Jensen's inequality

In mathematics, Jensen's inequality, named after the Danish mathematician Børge Jensen, relates the value of a convex function of an integral to the integral of the convex function. Formally, it states that when f is convex and y is a non-negative function of x fulfilling

<math>\int_a^b y(x)\, dx = 1 \,, <math>

then

<math>f\left(\int_a^b x\, y(x)\, dx\right) \le \int_a^b f(x)\,y(x)\, dx.<math>

For example,

<math>\left(\int_{-1}^1 \frac{x}{2}\, dx\right)^2 = 0 \le \frac{1}{3} = \int_{-1}^1 \frac{x^2}{2}\, dx.<math>

In probability theory, Jensen's inequality is often written using expectations. If f is a convex function and X is a random variable, then

<math>f\left(E\left[X\right]\right) \le E\left[f(X)\right].<math>

Jensen's inequality is also valid when applied to a sequence of positive numbers. Given numbers

<math>a_1, a_2, \ldots a_n<math>

and nonnegative weights

<math>\lambda_1, \ldots, \lambda_n<math>

such that

<math>\lambda_1 + \cdots + \lambda_n = 1<math>

as well as a convex function f, we have that

<math>f\left(\lambda_1a_1 + \lambda_2a_2 + \cdots + \lambda_na_n\right) \le \lambda_1f(a_1) + \lambda_2f(a_2) + \cdots + \lambda_nf(a_n). <math>

For a concave function f, we have the opposite inequality sign.

Jensen's inequality serves as logo for the mathematics department of Copenhagen University.