Let B be a positive diagonal matrix, and C a PSD symmetric matrix with eigenvalues $\Lambda$, with $\lambda_i = {0,1}$. I'm trying to find a reference for the following bounds or similar

$B^{1\over 2}DB^{1\over 2} \preceq \Lambda B$

$DB \preceq \Lambda B$.

I've checked this answer but they bound the eigenvalues according to the weak majorization order. Also, this answer shows a bound $\lambda_i(AB)\leq \lambda_i(A)\lambda_i(B)$, where $\lambda_i(A)$ denotes the ith eigenvalue of A, but it is only for positive definite matrices.