Let the function \(\varphi \) be holomorphic in the unit disk \({\mathbb {D}}\) of the complex plane \({\mathbb {C}}\) and let \(\varphi ({\mathbb {D}})\subset {\mathbb {D}}\). We study the level sets and the critical points of the hyperbolic derivative of \(\varphi \),

$$\begin{aligned} |D_{\varphi }(z)|:=\frac{(1-|z|^2)|\varphi '(z)|}{1-|\varphi (z)|^2}. \end{aligned}$$

In particular, we show how the Schwarzian derivative of \(\varphi \) reveals the nature of the critical points.

中文翻译：

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单位圆盘的解析自映射的双曲导数的水平集

让函数\（\ varphi \）在复平面\（{\ mathbb {C}} \\）的单位磁盘\（{\ mathbb {D}} \}中是全纯的，然后让\（\ varphi（{\ mathbb {D}}）\ subset {\ mathbb {D}} \）。我们研究\（\ varphi \）的双曲导数的水平集和临界点，

$$ \ begin {aligned} | D _ {\ varphi}（z）| :: = \ frac {（1- | z | ^ 2）| \ varphi'（z）|} {1- | \ varphi（z）| ^ 2}。\ end {aligned} $$

特别是，我们展示了\（\ varphi \）的Schwarzian导数如何揭示临界点的性质。