Consequences of the Schwarz Lemma

Abstract

The familiar classical Schwarz lemma deals with functions defined in the open unit disc U ⊂ ℂ, and asserts the following:

• (a) Iff: U → U is holomorphic, then |f′(0)| < 1, except when f(λ) = cλ for some c∈ℂ with |c| = 1.

• (b) If also f(0) = 0, then |f(λ)| < |λ| for every λ ∈ U\{0}, except when f(λ) = cλ, as in (a).