Abstract: Let $f$ be a twice continuously differentiable self-mapping of a unit disk satisfying Poisson differential inequality $|\Delta f(z)|\leq B\cdot|D f(z)|^{2}$ for some $B>0$ and $f(0)=0.$ In this note, we show that $f$ does not always satisfy the Schwarz-Pick type inequality $$\frac{1-|z|^{2}}{1-|f(z)|^{2}}\leq C(B),$$ where $C(B)$ is a constant depending only on $B.$ Moreover, a more general Schwarz-Pick type inequality for mapping that satisfies general Poisson differential inequality is established under certain conditions.

Key words: Schwarz-Pick inequality, Poisson differential inequality, hyperbolically Lipschitz continuity