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Let $f:D(0,1) \to \mathbb{C}$ be an analytic function such that $|f(z)| \leq M, ~\forall z \in D(0,1)$ and $f(z_1) = 0.$

Claim: The estimate

\begin{equation*} |f(z)| \leq M \left( \frac{|z-z_1|}{|1-\overline{z_1}z|}\right) \end{equation*}

holds.

Can anyone give suggestions on how to start this?

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  • $\begingroup$ Compose $f$ with $(z_1 - z)/(1-\bar {z_1}z).$ $\endgroup$ – zhw. May 12 '15 at 18:24
  • $\begingroup$ I think want you want is to use a certain Möbius transformation and then apply Schwarz lemma. $\endgroup$ – Suugaku May 12 '15 at 18:24
  • $\begingroup$ @zhw. Is $f\bigg(\frac{z_1 - z}{1-\overline{z_1}z}\bigg) \leq M$? I don't know where you are going with that. Is the thing I am composing with in D(0,1)? $\endgroup$ – Mr.Fry May 12 '15 at 18:28
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Hint. Set $$ \varphi(z)=\frac{z+z_1}{1+\overline{z}_1z} $$ then $$ \varphi^{-1}(z)=\frac{z-z_1}{1-\overline{z}_1z}. $$ Both, $\varphi$ and $\varphi^{-1}$ are bijections between $D$ and $D$.

In particular, if $g(z)=f\big(\varphi(z)\big)$, then $g(0)=0$, $\lvert g(z)\rvert\le M\lvert z\rvert$.

Then apply Schwarz lemma on $g$ and obtain that $\lvert g(z)\rvert \le M\lvert z\rvert$, and hence $$ \lvert\,f(z) \rvert=\left|g\big(\varphi^{-1}(z)\big)\right| \le M\lvert\varphi^{-1}(z)\rvert =M\left|\frac{z-z_1}{1-\overline{z}_1z}\right|, $$ for all $z\in D$.

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  • $\begingroup$ How did $M |\phi^{-1}(z)|$ come about? ( I understand everything else) $\endgroup$ – Mr.Fry May 12 '15 at 19:02
  • $\begingroup$ Also is $|g| \leq 1$? THis is required for the Lemma. $\endgroup$ – Mr.Fry May 12 '15 at 19:08
  • $\begingroup$ @Rod: See the updated last formula. $\endgroup$ – Yiorgos S. Smyrlis May 12 '15 at 19:09
  • $\begingroup$ I review this and schwarz lemma gives $|g| \leq|z|$ how do you get $|g| \leq M|z|$? $\endgroup$ – Mr.Fry May 13 '15 at 17:40
  • $\begingroup$ Apply Schwarz Lemma to $h(z)=g(z)/M$. $\endgroup$ – Yiorgos S. Smyrlis May 13 '15 at 17:52

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