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)

Comments

Schwarz ([1]) stated this result for univalent functions only. The formulation, designation and systematic use of this lemma in the general form stated above is due to C. Carathéodory [a2]. For the history of this result, see [a3], pp. 191-192.

The inequalities (2) and (3) are also known as the Schwarz–Pick lemma. Equality (2) can be written in the equivalent form

For an extensive treatment of the Schwarz lemma and its many generalizations and applications see [a1].

References

[a1]	S. Dineen, "The Schwarz lemma" , Oxford Univ. Press (1989)
[a2]	C. Carathéodory, "Untersuchungen über die konformen Abbildungen von festen und veränderlichen Gebieten" Math. Ann. , 72 (1912) pp. 107–144
[a3]	R.B. Burchel, "An introduction to classical complex analysis" , 1 , Birkhäuser (1979)
[a4]	A.I. Markushevich, "Theory of functions of a complex variable" , 1–2 , Chelsea (1977) pp. 381, Thm. 17.8 (Translated from Russian)
[a5]	L.V. Ahlfors, "Conformal invariants. Topics in geometric function theory" , McGraw-Hill (1973)
[a6]	J.-B. Garnett, "Bounded analytic functions" , Acad. Press (1981)
[a7]	W. Rudin, "Function theory in the unit ball in " , Springer (1980)