Jensens Inequality for g-Convex Function under g-Expectation

A real valued function defined on}$mathbb{R}$ {small is called}$g${small --convex if it satisfies the following textquotedblleft generalized Jensens inequalitytextquotedblright under a given}$g${small -expectation, i.e., }$h(mathbb{E}^{g}[X])leq mathbb{E}% ^{g}[h(X)]${small, for all random variables}$X$ {small such that both sides of the inequality are meaningful. In this paper we will give a necessary and sufficient conditions for a }$C^{2}${small -function being}$% g ${small -convex. We also studied some more general situations. We also studied}$g${small -concave and}$g${small -affine functions.