Lets say, $A\in \mathbb{R}^{m\times n}$, and $D\in \mathbb{R}^{m\times m}$, where $D$ is a diagonal matrix with positive diagonal elements, and all the elements are $\le 1$. For simplicity, assume that $A^TA$ is positive definite.

It is easy to see that $$\lambda_{\max}(A^TDA)\le \lambda_{\max}(A^TA)\cdot \max_{i}D_{ii},\\\lambda_{\min}(A^TDA)\ge \lambda_{\min}(A^TA)\cdot \min_{i}D_{ii}$$

However, are these bounds generally tight? For example, if $D$ is such that one of its elements is $1$, and all others are equal to some small number $\epsilon$, then does the lower bound on the least eigenvalue produce a sever underestimation of the lowest eigenvalue of $A^TDA$. Specifically,

Do there exist explicit upper bounds on $|\lambda_{\min}(A^T D A)-\lambda_{\min}(A^TA)\cdot \min_{i}D_{ii}|$, $|\lambda_{\max}(A^T D A)-\lambda_{\max}(A^TA)\cdot \max_{i}D_{ii}|$? Here $A$ is not a diagonal matrix, otherwise, the bounds would be trivially satisfied with equality. Also, to avoid triviality, $D$ is assumed not to be an identity matrix.

Any references on this subject? Thanks in advance.