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$f$ is analytic mapping of the unit disk into itself such that $f(a) = 0$. Show $|f(z)| \leq |\frac{z-a}{1-\bar{a}z}|$

I considered $F(z) =f(\phi_{-a}(z))$ where $\phi_a(z) = \frac{z-a}{1-\bar{a}z}$ maps unit disk to unit disk and $\phi_{-a}$ is the inverse of $\phi_a$

I get F is analytic mapping unit disk to unit disk, and $F(0) = 0$, so by Schwarz lemma I get $|F(z)|\leq |z|$ and $|F'(0)|\leq1$. And then I get stuck. I am not sure what I can do to get the final inequality. Or maybe I am thinking about the problem in a wrong way.

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You're almost there. Now, let $\xi=\phi_{-a}(z)$. We get $F(z)=f(\phi_{-a}(z))=f(\xi)$, for So the inequality would be, in this terms,

$$|f(\xi)|<|\phi_{a}(\xi)|=\left|\frac{\xi-a}{1-\bar a\xi}\right|$$

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  • $\begingroup$ hehe, that was simple! Thanks a lot! $\endgroup$ – aregak Mar 5 '17 at 19:36
  • $\begingroup$ You can obtain another condition if you write $F'(0)$ using the chain rule, I think. $\endgroup$ – A. Salguero-Alarcón Mar 5 '17 at 19:38

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