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

TL;DR

This paper surveys recent advances in the Nehari-Schwarz lemma, exploring its generalizations, boundary cases, and implications for zero sets and invariant subspaces in Bergman spaces, highlighting new boundary rigidity results.

Contribution

It provides a comprehensive overview of recent generalizations and sharpenings of the Nehari-Schwarz lemma, including boundary equality and critical point analysis.

Findings

Extended Nehari's lemma to cases with infinitely many critical points

Connected boundary equality cases to boundary rigidity of holomorphic functions

Linked zero sets and invariant subspaces in Bergman spaces to Nehari's extensions

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.