The Nehari--Schwarz lemma and infinitesimal boundary rigidity of bounded holomorphic functions

Abstract

We survey a number of recent generalizations and sharpenings of Nehari's  extension of Schwarz' lemma  for holomorphic self--maps of the unit disk. In particular, we discuss the case of infinitely many critical points and its relation to the zero sets and invariant subspaces for Bergman spaces, as well as the case of equality at the boundary.