Question

If z is a complex number such that $-\pi /2\le argz\le \pi /2$ then which of the following inequality is true?

• A. $|z-\overline{z}|\le |z|(argz-arg\overline{z})$
• B. $|z-\overline{z}|\ge |z|(argz-arg\overline{z})$
• C. $|z-\overline{z}|<|z|(argz-arg\overline{z})$
• D. None of these.

Answer 1

The correct option is A $|z-\overline{z}|\le |z|(argz-arg\overline{z})$

$|z-\overline{z}|$ is the length AB while |z| $(argz-arg\overline{z})$ is are length AB.
$\therefore |z-\overline{z}|\le |z|(argz-arg\overline{z})$