Optimal estimates for mappings admitting general Poisson representations in the unit ball

Suppose that $1<p\le \mathrm{\infty }$ and $\phi \in L^p({\mathbb{B}}^n,{\mathbb{R}}^n)\mathrm{.}$ In this note, we use Hölder inequality and some basic properties of hypergeometric functions to establish the sharp constant $C_p$ and function $C_p(x)$ in the following inequalities $$\mathrm{\mid }u(x)\mathrm{\mid }\le \frac{C_p}{(1-\mathrm{\mid }x{\mathrm{\mid }}^2{)}^{(n-1)\mathrm{/}p}}\cdot \mathrm{\mid }\mathrm{\mid }\phi \mathrm{\mid }{\mathrm{\mid }}_{L^p}$$ and $$\mathrm{\mid }u(x)\mathrm{\mid }\le \frac{C_p(x)}{(1-\mathrm{\mid }x{\mathrm{\mid }}^2{)}^{(n-1)\mathrm{/}p}}\cdot \mathrm{\mid }\mathrm{\mid }\phi \mathrm{\mid }{\mathrm{\mid }}_{L^p},$$ where $u$ are those mapping from the unit ball ${\mathbb{B}}^n$ into ${\mathbb{R}}^n$ admitting general Poisson representations. The obtained results generalize and extend some known results from harmonic mappings (\cite[Proposition 6.16]{ABR92} and \cite[Theorems 1.1 and 1.2]{DM12}) and hyperbolic harmonic mappings (\cite[Theorems 1.1 and 1.2]{CJLK20}).