jensen's inequality

Topics

• convex functions

Jensen’s inequality applies to convex functions. The basic form is as follows :

$$f(\alpha \mathbf{x}+(1-\alpha )\mathbf{y})\le \alpha f(\mathbf{x})+(1-\alpha )f(\mathbf{y})\forall \mathbf{x},\mathbf{y}\in X,\alpha \in [0,1]$$

The generalized form of Jensen’s Inequality is :

$$f(\sum _i{w}_i{x}_i)\le \sum _i{w}_{i}f(x_i)|w_i\ge 0,\sum _i{w}_i=1$$

Proposition: The arithmetic mean of a set of numbers is $\ge$ their geometric mean.
Proof : Choose $f(x):=-\ln x,w_i=\frac{1}{n}$ , Note that $-\ln x$ is a convex function, as a result the Jensen’s inequality holds true. This gives us

$$\begin{array}{rl}-\ln \left(\frac{1}{n}\sum _i{x}_i\right)& \le -\frac{1}{n}\sum _i\ln x_i\\ -\ln \left(\frac{1}{n}\sum _i{x}_i\right)& \le -\frac{1}{n}\ln \prod _i{x}_i\\ -\ln \left(\frac{1}{n}\sum _i{x}_i\right)& \le -\ln (\prod _i{x}_i{)}^{\frac{1}{n}}\\ \frac{1}{n}\sum _i{x}_i& \ge (\prod _i{x}_i{)}^{\frac{1}{n}}\end{array}$$

Tip

For concave functions like $\log (x),\ln (x)$, one can flip the inequality, or use negative of the function.

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