I need to show if the following inequality is true
$$ (A + B)^{-1}M (A + B)^{-1} - A^{-1} M A^{-1} \preceq 0$$
given that $(A,B)=(A^T,B^T) \succ 0$ and $M = M^T \succeq 0$ also we have that $A + B \succeq A$. If $M = I$, then, it is pretty clear that the above inequality holds, but can we find conditions on $A$ such that the above holds for any PSD $M$?
It's not true. When $M=I$, it reduces to $(A+B)^{-2}\preceq A^{-2}$ or equivalently, $A^2\preceq (A+B)^2$. So it's essentially saying that $0\preceq X\preceq Y$ implies $X^2\preceq Y^2$, which is false in general (see here for a counterexample).