Operator revision of a Ky Fan type inequality

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.