# 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 ).

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 ).

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