On generalized Powers-St$\o$rmer's Inequality

A generalization of Powers-St$\o$rmer's inequality for operator monotone functions on $[0, +\infty)$ and for positive linear functional on general $C^*$-algebras will be proved. It also will be shown that the generalized Powers-St$\o$rmer inequality characterizes the tracial functionals on $C^*$-algebras.
Comment: This is a revised version which was submitted to LAA