TITLE: f is analytic mapping of the unit disk into itself such that f(a) = 0. Show $|f(z)| \leq |\frac{z-a}{1-\bar{a}z}|$
QUESTION [3 upvotes]: $f$ 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.

REPLY [1 votes]: 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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