Operator Ky Fan type inequalities

In this paper, we extend some significant Ky Fan type inequalities in a large setting to operators on Hilbert spaces and derive their equality conditions. Among other things, we prove that if $f:[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ is an operator monotone function with $f(1)=1$, $f^{\mathrm{\prime }}(1)=\mu$, and associated mean $\sigma$, then for all operators $A$ and $B$ on a complex Hilbert space $\mathcal{H}$ such that $0<A,B\le \frac{1}{2}I$, we have \begin{equation*} A’\nabla_\mu B’-A’\sigma B’\leq A\nabla_\mu B-A\sigma B, \end{equation*} where $I$ is the identity operator on $\mathcal{H}$, $A^{\mathrm{\prime }}:=I-A$, $B^{\mathrm{\prime }}:=I-B$, and ${\mathrm{\nabla }}_{\mu }$ is the $\mu$-weighted arithmetic mean.