We prove an equivalent, more convenient formulation: Let $X$ be some random variable, and let $f(x)$ be a convex function (defined at least on a segment containing the range of $X$ . Then the expected value of $f(X)$ is at least the value of $f$ at the mean of $X$ $$ \Expect[f(X)] \ge f(\Expect [X]). $$

Indeed, let $c=\Expect [X]$ Since $f(x)$ is convex, there exists a supporting line for $f(x)$ at $c$ $$ \varphi(x)=\alpha (x-c) + f(c) $$ for some $\alpha$ and $\varphi(x)\le f(x)$ Then $$ \Expect[f(X)] \ge \Expect[\varphi(X)] = \Expect[\alpha (X-c) + f(c)] = f(c) $$ as claimed.