Pluriharmonic mappings in CnCn and complex Banach spaces

In this paper, we obtain a sufficient condition for pluriharmonic mappings on the Euclidean unit ball BnBn to be univalent, sense-preserving, quasiconformal and bi-Lipschitz diffeomorphisms on BnBn and to have linearly connected images. Also, we give a sufficient condition for pluriharmonic mappings on BnBn to have quasiconformal extensions to CnCn. Next, we generalize the harmonic Schwarz lemma to pluriharmonic mappings of the unit ball BXBX of a complex Banach space X   into the unit ball BnBn of CnCn with respect to an arbitrary norm. Further, we obtain a generalization of the harmonic Schwarz–Pick lemma to the case of pluriharmonic mappings of the homogeneous unit ball BXBX of a complex Banach space X   into the unit ball BnBn. We also obtain a version of the holomorphic Schwarz–Pick lemma for the Jacobian determinant on the Euclidean unit ball BnBn to the case of pluriharmonic mappings of the homogeneous unit ball BXBX into BnBn, in the case that BXBX is an open subset of CnCn. Finally, we obtain the Landau and the Bloch theorems for pluriharmonic or holomorphic mappings on finite dimensional homogeneous unit balls.