  In this paper, we prove an operator version of the Jensen's inequality and
its converse for $h$-convex functions. We provide a refinement of the Jensen
type inequality for $h$-convex functions. Moreover, we prove the
Hermite-Hadamard's type inequality and a multiple operator version of the
Jensen's inequality for $h$-convex functions. In particular, a result for
convex, $P$-class, $s$-convex, Godunova-Levin, and $s$-Godunova-Levin functions
can be deduced.
