  Inequalities play an important role in pure and applied mathematics. In
particular, Jensen's inequality, one of the most famous inequalities, plays a
main role in the study of the existence and uniqueness of initial and boundary
value problems for differential equations. In this work we prove some new
Jensen-type inequalities for $m$-convex functions, and we apply them to
generalized Riemann-Liouville-type integral operators. It is remarkable that,
if we consider $m=1$, we obtain new inequalities for convex functions.
