I am trying to prove the following problem from An Introduction to Complex Function Theory by Palka.

If $f$ is analytic in the unit disk $D(0,1)$, and if $f(0)=f'(0)=\ldots= f^{(k)}(0)=0$, and $|f^{(k)}(z)|\leq 1$ for every $z\in D(0,1)$, then show that $|f(z)|\leq |z|^{k+1}/(k+1)!$ in the disk.

I tried to use Schwarz lemma or maximum modulus principle, but what I get is something like $|f(z)|\leq |z|^{k+1}$, and I can't seem to use all the information given.



  1. By Schwarz Lemma, $|f^{(k)}(z)|\le|z|$.
  2. Using $g(z)-g(0)= \int_0^zg'(w)dw$ for any holomorphic $g$ defined on $D(0,1)$ or not, prove that $|f^{(k-j)}(z)|\le \frac{|z|^{j+1}}{(j+1)!}$ for $j\le k$ by induction on $j$.
| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.