I want to show the following:
$D \subset \mathbb{C}$ is simply connected $\Leftrightarrow$ every zero-free holomorphic bijective function in $D$ has a holomorphic square root.
I already proofed '$\Rightarrow$'.
I also have shown that $D$ is simply connected $\Leftrightarrow$ $D$ is homologically-connected (every loop in $D$ is nullhomologous in $D$).
So what I would like to do is to show that:
every zero-free holomorphic bijective function in $D$ has a holomorphic square root $\Rightarrow$ $D$ is homologically-connected.
Can you help me on that?