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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 [[#References|[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|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|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 [[#References|[4]]]).

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

====Comments====
Schwarz ([[#References|[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 [[#References|[a2]]]. For the history of this result, see [[#References|[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 [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Dineen,   "The Schwarz lemma" , Oxford Univ. Press  (1989)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C. Carathéodory,   "Untersuchungen über die konformen Abbildungen von festen und veränderlichen Gebieten"  ''Math. Ann.'' , '''72'''  (1912)  pp. 107–144</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R.B. Burchel,   "An introduction to classical complex analysis" , '''1''' , Birkhäuser  (1979)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A.I. Markushevich,   "Theory of functions of a complex variable" , '''1–2''' , Chelsea  (1977)  pp. 381, Thm. 17.8  (Translated from Russian)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  L.V. Ahlfors,   "Conformal invariants. Topics in geometric function theory" , McGraw-Hill  (1973)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J.-B. Garnett,   "Bounded analytic functions" , Acad. Press  (1981)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  W. Rudin,   "Function theory in the unit ball in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083560/s08356033.png" />" , Springer  (1980)</TD></TR></table>