Jensen's Inequality

Definition

Let $f$ be a Convex Function on an interval $I$, $X$ be a Random Variable with support $S\subset I$, and $E(X)<\infty$, then $f(E[X])\le E[f(X)]$

Facts

$\frac{1}{\frac{1}{n}\sum \frac{1}{a_i}}\le (\prod _{i=1}^n{a}_i{)}^{1/n}\le \frac{1}{n}\sum _{i=1}^n{a}_i$

By Jensen’s Inequality, the following relation is satisfied arithmetic mean $\le$ geometric mean $\le$ harmonic mean