I am working on a problem which requires me to bound $\|A^{1/2}(B^{-1}-A^{-1})A^{1/2}\|$ where $A$ and $B$ are $n \times n$ positive definite matrices with $\lambda_{max}(A) ,\lambda_{max}(B) < c$ and $\lambda_{min}(A) , \lambda_{min}(B)> d$. Here $\|\|$ means operator norm.
I am able to prove that $\|A^{1/2}(B^{-1}-A^{-1})A^{1/2}\|$ < $\frac{c}{d} \epsilon$. But is there a way to avoid $\frac{c}{d}$ and show that $\|A^{1/2}(B^{-1}-A^{-1})A^{1/2}\| < \epsilon$? I am able to show this when the matrices inside the norm commute but how to prove in general?