If $|Z|\le 1$, $|W| \le 1$, Show that $$|Z-W|^2 \le (|Z| - |W|)^2 + (\text{arg} Z - \text{arg} W)^2)$$
$Z \text{ and } W$ are Complex Numbers.

Question

If $|Z|\le 1$, $|W| \le 1$, Show that $$|Z-W|^2 \le (|Z| - |W|)^2 + (\text{arg} Z  - \text{arg} W)^2)$$  
$Z \text{ and } W$ are Complex Numbers.

anonymous · Answer

ahhh,very displeasing to eyes

anonymous · Answer

i know, that is why i didn't try it :-D

mr.math · Answer

Why did you capital letters? It makes it look not good.

anonymous · Answer

idk maybe because it;s the same notation used in my book

mr.math · Answer

Let $\large z=r_1e^{i\theta_1}$ and $\large w=r_2e^{i\theta_2}$, then we're asked to show
$$|r_1e^{i\theta_1}-r_2e^{i\theta_2}|^2\le (r_1-r_2)^2+(\theta_1-\theta_2)^2.$$

anonymous · Answer

For the geometers out there, start with the Cosine Law
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