In this paper, we give the Alzer inequality for Hilbert space operators as follows: Let A,B be two selfadjoint operators on a Hilbert space H such that 0<A,B≤21I
Nearby in the stack
, where
I
is identity operator on
H
. Also, assume that
A∇λB:=(1−λ)A+λB
and
A♯λB:=A21(A−21BA−21)λA21
are arithmetic and geometric means of
A,B
, respectively, where
0<λ<1
. We show that if
A
and
B
are commuting, then
B′∇λA′−B′♯λA′≤A∇λB−A♯λB,
where
A′:=I−A
,
B′:=I−B
and
0<λ≤21
. Also, we state an open problem for an extension of Alzer inequality.