‎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.‎

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