Schwarz-Pick type inequalities from an operator theoretical point of view

Abstract

We use (versions of) the von Neumann inequality for Hilbert space contractions to prove several Schwarz-Pick inequalities. Specifically, we derive an alternate proof for a multi-point Schwarz-Pick inequality by Beardon and Minda, along with a generalized version for operators. Connections with model spaces and Peschl's invariant derivatives are established. Finally, Schwarz-Pick inequalities for analytic functions on polydisks and for higher order derivatives are discussed. An enhanced version of the Schwarz-Pick lemma, using the notion of distinguished variety, is obtained for the bidisk.

arXiv comment29 pages. Minor modifications. This version is close to the published one

References in corpus (5)

• A Schwarz lemma on the polydisk
• Nevanlinna-Pick interpolation on distinguished varieties in the bidisk
• Crouzeix's Conjecture and related problems
• Eigenvalue estimates for the resolvent of a non-normal matrix
• The von Neumann inequality for 3x3 matrices

Cited by in corpus (1)

• Contractive upper triangular matrices with prescribed diagonals and superdiagonals