# Schwarz lemma

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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)
How to Cite This Entry:
Schwarz lemma. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Schwarz_lemma&oldid=48634
This article was adapted from an original article by G.V. Kuz'mina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article