Operator norm and numerical radius analogues of Cohen's inequality

Let $D$ be an invertible multiplication operator on $L^2(X,\mu )$, and let $A$ be a bounded operator on $L^2(X,\mu )$. In this note we prove that $|A|^2 \le |D A| , |D^{-1} A|$,where$|\cdot|$ denotes the operator norm. If, in addition, the operators $A$ and $D$ are positive, we also have $w(A)^2 \le w(D A) , w(D^{-1} A)$,where$w$ denotes the numerical radius.