On the Boundary Schwarz lemma and the rigidity theorem for certain mappings

In this article, we characterize the holomorphic mappings from $B_{{\mathrm{\ell }}_p^n}\times {\mathbb{D}}^m$ into ${\mathbb{D}}^m$ for $p\in \{2,\mathrm{\infty }\}$. In addition, we give a simple proof for the boundary Schwarz lemma for vector valued holomorphic functions, which also extends the existing result. Also, we obtain the boundary Schwarz lemma for pluriharmonic self-mappings of the unit ball $B_{{\mathrm{\ell }}_p^n}$, $p\in [2,\mathrm{\infty }]$. Furthermore, we establish the boundary rigidity theorem for holomorphic self-mappings of $B_{{\mathrm{\ell }}_p^n}$, $p\in (1,\mathrm{\infty })$.