prove $|\frac{z-a}{1-\bar{a}z}|\lt1$

Asked Jan 08 '19 at 13:53

Active Jan 08 '19 at 13:53

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$z,a$ are both complex numbers. And $|z|\lt1,|a|\lt1$, prove $$|\frac{z-a}{1-\bar{a}z}|\lt1$$

I tried by denoting $z=x+yi,a=t+si$, and substitute them in $|\frac{z-a}{1-\bar{a}z}|$, but make it seems too sophisticated. So I am thinking about if there are ways more elegant to solve this by using some properties about complex number.

asked Jan 08 '19 at 13:53

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Possible duplicate of Show that $\left|\frac{\alpha - \beta}{1-\bar{\alpha}\beta}\right| < 1$ when $|\alpha|,|\beta| < 1$ – Martin R Jan 08 '19 at 13:57
Also: https://math.stackexchange.com/q/734804, https://math.stackexchange.com/q/1141986, ... – all found with Approach0 – Martin R Jan 08 '19 at 13:58

prove $|\frac{z-a}{1-\bar{a}z}|\lt1$

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