Upper and lower bound for seminorm induced by matrix

Question

I have the following situation. Given a positive definite diagonal matrix $D\in\mathbb{R}^{n\times n}$ and a symmetric, positive semi-definite matrix $A$ I want to prove that \begin{align} c\langle Ax,x\rangle \le \langle DAx,x\rangle \le C\langle Ax,x\rangle, \end{align} where $c,C>0$ are such that for all $i=1,...,n$ we have $c\le D_{ii}\le C$. EDIT: THE FOLLOWING SENTENCE IS WRONF I already know that $DA$ is positive semi-definite, so the existence of suitable constants for the upper and lower bound is clear. END-EDIT But I need them to be controlled by the $D_{ii}$. Therefore, my "suggestion" for the constants is possible, but I am also happy if you replace $c,C$ by $c(D_{11},...,D_{nn}), C(D_{11},...,D_{nn})$. Thank you very much for your help in advance.