Schwarz lemma

Let $ f( z) $ be a holomorphic function on the disc $ E = \{ | z | < 1 \} $, with $ f( 0) = 0 $ and $ | f( z) | \leq 1 $ in $ E $; then

$$ \tag{1 } | f( z) | \leq | z | \ \textrm{ and } \ \ | f ^ { \prime } ( 0) | \leq 1 . $$

If equality holds for a single $ z \neq 0 $, then $ f( z) \equiv e ^ {i \alpha } z $, where $ \alpha $ 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 $ f( z) $ is holomorphic in the disc $ E $ and if $ | f( z) | \leq 1 $ in $ E $, then for any points $ z _ {1} , z _ {2} \in E $,

$$ \tag{2 } r _ {E} ( f( z _ {1} ), f( z _ {2} )) \leq r _ {E} ( z _ {1} , z _ {2} ), $$

where $ r _ {E} ( a, b) $ is the hyperbolic distance between two points $ a, b $ in $ E $( see Hyperbolic metric); further, for $ z \in E $ one has

$$ \tag{3 } \frac{| df( z) | }{1- | f( z) | ^ {2} } \leq \frac{| dz | }{1- | z | ^ {2} } . $$

Equality holds in (2) and (3) only if $ f( z) $ is a biholomorphic mapping of $ E $ 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 $ E $ is transformed by a holomorphic function $ f( z) $ such that $ | f( z) | < 1 $ for $ z \in E $, then the hyperbolic length of an arbitrary arc in $ E $ decreases, except in the case when $ f( z) $ is a univalent conformal mapping of $ E $ 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 $ n $- dimensional complex space $ \mathbf C ^ {n} $ 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

$$ \frac{| f ( z) - f( w ) | }{| 1- f( z) f( \overline{w)}\; | } \leq \frac{| z- w | }{| 1- z \overline{w}\; | } . $$

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)