View source for Lindelöf principle

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

<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>

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.

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.

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

<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>

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

<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>

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" />.

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]].

====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>