Complex Numbers - Printable Version +- OMath! (http://math.elinkage.net) +-- Forum: Math Forums (/forumdisplay.php?fid=4) +--- Forum: Analysis (/forumdisplay.php?fid=10) +--- Thread: Complex Numbers (/showthread.php?tid=717) 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 $(|z|=1)\vee(|w|=1)$