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 proven 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 applied 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 (where t ∈ [0,1]),

while the graph of the function is the convex function of the weighted means,

Thus, Jensen's inequality is

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

The classical form of Jensen's inequality involves several numbers and weights. The inequality can be stated quite generally using either the language of measure theory or (equivalently) probability. In the probabilistic setting, the inequality can be further generalized to its full strength.

For a real convex function ${\displaystyle \varphi }$, numbers ${\displaystyle x_{1},x_{2},\ldots ,x_{n}}$ in its domain, and positive weights ${\displaystyle a_{i}}$, Jensen's inequality can be stated as:

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