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Define $\mathcal{F}$ to be the family of holomorphic functions which map the open unit disc to itself and which together with their first three derivatives vanish at $0$. Find $\sup_{f \in \mathcal{F}} |f(1/2)|$.

One direct estimate can be obtained from the Schwarz's lemma as $|f(z)| < |z|$ with no equality as the functions cannot be rotations. But how to use the information about the order of the zero at the origin to refine the estimate?

For example the general form of such a function would be $f(z) = z^4 g(z)$ where $g(Z)$ is a holomorphic function in the unit disc satisfying the following - $g(0) \neq 0$, $|g(z)|<1$ when $|z|=1$. From the maximum modulus principle for $g(z)$, we can conclude that $|g|<1$ for all $z \in \mathbb{D}$, which implies that $|f|<|z|^4$. Am I missing something here?

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You are correct that $|f(z)| \le |z|^4$, so in particular $|f(1/2)| \le 1/16$ for all $f\in\mathcal{F}$. On the other hand $f(z) = z^4$ belongs to $\mathcal{F}$ which gives the opposite inequality.

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  • $\begingroup$ That sounds more convincing. I was somehow not able to complete the argument. Thank you. $\endgroup$ – Sourav D Dec 23 '13 at 0:00
  • $\begingroup$ There is one small problem about which I am not being able to find a satisfactory explanation. I assumed that $|g(z)|< 1 \text{ when } |z| = 1$. How is this valid since we only know about holomorphicity in the open unit disc? $\endgroup$ – Sourav D Dec 24 '13 at 1:02
  • $\begingroup$ @SouravD: look at the max of $|g|$ on the circle $|z|=r$ for $r<1$, use the maximum modulus principle to get information about $|g|$ on $|z|<r$ and finally let $r\to 1$. $\endgroup$ – mrf Dec 24 '13 at 8:20
  • $\begingroup$ Oh Yes! I get it now. It's similar to the strategy that we use to prove the Schwarz lemma in the open unit disc. Thank you. $\endgroup$ – Sourav D Dec 24 '13 at 11:43

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