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. It was proved by Jensen in 1906. Given its generality, the inequality appears in many forms depending on the context, some of which are presented below. In its simplest form the inequality states that the convex transformation of a mean is less than or equal to the mean after convex transformation; it is a simple corollary that the opposite is true of concave transformations.

Jensen's inequality generalizes the statement that the secant line of a convex function lies above the graph of the function, which is Jensen's inequality for two points: the secant line consists of weighted means of the convex function, while the graph of the function is the convex function of the weighted means,

There are also converses of the Jensen's inequality, which estimate the upper bound of the integral of the convex function.

In the context of probability theory, it is generally stated in the following form: if X is a random variable and is a convex function, then