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

Jensen's Inequality for g-Convex Function under g-Expectation