A SHARP SCHWARZ INEQUALITY ON THE BOUNDARY

ROBERT OSSERMAN

Abstract. A number of classical results reflect the fact that if a holomorphic function maps the unit disk into itself taking the origin into the origin, and if some boundary point b maps to the boundary, then the map is a magnification at b. We prove a sharp quantitative version of this result which also sharpens a classical result of Loewner, and which implies that the map is a strict magnification at b unless it is a rotation.