I am looking for some generalization of Jensen's inequality for functions $g:\mathbb{R}^n \rightarrow \mathbb{R}$ where $g(x)$ is quasiconvex (or not convex). We known that for convex functions, $$\mathbb{E}\left[g(x)\right] \ge g\left(\mathbb{E}[x]\right).$$ Is there a generalization of this result for quasiconvex functions or functions which are not convex.
Thanks