Given two numbers $z_1, z_2 \in \mathbb{C}$, let $|z_1| < 1$ and $|z_2|$ < 1. Prove that $| \frac{z_1-z_2}{1 - \overline{z_1}z_2} | < 1 $.

Asked Jul 07 '21 at 14:23

Active Jul 07 '21 at 14:23

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I tried expanding into numbers $x_1, y_1, x_2, y_2 \in \mathbb{R}$, but the algebra became suspiciously complicated and I'm not sure I got any promising results when I stopped.

I also know that there's a similar question out there on here but it's not exactly the same. Maybe I could use similar principles from it, though.

I'd appreciate any help! Thank you!

asked Jul 07 '21 at 14:23

chuloobydooby

Could you link that "similar question" to this? – Bumblebee Jul 07 '21 at 14:28
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https://www.sarthaks.com/260927/lf-z1-and-z2-are-two-complex-numbers-such-that-z1-1-z2-then-prove-that-1-z1-bar-z2-z1-z2-1 it can help you it is similar – Mridul Kumar Jul 07 '21 at 14:28
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Does this answer your question? Show that $\left|\frac{\alpha - \beta}{1-\bar{\alpha}\beta}\right| < 1$ when $|\alpha|,|\beta| < 1$ – Arctic Char Jul 07 '21 at 14:34

Given two numbers $z_1, z_2 \in \mathbb{C}$, let $|z_1| < 1$ and $|z_2|$ < 1. Prove that $| \frac{z_1-z_2}{1 - \overline{z_1}z_2} | < 1 $.

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