  A real valued function defined on}$\mathbb{R}$ {\small is called}$g${\small
--convex if it satisfies the following \textquotedblleft generalized Jensen's
inequality\textquotedblright 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.
