Abstract :

A Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex analysis about holomorphic functions from the open unit disk to itself. The lemma is less celebrated than stronger theorems, such as the Riemann mapping theorem, which it helps to prove. It is, however, one of the simplest results capturing the rigidity of holomorphic functions. A variant of the Schwarz lemma, known as the Schwarz–Pick theorem (named after Georg pick), characterizes the analytic automorphisms of the unit disc. In this lecture, we shall discuss Schwarz lemma and Schwarz-Pick lemma in several complex variable cases and we shall characterize all bijective holomorphic functions from the unit ball (in n-dimensional complex Hilbert space) to itself.