  Suppose that $1<p\leq\infty$ and $\varphi\in
L^{p}(\mathbb{B}^{n},\mathbb{R}^{n}).$ In this note, we use H\"{o}lder
inequality and some basic properties of hypergeometric functions to establish
the sharp constant $C_{p}$ and function $C_{p}(x)$ in the following
inequalities $$|u(x)|\leq
\frac{C_{p}}{(1-|x|^{2})^{(n-1)/p}}\cdot||\varphi||_{L^{p}}$$
  and
  $$|u(x)|\leq
\frac{C_{p}(x)}{(1-|x|^{2})^{(n-1)/p}}\cdot||\varphi||_{L^{p}},$$
  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}).
