Jensen's Inequality

In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function.

General form

The inequality can be stated quite generally using measure theory. It can also be stated equally generally in the language of probability theory. The two statements say exactly the same thing.

In the language of measure theory

Let μ be a positive measure on a set Ω, such that μ(Ω) = 1. If g is a real-valued function that is Lebesgue integrable, and if φ is convex on the range of g, then
\varphi\left(\int_{\Omega} g\, d\mu\right) \le \int_\Omega \varphi \circ g\, d\mu.

In the language of probability theory

In the terminology of probability theory, μ is a probability measure. The function g is replaced by a real-valued random variable X (just another name for the same thing, as long as the context remains one of "pure" mathematics). The integral of any function over the space Ω with respect to the probability measure μ becomes an expected value. The inequality then says that if φ is any convex function, then
\varphi\left(E(X)\right) \leq E(\varphi(X)).\,

Special cases

Form involving a probability density function

Suppose Ω is a measurable subset of the real line and f(x) is a non-negative function such that
\int_{-\infty}^\infty f(x)\,dx = 1.
In probabilistic language, f is a probability density function. Then Jensen's inequality becomes the following statement about convex integrals: If g is any real-valued measurable function and φ is convex over the range of g, then
\varphi\left(\int_{-\infty}^\infty g(x)f(x)\, dx\right) \le \int_{-\infty}^\infty \varphi(g(x)) f(x)\, dx.
If g(x) = x, then this form of the inequality reduces to a commonly used special case:
\varphi\left(\int_{-\infty}^\infty x\, f(x)\, dx\right) \le \int_{-\infty}^\infty \varphi(x)\,f(x)\, dx.

Finite form

If \Omega is some finite set \{x_1,x_2,\ldots,x_n\}, and if \mu is a normalized counting measure on \Omega, then the general form reduces to a statement about sums:
\varphi\left(\sum_{i=1}^{n} g(x_i)\lambda_i \right) \le \sum_{i=1}^{n} \varphi(g(x_i))\lambda_i,
provided that \lambda_1 + \lambda_2 + \cdots + \lambda_n = 1, \lambda_i \ge 0. Suppose x_1, x_2, \ldots are real, g(x)=x, \lambda_i = 1/n and \varphi(x) = \exp(x). The above sum becomes
\exp\left(\sum_{i=1}^{n} \frac{x_i}{n} \right) \le \sum_{i=1}^{n} \frac{\exp(x_i)}{n},
Taking the natural logarithm of both sides gives the familiar AM-GM inequality:
\frac{x_1 + x_2 + \cdots + x_n}{n} \le \sqrtn{x_1 x_2 \cdots x_n}.
If φ is concave, each of the inequalities is simply reversed. There is also an infinite discrete form.

University logo

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

References

External links

 

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