Minimum and Maximum value of |z| This is a question that I came across today:

If $|z-(2/z)|=1$...(1) find the maximum and minimum value of |z|, where z represents a complex number.

This is my attempt at a solution:

Using the triangle inequality, we can write:
$||z|-|2/z||≤|z+2/z|≤|z|+|2/z|$
Let $|z|=r$ which implies that $|r-2/r|≤1≤r+2/r$ (From (1))

How must I proceed to find the value of |z|? Please help! Much thanks in advance :)

Using Complex Inequalities,   $$||z|-|w||\le|z+w|\le|z|+|w|$$
$w=-\dfrac2z$ and writing $|z|=r$
$$\left|r-\dfrac2r\right|\le\left|z-\dfrac2z\right|\le r+\dfrac2r$$
$$\implies\left|r-\dfrac2r\right|\le1\le r+\dfrac2r$$
Now as $r>0,$ $$\dfrac{r+\dfrac2r}2\ge\sqrt{r\cdot\dfrac2r}=\sqrt2\iff r+\dfrac2r\ge2\sqrt2>1$$
So, we need $$\left|r-\dfrac2r\right|\le1\iff-1\le r-\dfrac2r\le1$$
$$r-\dfrac2r\le1\iff r^2-r-2\le0$$
Now we know $(x-a)(x-b)\le0$ with $a\le b;a\le x\le b$
Here $-1\le r\le2$ But $r>0\implies0<r\le2$
Can check for $-1\le r-\dfrac2r$  to find $r\ge1$
So, the final range $$1\le r\le2$$
