How sharp is the Jensen inequality?

We study how good is Jensen's inequality, that is the discrepancy between $\int_0^1 \varphi(f(x)) \, dx$, and $\varphi \left( \int_0^1 f(x) \, dx \right)$, $\varphi$ 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 $\varphi$, 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.

We study how good the Jensen inequality is, that is, the discrepancy between (Formula presented.) , φ being convex and f(x) a nonnegative L1 function. Such an estimate can be useful to provide error bounds for certain approximations in Lp, or in Orlicz spaces, where convex modular functionals are often involved. Estimates for the case of C2 functions, as well as for merely Lipschitz continuous convex functions φ, 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.