I am trying to show if $|f(z)| \leq 1$, $|z| \leq 1$, then \begin{equation} \frac{|f^{'}(z)|}{1-|f(z)|^{2}} \leq \frac{1}{1-|z|^{2}} \end{equation}. I have used Cauchy's Inequality to derive |f^{'}(z)| \leq \frac{1}{1-|z|} yet I still couldn't get the result I need.
Also I am trying to find when equality would hold. Any tips or help would be much appreciated. Thanks!