one-one analytic function don't have vanishing derivative I'm not sure,is it possible a one-one analytic function on a simply connected region has a vanishing first order derivative at some point.I find a counter example in the real case,$y=x^3$,but I did't find a counter example in the complex case.

As a further hint in addition to the comments, consider the set $z^n = \operatorname{Re} z^n$.
It looks locally like $n$ lines intersecting each other. This can be made into a proof that conformal maps $D \rightarrow D$ must have non-vanishing derivative. As you know that is not sufficient though, consider the exponential function on $D = \mathbb{C}$.
I am sure that weird things can happen at the boundary, but domain implies open by definition, or not?
