D is a $n \times n$ diagonal matrix whose diagonal entries lies in $(0,1]$.
B is any $n \times n$ n.n.d. matrix.
What will be the sharpest upper bound on the largest eigenvalue of:
$(D+B)^{-1}D^2(D+B)^{-1}$ ?
A naive upper bound will be $\{\min_{1\leq i \leq n}D_{ii}\}^{-1}$. Does there exist a better upper bound? As the eigenvalues of $(D+B)^{-1}D$ are real and lies in $[0,1]$, I was wondering if a better bound for $\lambda_1\{(D+B)^{-1}D^2(D+B)^{-1}\}$ can be obtained. Else, is it true that the aforementioned bound is the strongest?