Hyperbolic metric, principle of the

Suppose that domains and lie in the -plane and -plane, respectively, and suppose they have at least three boundary points each; let be a holomorphic function in taking values in , and let and be the line elements of the hyperbolic metric of and at points and , respectively. The following inequality will then be true:

At any point equality holds if in , where the function maps conformally onto the disc , while the function maps conformally onto . 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 in by a more general assumption, i.e. that is an analytic function which is defined in by any one of its elements and which can be analytically continued in 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 is a rectifiable curve in , then (for the meaning of the symbols see Hyperbolic metric)

If and are two points in , then

If is a domain in , then

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 the image of the hyperbolic disc with centre at the point is contained in the hyperbolic disc with its centre at the point 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 by a regular function

in , the hyperbolic distance between the images of the points and of does not exceed the hyperbolic distance between and , and is equal to that distance only for a bilinear transformation of onto itself.

The principle of the hyperbolic metric is connected with the Lindelöf principle as follows. If the domains and have a Green function and are simply connected, both these principles are identical. If is simply connected, while 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 , defined by an inequality under the mapping , where denotes the Green function of with logarithmic pole at . 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.