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The main purpose of this paper is to develop some methods to investigate theSchwarz type lemmas of holomorphic mappings and pluriharmonic mappings inHilbert and Banach spaces. Initially, we extend the classical Schwarz lemmas ofholomorphic mappings to Hilbert and Banach spaces. Furthermore, we improve andgeneralize the classical Schwarz lemmas of planar harmonic mappings into thesharp forms of Banach spaces, and present some applications to sharp boundarySchwarz type lemmas for pluriharmonic mappings in Hilbert and Banach spaces andthe obtained results provide the improvements and generalizations of somelatest corresponding results in this subject. In the last section we studiedharmonic and hyperbolical-harmonic function mapping unit ball in Euclidialn-space, to unit ball in Euclidial m-space, taking prescribed value in theorigin, which improved some previous results.

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