Simplest case: For a concave up (convex) function $f$, the average of outputs is *never* surpassed by the function at the average of the inputs. $f\left(\frac{\sum x_{i}}{n}\right) \leq \frac{\sum f\left(x_{i}\right)}{n}$ For a concave down (concave) function $f$, the average of outputs is *always* surpassed by the function at the average of the inputs. $f\left(\frac{\sum x_{i}}{n}\right) \geq \frac{\sum f\left(x_{i}\right)}{n}$ When applied to $f(x)=\log x$, the inequality leads the [[AM-GM Inequality]]