# 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)
How to Cite This Entry:
Lindelöf principle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lindel%C3%B6f_principle&oldid=47642
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article