Question

If $z$ is a complex number such that $-\frac{\pi }{2}\le \arg \left(z\right)\le \frac{\pi }{2}$, then which of the following inequality is always true?

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

Answer 1

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


From the above diagram,
$|z-\overline{z}|$ is length of chord $AB$ while $|z|\left(\text{arg z - arg}\overline{z}\right)$ is arc length $AB.$
$\therefore |z-\overline{z}|\le |z|\left(\text{arg z - arg}\overline{z}\right)$