Let $f$ be analytic on the unit disc $D$ and bounded in modulus by $M$ there. I want to show that $|f'(z)|\le \frac{M}{1-|z|}$ for all $z\in D$.
I want to use Schwarz's lemma here after some suitable FLTs, as in the proof of Pick's lemma, but I haven't made progress. Does anyone have an idea?
Edited: Forgot a factor of $M$ in the inequality originally.