I have a set of convex functions $f_{ij}:\mathbb{R}_+^n\mapsto\mathbb{R}$ for all $i,j\in \{1,\ldots,n\}$.
If I defined the following functions $g_{ij}:\mathbb{R}_+^{n\times n}\times\mathbb{R}_+^n\mapsto\mathbb{R}$ by
$$ g_{ij}(\mathbf{x},\mathbf{y})=x_{ij}f_{ij}(\mathbf{y}), $$
can this function be convex?
Consider $f(y)=y$ on $\mathbb R_+$, which is obviously convex and $$ g(x,y) = xf(y) = xy $$ on $\mathbb R^2_+$. Then, $g$ is not convex, as its hessian is not positive semidefinite.
Or a direct proof: $$ g(1\pm x, 1\mp x) = 1 - x^2 < 1 = g(1,1). $$