  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 $f:[0,\infty)\rightarrow[0,\infty)$ is an
operator monotone function with $f (1) = 1$, $f'(1)=\mu$, and associated mean
$\sigma$, then for all operators $A$ and $B$ on a complex Hilbert space
$\mathscr{H}$ such that $0<A,B\leq\frac{1}{2}I$, 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 $\mathscr{H}$, $A':=I-A$, $B':=I-B$,
and $\nabla_\mu$ is the $\mu$-weighted arithmetic mean.
