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Complex Numbers - Printable Version

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Complex Numbers - elim - 07-04-2016 05:33 PM

Suppose that$\,\;z,\,w\in\mathbb{C},\;|z|\le 1\ge |w|,\;z\bar{w}\ne 1.\;$
Prove that $\left|\dfrac{z-w}{1-\bar{z}w}\right|\le 1$ and determine when equality holds.


Sol. - elim - 07-04-2016 05:48 PM

$\because\;\;|1-z\bar{w}|^2 -|z-w|^2$
$\qquad = 1+|z|^2|w|^2-2\text{re}(z\bar{w})-|z|^2-|w|^2+2\text{Re}(z\bar{w})$
$\qquad =(1-|z|^2)(1-|w|^2)\ge 0$

$\therefore \;\;\left|\dfrac{z-w}{1-\bar{z}w}\right|\le 1.\quad$Equality holds iff $(|z|=1)\vee(|w|=1)$