Imagine a probability density function $f(x)$, and let's note its $n$th non-centred moment $x_{n}$.
The mean $x_{1}$ is fixed.
How can I find $f(x)$ so that it minimises some given function of the moments? In my case $x_{3}-x_{1}x_{2}$.
This is like minimising a functional, but as the functional in question doesn't depend on the derivatives, I don't know how to do.
Many thanks!