöf principle states that under these conditions: 1) the inverse image $ \widetilde{D} _ \rho $ of the domain $ | w | < \rho $, $ 0 < \rho < 1 $, under the mapping $ w = \widetilde{f} ( z) $ lies inside the inverse image $ D _ \rho $ of the same domain $ | w | < \rho $ under the mapping $ w = f ( z) $, and their boundaries $ \widetilde \Gamma _ \rho $ and $ \Gamma _ \rho $ can be in contact only if $ \widetilde{D} = D $; 2) $ | \widetilde{f} {} ^ { \prime } ( z _ {0} ) | \geq | f ^ { \prime } ( z _ {0} ) | $, equality being possible only if $ \widetilde{D} = D $; and 3) if there is a common point $ z _ {1} $ of $ \widetilde \Gamma $ and $ \Gamma $, then

$$ | \widetilde{f} {} ^ { \prime } ( z _ {1} ) | \leq | f ^ { \prime } ( z _ {1} ) | , $$

equality being possible only if $ \widetilde{D} = D $. In other words, if the boundary $ \Gamma $ of $ D $ is pushed inward, then: a) all level curves $ \Gamma _ \rho $, that is, the inverse images of the circles $ | w | = \rho $, are contracted; b) the stretching at the point $ z _ {0} $ increases; and c) the stretching at fixed points $ z _ {1} $ of the boundary decreases.

From the information given in the Lindelöf principle it also follows that the length of the image of an arc $ \gamma $ of the boundary $ \Gamma $ subject to indentation along $ \widetilde \gamma $ never exceeds the length of the image of $ \widetilde \gamma $( $ \textrm{ length } f ( \gamma ) \leq \textrm{ length } \widetilde{f} ( \widetilde \gamma ) $), and equality holds only in case $ \widetilde \Gamma = \Gamma $. This consequence of the Lindelöf principle is also known as Montel's principle.

In the more general situation when $ \widetilde{D} $ and $ D $ are finitely-connected domains bounded by finitely many Jordan curves situated in the $ z $- plane and $ w $- plane, respectively, and $ w = f ( z) $ is a meromorphic function in $ \widetilde{D} $ with values in $ D $, the Lindelöf principle consists of the following. If $ w _ {0} $ is a point in the image $ f ( \widetilde{D} ) $ of $ \widetilde{D} $, $ \{ z _ \nu \} $, $ \nu = 0 , 1 \dots $ is the set of points of $ \widetilde{D} $ for which $ f ( z _ \nu ) = w _ {0} $, $ m _ \nu $ is the multiplicity of the zero $ z _ \nu $ of the function $ f ( z) - w _ {0} $, $ \widetilde{g} ( z , z _ \nu ) $ is the Green function of $ \widetilde{D} $ with pole $ z _ \nu $, and $ g ( w , w _ {0} ) $ is the Green function of $ D $ with pole $ w _ {0} $, then the inequality

$$ \tag{1 } g ( w , w _ {0} ) \geq \sum _ {\nu = 0 } ^ \infty m _ \nu \widetilde{g} ( z , z _ \nu ) $$

holds for all $ z \in \widetilde{D} $, $ w= f ( z) $. If equality holds in (1) for at least one point $ z \in \widetilde{D} $, then it holds everywhere in $ \widetilde{D} $. In particular, the inequality

$$ \tag{2 } g ( w , w _ {0} ) \geq \widetilde{g} ( z , z _ {0} ) , $$

which follows from (1), was obtained by Lindelöf in [1]. It implies that the image of the domain $ \{ {z } : {\widetilde{g} ( z , z _ {0} ) > \lambda } \} $ always lies inside the domain $ \{ {w } : {g ( w , w _ {0} ) > \lambda } \} $.

The Lindelöf principle in the general form (1) can be applied to arbitrary domains $ \widetilde{D} $ and $ D $, but the conclusion concerning equality is not generally true here. The Lindelöf principle makes it possible to obtain many quantitative estimates for the variation of a conformal mapping under variation of the domain (see [3]). It is closely connected with the subordination principle, and it can also be regarded as a generalization of the Schwarz lemma.

References

[1] E. Lindelöf, "Mémoire sur certaines inegualités dans la théorie des fonctions monogènes et sur quelques propriétes nouvelles de ces fonctions dans la voisinage d'un point singulier essentiel" Acta. Soc. Sci. Fennica , 35 : 7 (1909) pp. 1–35
[2] G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)
[3] M.A. Lavrent'ev, B.V. Shabat, "Methoden der komplexen Funktionentheorie" , Deutsch. Verlag Wissenschaft. (1967) (Translated from Russian)
[4] S. Stoilov, "The theory of functions of a complex variable" , 1–2 , Moscow (1962) (In Russian; translated from Rumanian)
[5] M.A. Lavrent'ev, "Variational methods for boundary value problems for systems of elliptic equations" , Noordhoff (1963) (Translated from Russian)
How to Cite This Entry:
Lindelöf principle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lindel%C3%B6f_principle&oldid=22753
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article