On pluriharmonic $$\nu $$-Bloch-type mappings and hyperbolic-harmonic mappings

Abstract

In this paper, we first establish a new version of Landau-type theorem of pluriharmonic mappings in the unit ball of $${\mathbb {R}}^{2n}$$ . Next we obtain a Bloch theorem of pluriharmonic $$\nu $$ -Bloch-type mappings. Then, we provide a necessary condition for the hyperbolic-harmonic $$\nu $$ -Bloch mappings in the unit ball of $${\mathbb {C}}^n$$ . Finally, we obtain a sufficient and necessary condition for the hyperbolic-harmonic $$\nu $$ -Bloch mappings for the case of $$0<\nu \le 1$$ , which generalizes a result of Chen et al. (Math Model Anal 18:66–79, 2012).