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Let $\Omega \subset \mathbb{C}$ a simply connected subset of $\mathbb{C}$ (not equal to the complex plane).

Can we find a conformal bijection between $\Omega$ and the unit Disk with:

  • $\varphi(a) =0$ and $\varphi'(a)>0$

The first condition is clear, I can compose the conformal bijection given by the Riemann mapping theorem with an automorphism of the unit disk and it's done.

But how can I make the derivative positive?

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Note that certainly $\phi(a)\ne 0$. Hence appending a rotation solves the problem.

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  • $\begingroup$ I think you mean $\phi'(a)\neq 0$. $\endgroup$ – mrf Jun 12 '15 at 22:06

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