Let $\mathscr{ H}$ be a complex Hilbert space and $A,B\in \mathbb{ B}(\mathscr{ H})$ such that $0<A,B\leq\frac{ 1}{ 2}I$. Setting $A':=I-A$ and $B':=I-B$, we prove $$ A'\nabla_\lambda B'-A'!_\lambda B' \leq A\nabla_\lambda B-A!_\lambda B, $$ where $\nabla_\lambda$ and $!_\lambda$ denote the weighted arithmetic and harmonic operator means, respectively. This inequality is the natural extension of a Ky Fan type inequality due to H. Alzer. Some parallel and related results are also obtained.
Categories: math.FA