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.