I am interested in demonstrating the following:
$$ \frac{\partial^2 log(P_{3,n}) }{\partial x_1 \partial x_2} \leq 0 $$
and
$$ \frac{\partial^3 log(P_{3,n}) }{\partial x_1 \partial x_2 \partial x_3} \geq 0 $$
where $P_{3,n}$ is the n-degre complete symmetric polynomial on the variables $x_1, x_2, x_3$. I have checked that the above inequalities hold for $n=2,3,4,5$ and $6$ but I am not able to find a general proof for all $n$ or at least for a set of values of n. Thank you for your help!
