Schwarz's lemma asserts that analytic mappings from
the unit disc into itself decrease hyperbolic
distances. In this paper, inner functions which
decrease hyperbolic distances as much as possible,
when one approaches the unit circle, are constructed.
Actually, it is shown that a quadratic condition
governs the best decay of the hyperbolic derivative
of an inner function. This is related to a result
of L. Carleson on the existence of singular symmetric
measures. As a consequence, some results on composition
operators are obtained, bringing out the importance
of the Bloch spaces in this connection. Another
consequence is a uniform way of producing singular
measures which are simultaneously symmetric and Kahane.
1991 Mathematics Subject Classification: primary 30D50; secondary 30D45, 26A30, 47B38.