  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 $\mathcal H$ such
that $0 < A, B \le \frac{1}{2}I$, where $I$ is identity operator on $\mathcal
H$. Also, assume that $A \nabla_\lambda B:=(1-\lambda)A+\lambda B$ and $A
\sharp_\lambda
B:=A^{\frac{1}{2}}\left(A^{-\frac{1}{2}}BA^{-\frac{1}{2}}\right)^\lambda
A^{\frac{1}{2}}$ are arithmetic and geometric means of $A, B$, respectively,
where $0 < \lambda < 1$. We show that if $A$ and $B$ are commuting, then $$
B'~\nabla_\lambda~A' - B'~\sharp_\lambda~A' \le A~\nabla_\lambda~B -
A~\sharp_\lambda~B\,, $$ where $A':=I-A$, $B':=I-B$ and $0 < \lambda \le
\frac{1}{2}$. Also, we state an open problem for an extension of Alzer
inequality.
