Finding the minimum value of a complex number If $z$ is a complex number satisfying $|z^2+1| = 4|z|$ . Then prove that the minimum value of $|z|$ is $4$
This is how I attempted the problem ,
$\frac{|z^2+1|}{|z|} = 4$
Therefore , 
$|z + \frac{1}{z}| = 4$
How do I proceed from here ? 
According to the solution of the above problem the next step would be as follows 
$||z| - |\frac{1}{z}|| ≤ 4$
We can then form a quadratic in $z$ to find out the least value of $z$. However , I’m stuck at the first step of the solution . How does $|z + \frac{1}{z}| = 4$ imply $||z| - |\frac{1}{z}|| ≤ 4$ ? Please help .

This follows from a version of the reverse triangle inequality, in your case $$\lvert\lvert z\rvert-\lvert\frac{1}{z}\rvert\rvert\leq\lvert z+\frac{1}{z}\rvert.$$
