How sharp is the Jensen inequality?

We study how good the Jensen inequality is, that is, the discrepancy between integral(1)(0)phi(f (x)) dx, and phi(integral(1)(0) f(x) dx), phi being convex and f (x) a nonnegative L-1 function. Such an estimate can be useful to provide error bounds for certain approximations in L-p, or in Orlicz spaces, where convex modular functionals are often involved. Estimates for the case of C-2 functions, as well as for merely Lipschitz continuous convex functions phi, are established. Some examples are given to illustrate how sharp our results are, and a comparison is made with some other estimates existing in the literature. Finally, some applications involving the Gamma function are obtained.