Some inequalities for self-mappings of unit ball satisfying the invariant Laplacians

In this paper, we study those mappings in unit ball satisfying the Dirichlet problem of the following differential operators

$$\begin{aligned} \Delta _{\gamma }=\big (1-|x|^{2}\big )\cdot \left[ \frac{1-|x|^{2}}{4}\cdot \sum _{i}\frac{\partial ^{2}}{\partial x_{i}^{2}}+\gamma \sum _{i}x_{i}\cdot \frac{\partial }{\partial x_{i}}+\gamma \left( \frac{n}{2}-1-\gamma \right) \right] . \end{aligned}$$

Our aim is to establish the Schwarz type inequality, Heinz-Schwarz type inequality and boundary Schwarz inequality for those mappings.