Jensen’s inequality for g-convex function under g-expectation

A real valued function h defined on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}}$$\end{document} is called g-convex if it satisfies the “generalized Jensen’s inequality” for a given g-expectation, i.e., \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${h(\mathbb{E}^{g}[X])\leq \mathbb{E}^{g}[h(X)]}$$\end{document} 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 C2-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.