The Boundary Schwarz Lemma for Harmonic and Pluriharmonic Mappings and Some Generalizations

We use the improvement of the classical Schwarz lemmas for planar harmonic mappings into the sharp form, in order to provide some applications to sharp boundary Schwarz type lemmas for holomorphic and in particular pluriharmonic mappings between the unit balls in Hilbert and Banach spaces. In the second part of this article, using Burget’s estimate we establish the sharp boundary Schwarz type lemmas for harmonic mappings between finite dimensional balls. Since here we do not suppose in general that maps fix the origin this is a generalization of the result, previously established by David Kalaj, for harmonic functions. At the end of this section, we derived some interesting conclusion considering hyperbolic-harmonic functions in the unit ball, which shows that Hopf’s lemma is not applicable for those functions.