Lindelöf principle

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)