  In this paper, we introduce the notion of conditional $h$-convex functions
and we prove an operator version of the Jensen inequality for conditional
$h$-convex functions. Using this type of functions, we give some refinements
for Ky-Fan's inequality, arithmetic-geometric mean inequality, Chrystal
inequality, and H$\ddot{o}$lder-McCarthy inequality. Many of the other
inequalities can be refined by applying this new notion.
