  In this paper, we state some characterizations of $h$-convex function is
defined on a convex set in a linear space. By doing so, we extend the
Jensen-Mercer inequality for $h$-convex function. We will also define
$h$-convex function for operators on a Hilbert space and present the operator
version of the Jensen-Mercer inequality. Lastly, we propound the complementary
inequality of Jensen's inequality for $h$-convex functions.
