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Suppose that domains  $  D $
and  $  G $
lie in the  $  z $-
plane and  $  w $-
plane, respectively, and suppose they have at least three boundary points each; let  $  w = f( z) $
be a holomorphic function in  $  D $
taking values in  $  G $, 
and let  $  d \sigma _ {z} $
and  $  d \sigma _ {w} $
be the line elements of the hyperbolic metric of  $  D $
and  $  G $
at points  $  z $
and  $  w = f( z) $, 
respectively. The following inequality will then be true:

$$ 
d \sigma _ {w}  \leq   d \sigma _ {z} .
$$

At any point  $  z _ {0} \in D $
equality holds if  $  f( z) \equiv w [ \zeta ( z)] $
in  $  D $, 
where the function  $  \zeta = \zeta ( z) $
maps  $  D $
conformally onto the disc  $  E = \{  \zeta  : {| \zeta | < 1 } \} $, 
while the function  $  w = w ( \zeta ) $
maps  $  E $
conformally onto  $  G $. 
The principle of the hyperbolic metric generalizes the [[Schwarz lemma|Schwarz lemma]] to multiply-connected domains in which a hyperbolic metric can be defined.

In formulating the principle of the hyperbolic metric it is permissible to replace the assumption on the analyticity of the function  $  f( z) $
in  $  D $
by a more general assumption, i.e. that  $  f( z) $
is an analytic function which is defined in  $  D $
by any one of its elements and which can be analytically continued in  $  D $
along any path.

The same principle can also be formulated about the behaviour of the hyperbolic length of curves, the hyperbolic distance or the hyperbolic area for a given mapping. In fact, if  $  L $
is a rectifiable curve in  $  D $, 
then (for the meaning of the symbols see [[Hyperbolic metric|Hyperbolic metric]])

$$ 
\mu _ {G} ( f ( L))  \leq   \mu _ {D} ( L).
$$

If  $  z _ {1} $
and  $  z _ {2} $
are two points in  $  D $, 
then

$$ 
r _ {G} ( f ( z _ {1} ), f ( z _ {2} ))  \leq  \ 
r _ {D} ( z _ {1} , z _ {2} ).
$$

If  $  B $
is a domain in  $  D $, 
then

$$ 
\Delta _ {G} ( f ( B))  \leq   \Delta _ {D} ( B).
$$

Equality in these inequalities holds only in the above-mentioned case.

The above result as applied to the hyperbolic distance shows that under the mapping  $  w = f( z) $
the image of the hyperbolic disc with centre at the point  $  z _ {0} \in D $
is contained in the hyperbolic disc with its centre at the point  $  w _ {0} = f( z _ {0} ) $
of the same hyperbolic radius.

This result is a generalization to the case of multiply-connected domains of the following fact in the theory of [[Conformal mapping|conformal mapping]] (the invariant form of Schwarz' lemma): Under the mapping of the disc  $  E $
by a regular function

$$ 
w  =  f ( z),\ \ 
| f ( z) |  <  1
$$

in  $  E $, 
the hyperbolic distance between the images of the points  $  z _ {1} $
and  $  z _ {2} $
of  $  E $
does not exceed the hyperbolic distance between  $  z _ {1} $
and  $  z _ {2} $, 
and is equal to that distance only for a bilinear transformation of  $  E $
onto itself.

The principle of the hyperbolic metric is connected with the [[Lindelöf principle|Lindelöf principle]] as follows. If the domains  $  D $
and  $  G $
have a Green function and are simply connected, both these principles are identical. If  $  D $
is simply connected, while  $  G $
is multiply connected, the principle of the hyperbolic metric yields a more precise estimate of the domain containing the image of a hyperbolic disc in  $  D $, 
defined by an inequality  $  g _ {D} ( z, z _ {0} ) > \lambda $
under the mapping  $  w = f( z) $, 
where  $  g _ {D} ( z, z _ {0} ) $
denotes the Green function of  $  D $
with logarithmic pole at  $  z _ {0} \in D $. 
The principle of the hyperbolic metric is also applicable to cases in which Lindelöf's principle does not apply — e.g. to domains having at least three boundary points but not having a Green function.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  L.V. Ahlfors,   "Conformal invariants. Topics in geometric function theory" , McGraw-Hill  (1973)</TD></TR>
</table>