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Can anyone help me to find a concrete example of an analytic self map $\varphi$ (not identity) of the open unit disc $\mathbb{D}\subset\mathbb{C}$ such that $$\sup\limits_{z\in\mathbb{D}}\frac{|\varphi(z)-z|}{(1-\max{\{|z|,|\varphi(z)|\}})^2}<\infty.$$

Further, how to find an example of $\varphi$ (not identity and having boundary contact $\overline{\partial\varphi(\mathbb{D}})\cap\partial\mathbb{D}\neq\emptyset$) such that $$\lim\limits_{|\varphi(z)|\rightarrow1} \frac{|\varphi(z)-z|}{(1-\max{\{|z|, |\varphi(z)|\}})^2}=0?$$

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