<s>[INST] Minimum value of magnitude of complex expression 
If $|2z_1 + \bar z_2| = 2\sqrt2$ and   $|1 + 2z_1z_2 | = 3 $ then minimum value of $(|z_1|^2 + 4|z_2|^2)$, is:-

Now
$|2z_1 + \bar z_2|^2 = 8$
$(2 z_1 + \bar z_2)(2\bar z_1 + z_2) = 8$
$4z_1\bar z_1 + 2 z_1z_2 + 2\bar z_1\bar z_2 + z_2\bar z_2 = 8$

$4|z_1|^2 + |z_2|^2 + 2 z_1z_2 + 2\bar z_1\bar z_2 = 8.......(1)$

Now
$|1 + 2z_1z_2 |^2 = 9 $
$(1 + 2z_1z_2)(1 + 2\bar z_1\bar z_2) = 9 $

$1 + 4|z_1|^2|z_2|^2 + 2z_1z_2 + 2\bar z_1\bar z_2= 9........(2)$

$(1) - (2)$
$4|z_1|^2 + |z_2|^2 = 4|z_1|^2|z_2|^2$
I am not able to make any progress after this step.
 [/INST] $$4|z_1|^2 + |z_2|^2 = 4|z_1|^2|z_2|^2$$
$$\dfrac{1}{|z_1|^2} + \dfrac{4}{|z_2|^2} = 4$$
By Cauchy-Schwarz,
$$\Bigl(\dfrac{1}{|z_1|^2} + \dfrac{4}{|z_2|^2} \Bigr)\Bigl(|z_1|^2 + 4|z_2|^2\Bigr) \ge (1 + 2\cdot2)^2 = 25 $$
$$|z_1|^2 + 4|z_2|^2 \ge \dfrac{25}{4} $$
with equality for
$$\dfrac{1}{|z_1|}\cdot2|z_2| - 2\dfrac{1}{|z_2|}\cdot|z_1|=0$$
$$\Rightarrow |z_1|=|z_2|=\dfrac{\sqrt5}{2}$$
</s>