In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Jensen's inequality, named after the Danish mathematician
Johan Jensen, relates the value of a
convex function
In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigraph (the set of poi ...
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 Chri ...
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
In mathematics and logic, a corollary ( , ) is a theorem of less importance which can be readily deduced from a previous, more notable statement. A corollary could, for instance, be a proposition which is incidentally proved while proving another ...
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 (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 ...
, 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 po ...
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
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simila ...
or (equivalently) probability. In the probabilistic setting, the inequality can be further generalized to its ''full strength''.
Finite form
For a real
convex function
In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigraph (the set of poi ...
, numbers
in its domain, and positive weights
, Jensen's inequality can be stated as:
and the inequality is reversed if
is
concave
Concave or concavity may refer to:
Science and technology
* Concave lens
* Concave mirror
Mathematics
* Concave function, the negative of a convex function
* Concave polygon, a polygon which is not convex
* Concave set
In geometry, a subset o ...
, 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
Concave or concavity may refer to:
Science and technology
* Concave lens
* Concave mirror
Mathematics
* Concave function, the negative of a convex function
* Concave polygon, a polygon which is not convex
* Concave set
In geometry, a subset o ...
'', so substituting
in the previous formula () establishes the (logarithm of the) familiar
arithmetic-mean/geometric-mean inequality:
: