# Schwarz lemma

Let be a holomorphic function on the disc , with and in ; then

 (1)

If equality holds for a single , then , where is a real constant (the classical form of the Schwarz lemma). This lemma was proved by H.A. Schwarz (see [1]).

Various versions of the Schwarz lemma are known. For instance, the following invariant form of the Schwarz lemma: If a function is holomorphic in the disc and if in , then for any points ,

 (2)

where is the hyperbolic distance between two points in (see Hyperbolic metric); further, for one has

 (3)

Equality holds in (2) and (3) only if is a biholomorphic mapping of onto itself.

Inequality (3) is also called the differential form of the Schwarz lemma. Integrating this inequality leads to the following formulation of the Schwarz lemma: If the disc is transformed by a holomorphic function such that for , then the hyperbolic length of an arbitrary arc in decreases, except in the case when is a univalent conformal mapping of onto itself; in this case hyperbolic distances between points are preserved.

The principle of the hyperbolic metric (cf. Hyperbolic metric, principle of the) is a generalization of the invariant form of the Schwarz lemma to multiply-connected domains on which a hyperbolic metric can be defined. Analogues of the Schwarz lemma for holomorphic mappings in the -dimensional complex space are known (see [4]).

#### References

 [1] H.A. Schwarz, "Gesamm. math. Abhandl." , 1–2 , Springer (1890) [2] G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian) [3] R. Nevanilinna, "Analytic functions" , Springer (1970) (Translated from German) [4] B.V. Shabat, "Introduction of complex analysis" , 2 , Moscow (1976) (In Russian)