I would have a question that is a specific case of this post - Bounding the ratio of the $l^1$ norms of two vectors to the ratio of their $l^2$ norms - with $x_j=y_j^2$ and $c_1=1$, $c_2=1/2$.
Specifically, I would like to show that, given any real strictly positive vector $\mathbf{y}=(y_1,\dots,y_n)>\mathbf{0}$, the following holds: $$ \left(\frac{\sum_{i=1}^n y_i^4}{\sum_{i=1}^n y_i^2} \right)^{1/2} \geq \frac{\sum_{i=1}^n y_i^2}{\sum_{i=1}^n y_i}. $$
The inequality seems to always hold numerically, but any help on a proof would be greatly appreciated.
By Holder $$\sum_{i=1}^ny_i^4\left(\sum_{i=1}^ny_i\right)^2\geq\left(\sum_{i=1}^ny_i^2\right)^3,$$ which gives your inequality.