Let $A,B \in M_p(R)$ be symmetric matrices, $A$ is given and non-singular, where $\|.\|$ is Frobenius norm.
I would like to find the lower bound of $\|A^{-1}.B.(A+B)^{-1}\|$:
I have the result: $$\|A^{-1}.B.(A+B)^{-1}\| \ge \frac{1}{\|A\|} - \frac{1}{\|A\|+\|B\|} $$
I think that lower bound can be greater is $p(\frac{1}{\|A\|} - \frac{1}{\|A\|+\|B\|})$ but I can't prove. Can I find the greatest lower bound? You can have some mild assumptions if needed.