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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!

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up vote 6 down vote accepted

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

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