Hyperbolic metric, principle of the

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 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)

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

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 (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 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

[a1]	L.V. Ahlfors, "Conformal invariants. Topics in geometric function theory" , McGraw-Hill (1973)