Conformal map betwen unit disk and simply connected subset of $\mathbb{C}$ with positive derivative at $a$

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?

Note that certainly $\phi(a)\ne 0$. Hence appending a rotation solves the problem.
• I think you mean $\phi'(a)\neq 0$. – mrf Jun 12 '15 at 22:06