Given a univalent function on the disk satisfying $f(0)=0$ and f'(0)=1, Koebe Distortion theorem says that
\begin{equation} \frac{1-|z|}{1+|z|}\le \left|z\frac{f'(z)}{f(z)}\right|\le \frac{1+|z|}{1-|z|}. \end{equation}
I'm wondering if there is some kind of a converse statement:
Given a holomorphic function on the unit disk with $f(0) = 0$ and f'(0)=1 satisfying the inequality above, what are some minimal set of conditions that would make $f$ univalent?
I will be very grateful if someone could point me in the right direction in terms of finding appropriate references.
Thank you in advance.