The original question is asked here by myself. But seems like nobody give an answer. So I decide to ask this question in mathoverflow. Sorry for duplication.

I want to prove the following results,

Given two non-negative definite matrices $A$ and $B$, suppose vector $a \in C(A)\cap C(B)$ , where $C(A)$ and $C(B)$ are column spaces of $A$ and $B$ separately.

Then

$(a' A^- a)(a' B^- a) \geq (a' (A+B)^- a) ( a' A^- a+ a' B^- a)$

where $A^-, B^-, (A+B)^-$ are generalized inverses, i.e. , $A A^-A=A$.

I start from the simplest case $A=B$ etc. But for general case, I can't prove it. Any hints or ideas?

Thanks in advance!