### Reciprocal Convexity to reverse the Jensen Inequality

Jensen's inequality is a powerful tool often used in mathematical derivations and analyses. It states that for a convex function $f(x)$ and an arbitrary random variable $X$ we have the following *upper* bound:
$$
f\left(\E X\right)
\le
\E f\left(X\right)
$$

However, oftentimes we want the inequality to work in the other direction, to give a *lower* bound. In this post I'll outline one possible approach to this.