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

In this article, we characterize the holomorphic mappings from $B_{\ell_p^n}\times\mathbb{D}^{m}$ into $\mathbb{D}^{m}$ for $p\in \{2,\infty\}$. 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 $B_{\ell_p^n}$, $p \in [2,\infty]$. Furthermore, we establish the boundary rigidity theorem for holomorphic self-mappings of $B_{\ell_p^n}$, $p \in (1,\infty)$.