The Schwarz lemma at the boundary
Steven G. Krantz
TL;DR
This paper explores boundary versions of the Schwarz lemma for holomorphic functions, providing estimates on derivatives and reviewing recent advances along with new theorems.
Contribution
It introduces new boundary Schwarz lemmas and derivative estimates, extending classical results to boundary points of general domains.
Findings
Derived new boundary Schwarz lemmas.
Established derivative estimates at boundary points.
Reviewed recent developments in boundary Schwarz lemmas.
Abstract
The most classical version of the Schwarz lemma involves the behavior at the origin of a bounded, holomorphic function on the disc. Pick's version of the Schwarz lemma allows one to move the origin to other points of the disc. In the present paper we explore versions of the Schwarz lemma at a boundary point of a domain (not just the disc). Estimates on derivatives of the function, and other types of estimates as well, are considered. We review recent results of several authors, and present some new theorems as well.
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Taxonomy
TopicsHolomorphic and Operator Theory · Analytic and geometric function theory · Geometric and Algebraic Topology