Prove the following:
$$ \frac{n^2}{4} \log ^2 \left(\frac{n}{4}\right) = \Omega \left(\frac{n^2}{2} \log ^2 \left(\frac{n}{2}\right)\right)$$
Whatever I try to do gets me nowhere, I tried multiplying both sides by $\dfrac{2}{n^2}$ and got to the inequality: $$\frac{\log^2 (\frac{n}{4})}{\log^2 (\frac{n}{2})} \geq 2c$$ $$\frac{(\log n-2)^2}{(\log n-1)^2} \geq 2c$$
But that leads me nowhere.
Denote for simplicity $\log n =t$. From your last expression we need to find for what $t$ this holds. Solve the inequality $$ (1-2 c_1) t^2 +(4c_1-2)t +4 -2 c_1 \geq 0 $$ and select the positive root