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I'm really struggling with möbius transformations. If $\mathbb{D}$ is the unit disc, $R >0$, $c \in \mathbb{C}$, $D_R(c) = \{z \in \mathbb{C}: |z-c| < R\}$ and $w_0 \in D_R(c)$, I want to find a biholomorphic map $g$ from $\mathbb{D}$ to $D_R(c)$ such that $g(0) = w_0$ and $g'(0) < 0.$ I also have the function $f = Rz+c.$

So $f^{-1}(w_0) = \frac{1}{R}(w_0 - c)$ which I denote by $\alpha.$ Then I let $g = f \circ \psi,$ where $\psi$ is the möbius transformation interchanging $0$ and $\alpha$. Then $g(0) = f(\alpha) = w_0,$ which was the first condition.

But how do I show that $g'(0) = f'(\alpha) \psi'(0) < 0$?

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    You may need a rotation in addition to $\psi$ and $f$. – 2017-01-03
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    To map a circle to a circle via the mobius transform it can be useful to look into the concept of cross ratios. – 2017-01-03

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