Let $A,B \in M_p(R)$ be symmetric matrices, $A$ is given and non-singular, $\|A^{-1}\|.\|B\| < 1$ where $\|.\|$ is Frobenius norm.
I would like to find the lower bound and upper bound of $\|A^{-1}.B.(A+B)^{-1}\|$ such as these bounds just depend on $\|B\|$.
I have the result: $$\frac{1}{\|A\|} - \frac{1}{\|A\|+\|B\|} \le \|A^{-1}.B.(A+B)^{-1}\| \le \frac{\| A^{-1}||^2.\|B\|}{1-\|A^{-1}\|.\|B\|}$$
I think that lower bound can be greater is $p^2(\frac{1}{\|A\|} - \frac{1}{\|A\|+\|B\|})$ but I can't prove. Can I find the greatest lower bound and smallest upper bound? You can have some mild assumptions if needed.
