A remark on normal derivations

Given a Hilbert space $H$, let $A,S$ be operators on $H$. Anderson has proved that if $A$ is normal and $AS=SA$, then $\|AX-XA+S\|\ge\|S\|$ for all operators $X$. Using this inequality, Du Hong-Ke has recently shown that if (instead) $ASA=S$, then $\|AXA-X+S\|\ge\|A\|^{-2}\|S\|$ for all operators $X$. In this note we improve the Du Hong-Ke inequality to $\|AXA-X+S\|\ge\|S\|$ for all operators $X$. Indeed, we prove the equivalence of Du Hong-Ke and Anderson inequalities, and show that the Du Hong-Ke inequality holds for unitarily invariant norms.