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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.

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    Oh! I didn't realize you wrote the paper yourself! Thank you for the link :)2011-11-12

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