Derivative of an univalent function I know that the derivative of a univalent function cannot be zero. But I am wondering if a function has non zero derivative, then is it univalent? Under what conditions the converse is true?

Let $f$ analytic in the unit disc and locally univalent ($f'(z) \
e 0$ anywhere); then consider the Schwarzian derivative $S(f)(z)= \frac{d}{dz}({\frac{f''(z)}{f'(z)}})-\frac{1}{2}(\frac{f''(z)}{f'(z)})^2$.

Nehari's theorem says that if $|S(f)(z)| \le 2(1-|z|^2)^{-2}$ for all $|z|<1$, then $f$ is univalent and $2$ is best constant for which the result holds (there are counterexamples for all constants bigger than $2$)

(Conversely, it is known that for all univalent functions in the disc, the inequality above is satisfied with $6$ instead of $2$ and again that is best possible)