The original Jensen's inequality in probability theory is generally stated in the following form: if $X$ is a random variable and $f$ is a convex function, then $f \left(\mathbb{E}[X]\right) \leq \mathbb{E}\left[f(X)\right]$ holds, where $\mathbb{E}$ means the expectation of $X$.
Then I have another function $g(\cdot)$ and I do not know if it is convex or not. Does $f \left(\mathbb{E}[g(X)]\right) \leq \mathbb{E}\left[f(g(X))\right]$ hold? Why?
It follows from an application of Jensen's inequality to the random variable $g(X)$ (provided that $g$ is Borel measurable) instead of $X$.