The Boundary Schwarz lemma for harmonic and pluriharmonic mappings and some generalisations

Abstract

We improve and generalize the classical Schwarz lemmas for planar harmonic mappings into the sharp forms for harmonic mappings between finite dimensional Euclidean unit ball and the unit ball in Hilbert space, and present 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 addition using Burget's estimate we establish the sharp boundary Schwarz type lemmas for harmonic mappings between finite dimensional balls.