I’ve been trying to decide, unsuccessfully, if the symmetric power sum functions, corresponding to partitions with even parts, are convex. In other words, is the polynomial $p_\lambda(x)$ convex where $x\in\mathbb{R}^n$ and $\lambda$ is a partition with only even parts? I appreciate your help.
Recall that $p_\lambda(x)=\prod_{i=1}^k{x_1^{\lambda_i}+\dots+x_n^{\lambda_i}}$ and in this case the $\lambda_i$ are even positive integers.
If $\lambda$ consists of one part then it’s true since the Hessian is obviously positive semidefinite.
Note $\log(p_\lambda)$ is not necessarily convex.