Given $f$ holomorphic on the unit disk, $|f(z)|\leq M$, prove $|f'(z)|\leq M(1-|z|)^{-1}$.

Asked Mar 18 '13 at 15:54

Active Mar 18 '13 at 15:54

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Problem: Let $f$ be a holomorphic function on the unit disk and $|f(z)|\leq M$ for all $z$ in the unit disk. Prove that $|f'(z)|\leq M(1-|z|)^{-1}$ for all $z$ in the unit disk.

Using a generalized version of the Schwarz lemma, I can prove that $|f'(z)|\leq M(1-|z|^2)^{-1}$ on the unit disk, however I cannot see a way to prove the inequality that is asked for. Any suggestions?

asked Mar 18 '13 at 15:54

Elise delMas

Did you try using Schwarz-Pick Lemma? – Myshkin Mar 18 '13 at 16:00
Cauchy integral formula – Ehsan M. Kermani Mar 18 '13 at 16:00
isn't this Koebe's distortion theorem ? – Albert Mar 18 '13 at 16:01
Thanks, the Cauchy integral formula did it. Don't know why I didn't think of that earlier... – Elise delMas Mar 19 '13 at 20:17

Given $f$ holomorphic on the unit disk, $|f(z)|\leq M$, prove $|f'(z)|\leq M(1-|z|)^{-1}$.

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