Operator norm and numerical radius analogues of Cohen's inequality

Abstract: 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.