Showing the inequality $\frac{|f^{'}(z)|}{1-|f(z)|^{2}} \leq \frac{1}{1-|z|^{2}}$ 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!

That's the Schwarz–Pick theorem. The wikipedia page contains a proof.
