“Jensen Inequality”
If \(p_{1}, \dots, p_{n}\) are positive numbers which sum to 1 and \(f\) is a real continuous function that is convex, then
[f\left(\sum_{i=1}^{n} p_{i} x_{i}\right) \leq \sum_{i=1}^{n} p_{i} f\left(x_{i}\right)]
If f is concave, then the inequality reverses, giving
[f\left(\sum_{i=1}^{n} p_{i} x_{i}\right) \geq \sum_{i=1}^{n} p_{i} f\left(x_{i}\right)]
One special case:
when \(p_{i}=\frac{1}{n}\) with the concave function \(\ln(x)\) gives
[\ln \left(\frac{1}{n} \sum_{i=1}^{n} x_{i}\right) \geq \frac{1}{n} \sum_{i=1}^{n} \ln x_{i}]
which can be exponentiated to give the arithmetic mean-geometric mean inequality
[\frac{x_{1}+x_{2}+\ldots+x_{n}}{n} \geq \sqrt[n]{x_{1} x_{2} \cdots x_{n}}]
Here, equality holds iff \(x_{1}=x_{2}=\ldots=x_{n}\)
