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