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 and in the complex -plane are such that their boundaries and , respectively, consist of finitely many Jordans arcs, is contained in , and suppose that the point . Suppose also that and are functions that realise a conformal mapping of and , respectively, onto the unit disc and that , . The Lindelöf principle states that under these conditions: 1) the inverse image of the domain , , under the mapping lies inside the inverse image of the same domain under the mapping , and their boundaries and can be in contact only if ; 2) , equality being possible only if ; and 3) if there is a common point of and , then

equality being possible only if . In other words, if the boundary of is pushed inward, then: a) all level curves , that is, the inverse images of the circles , are contracted; b) the stretching at the point increases; and c) the stretching at fixed points of the boundary decreases.

From the information given in the Lindelöf principle it also follows that the length of the image of an arc of the boundary subject to indentation along never exceeds the length of the image of (), and equality holds only in case . This consequence of the Lindelöf principle is also known as Montel's principle.

In the more general situation when and are finitely-connected domains bounded by finitely many Jordan curves situated in the -plane and -plane, respectively, and is a meromorphic function in with values in , the Lindelöf principle consists of the following. If is a point in the image of , , is the set of points of for which , is the multiplicity of the zero of the function , is the Green function of with pole , and is the Green function of with pole , then the inequality

(1)

holds for all , . If equality holds in (1) for at least one point , then it holds everywhere in . In particular, the inequality

(2)

which follows from (1), was obtained by Lindelöf in [1]. It implies that the image of the domain always lies inside the domain .

The Lindelöf principle in the general form (1) can be applied to arbitrary domains and , 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)