I'm have some difficulties in bounding the following inequality:

I want to find a c as small as possible s.t. $$\sum_{i=1}^nx_i^4\sum_{i=1}^nx_i^2 -\sum_{i=1}^nx_i^6 \leq c\left(\sum_{i=1}^nx_i^3\right)^2$$

where $x_i$ are all non-negative

I know from the cauchy-inequality that

$$\sum_{i=1}^nx_i^4\sum_{i=1}^nx_i^2 \geq \left(\sum_{i=1}^nx_i^3\right)^2$$

But I think it useless in my question..

And more generally for some k and l, find out a small c s.t. $$\sum_{i=1}^nx_i^{2k-l}\sum_{i=1}^nx_i^l -\sum_{i=1}^nx_i^{2k} \leq c\left(\sum_{i=1}^nx_i^k\right)^2$$

Anyone help with out? Thanks!!