Let $x\in\mathbb{R}^n$ be a column vector such that $0\leq x_j\leq 1$ for all $j$ and define $X$ as the matrix with $x$ on the diagonal. Let also $\Omega$ be an real $n\times n$ matrix such that $0\leq \Omega_{ij}\leq 1$. Finally, let $A$ be a real $n\times n$ diagonal matrix such that all diagonal elements are strictly bigger than 0.
Consider the function $f:\mathbb{R^n}\rightarrow \mathbb{R}$ defined as $$ f(x) = 1'\left( I-AX\Omega'\right)^{-1}AX1 $$ where 1 denotes a column vector full of ones and where $I$ is the $n\times n$ identity matrix.
I am interested in the concavity properties of $f$? Is it concave, log-concave, quasi-concave? Do these properties depend on $\Omega$? I believe that $f$ is (at least) log-concave for $\Omega_{ij}=1$ for instance. I was thinking of using Neumann series to make the problem more tractable but so far no luck.