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For pedagogical purposes, I am looking for an elementary proof (i.e. without resorting to the maximum principle) that $f_a(z):=\frac{z-a}{1-\overline{a}z}$ maps the unit disk into itself when $|a|<1$.

The usual argument (at least usual for me) is to look at $f_a(e^{it})$ and check that $|f_a(e^{it})|=1$. Then we are done by the maximum principle and the fact that $f_a(a)=0$.

However, I cannot help but think that there should be an elementary way to do this, using only basic facts about complex numbers such as the triangle inequality. Unfortunately I am running into circles.

Albert
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    Use $|w|^2 = w \overline w$ to evaluate $1 - \left|\frac{z-a}{1-\overline{a}z}\right|^2$ and show that it is positive for $|z| < 1$. – Martin R Feb 15 '21 at 10:30
  • @MartinR thanks! as expected, not too difficult, but I wasn't going at it right (and it was getting really annoying). If you want to put your comment in an answer I'll accept it – Albert Feb 15 '21 at 10:36

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