Alzer Inequality for Hilbert Spaces Operators

Abstract

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 $A{♯}_{\lambda }B:=A^{\frac{1}{2}}{\left(A^{-\frac{1}{2}}B{A}^{-\frac{1}{2}}\right)}^{\lambda }{A}^{\frac{1}{2}}$ are arithmetic and geometric means of $A,B$, respectively, where $0<\lambda <1$. We show that if $A$ and $B$ are commuting, then $$B^{\prime }{\nabla }_{\lambda }{A}^{\prime }-B^{\prime }{♯}_{\lambda }{A}^{\prime }\le A{\nabla }_{\lambda }B-A{♯}_{\lambda }B,$$ where $A^{\prime }:=I-A$, $B^{\prime }:=I-B$ and $0<\lambda \le \frac{1}{2}$. Also, we state an open problem for an extension of Alzer inequality.