“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 \(x_{1}=x_{2}=\ldots=x_{n}\)

YONG HUANG

YONG HUANG

Hi, I'm Yong Huang. I've recently graduated from Cornell Tech and obtained my master's degree, I shall start my Ph.D. in Computer Science this fall at UC Irvine. Thank you for visiting my site.