A real valued function h defined on \mathbbR{\mathbb{R}} is called g-convex if it satisfies the “generalized Jensen’s inequality” for a given g-expectation, i.e., h(\mathbbE^g[X]) £ \mathbbE^g[h(X)]{h(\mathbb{E}^{g}[X])\leq \mathbb{E}^{g}[h(X)]} holds for all random variables X such that both sides of the inequality are meaningful. In this paper we will give a necessary and sufficient condition for a C ²-function being g-convex, and study some more general situations. We also study g-concave and g-affine functions, and a relation between g-convexity and backward stochastic viability property.