Question

Let u, v, w, z be complex number such that |u| < 1, |v| = 1 and $w=\frac{v(u-z)}{(\overline{u}z-1)}$ then

• A. $|w|\le 1\Rightarrow |z|\le 1$
• B. $|w|\ge 1\Rightarrow |z|\ge 1$
• C. $|w|\le 1\Rightarrow |z|\ge 1$
• D. $|w|\ge 1\Rightarrow |z|\le 1$

Answer 1

Answer: (A), (B)

The correct options are
A

$|w|\le 1\Rightarrow |z|\le 1$

B

$|w|\ge 1\Rightarrow |z|\ge 1$

$|w|=\frac{|v||u-z|}{|\overline{u}z-1|}=\frac{|u-z|}{|\overline{u}z-1|}$
Let $|w|\le 1$
$|u-z|\le |\overline{u}z-1|(u-z)(\overline{u}-\overline{z})\le (\overline{u}z-1)(u\overline{z}-1)(|u{|}^2-1)(|z{|}^2-1)\ge 0|z{|}^2-1\le 0$
Similarly for $|w|\ge 1,|z|\ge 1.$