Question

If $z$ is a complex number such that $0\le \arg z\le \frac{\pi }{2}$, then which of the following inequality is true?

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

Answer 1

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

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