In
mathematics, Jensen's inequality, named after the Danish mathematician
Johan Jensen, relates the value of a
convex function of an
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
to the integral of the convex function. It was
proved by Jensen in 1906, building on an earlier proof of the same inequality for doubly-differentiable functions by
Otto Hölder
Ludwig Otto Hölder (December 22, 1859 – August 29, 1937) was a German mathematician born in Stuttgart.
Early life and education
Hölder was the youngest of three sons of professor Otto Hölder (1811–1890), and a grandson of professor Chris ...
in 1889. 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
Secant is a term in mathematics derived from the Latin ''secare'' ("to cut"). It may refer to:
* a secant line, in geometry
* the secant variety, in algebraic geometry
* secant (trigonometry) (Latin: secans), the multiplicative inverse (or recip ...
of a convex function lies ''above'' the
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
of the
function, which is Jensen's inequality for two points: the secant line consists of weighted means of the convex function (for ''t'' ∈
,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
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
, it is generally stated in the following form: if ''X'' is a
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...
and is a convex function, then
:
The difference between the two sides of the inequality,
, is called the
Jensen gap.
Statements
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''.
Finite form
For a real
convex function , numbers
in its domain, and positive weights
, Jensen's inequality can be stated as:
and the inequality is reversed if
is
concave, which is
Equality holds if and only if
or
is linear on a domain containing
.
As a particular case, if the weights
are all equal, then () and () become
For instance, the function is ''
concave'', so substituting
in the previous formula () establishes the (logarithm of the) familiar
arithmetic-mean/geometric-mean inequality:
: