Let f be a twice continuously differentiable self-mapping of a unit disk satisfying Poisson differential inequality ∣Δf(z)∣ ≤ B · ∣Df (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

$$\frac{1-\vert z\vert^{2}}{1-\vert f(z)\vert^{2}}\leq\ C(B),$$

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.

中文翻译：

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满足Poisson微分不等式的映射的Schwarz-Pick型不等式

让˚F是满足泊松微分不等式|Δ一个单元盘的两次连续可微自映射˚F（ż）|≤乙·| Df的（ż）| ²对于某些B> 0和˚F（0）= 0。在此注意，我们证明f并不总是满足Schwarz-Pick型不等式

$$ \ frac {1- \ vert z \ vert ^ {2}} {1- \ vert f（z）\ vert ^ {2}} \ leq \ C（B），$$

其中Ç（乙）是仅依赖于一个恒定乙。此外，在某些条件下，可以建立满足一般Poisson微分不等式的更一般的Schwarz-Pick型不等式。