Alzer Inequality for Hilbert Spaces Operators

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.