We know that
absolute value of(|z| -|1/z|) < |z +1/z| < |z| + |1/z|
|z| + |1/z| takes the highest value when |z| or |1/z| is the highest. From the inequality of |z|, it is obvious that |z| + |1/z| takes highest value when |z| is substituted by 4, which gives
4 + 1/4 = 17/4 which is the highest value.
absolute (|z| - |1/z|) takes the least value when |z| = |1/z| which means |z|=1, which also satisfies the inequality.
Therefore least value is 1 - 1/1 = 0.
I have mentioned absolute value, which means the positive value of the expression, while | means absolute magnitude of the complex number.
Cheers !! 