Schwarz–Ahlfors–Pick theorem

The lemma of Schwarz -Pick (after Hermann Schwarz and Georg Alexander Pick) is a statement from the function theory of holomorphic endomorphisms of the unit circle, which generalizes the Schwarz lemma. In the context of hyperbolic geometry, it means that holomorphic endomorphisms are contractions.

Statement

Denote the unit disk and is a holomorphic function. Then for all

And for all

The second statement follows from the first by dividing by and then lets go against.

Applications

In hyperbolic geometry

The hyperbolic distance. The first inequality of the lemma of Schwarz -Pick indicates therefore that holomorphic functions with respect to this metric are contractions.

If and are employed in the first inequality, we obtain as a special case the statement of Schwarz's lemma.