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Let $a$ and $b$ be complex numbers with modulus $< 1$. How can I prove that $\left | \frac{a-b}{1-\bar{a}b} \right |<1$ ? Thank you

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Is it homework? What did you try? Where are you stuck? –  Davide Giraudo Jan 11 '12 at 8:56
Not exactly homework. It is an oral question given at the entry test to the French school Ecole Polytechnique. –  user20010 Jan 11 '12 at 10:04

1 Answer 1

Here are some hints: Calculate $|a-b|^2$ and $|1-\overline{a}b|^2$ using the formula $|z|^2=z\overline{z}$. To show that $\displaystyle\left | \frac{a-b}{1-\bar{a}b} \right |<1$, it's equivalent to show that $$\tag{1}|1-\overline{a}b|^2-|a-b|^2>0.$$ To show $(1)$, you need to use the fact that $|a|<1$ and $|b|<1$.

If you need more help, I can give your more details.

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