
In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Young's inequality for products is a
mathematical inequality about the product of two numbers. The inequality is named after
William Henry Young
William Henry Young FRS (London, 20 October 1863 – Lausanne, 7 July 1942) was an English mathematician. Young was educated at City of London School and Peterhouse, Cambridge. He worked on measure theory, Fourier series, differential calcu ...
and should not be confused with
Young's convolution inequality.
Young's inequality for products can be used to prove
Hölder's inequality
In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality (mathematics), inequality between Lebesgue integration, integrals and an indispensable tool for the study of Lp space, spaces.
The numbers an ...
. It is also widely used to estimate the norm of nonlinear terms in
PDE theory, since it allows one to estimate a product of two terms by a sum of the same terms raised to a power and scaled.
Standard version for conjugate Hölder exponents
The standard form of the inequality is the following, which can be used to prove
Hölder's inequality
In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality (mathematics), inequality between Lebesgue integration, integrals and an indispensable tool for the study of Lp space, spaces.
The numbers an ...
.
A second proof is via
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, building on an earlier p ...
.
Yet another proof is to first prove it with
an then apply the resulting inequality to
. The proof below illustrates also why Hölder conjugate exponent is the only possible parameter that makes Young's inequality hold for all non-negative values. The details follow:
Young's inequality may equivalently be written as
Where this is just the concavity of the
logarithm
In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is , because is to the rd power: . More generally, if , the ...
function.
Equality holds if and only if
or
This also follows from the weighted
AM-GM inequality.
Generalizations
Elementary case
An elementary case of Young's inequality is the inequality with
exponent
In mathematics, exponentiation, denoted , is an operation involving two numbers: the ''base'', , and the ''exponent'' or ''power'', . When is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, i ...
which also gives rise to the so-called Young's inequality with
(valid for every
), sometimes called the Peter–Paul inequality.
This name refers to the fact that tighter control of the second term is achieved at the cost of losing some control of the first term – one must "rob Peter to pay Paul"
Proof: Young's inequality with exponent
is the special case
However, it has a more elementary proof.
Start by observing that the square of every real number is zero or positive. Therefore, for every pair of real numbers
and
we can write:
Work out the square of the right hand side:
Add
to both sides:
Divide both sides by 2 and we have Young's inequality with exponent
Young's inequality with
follows by substituting
and
as below into Young's inequality with exponent
Matricial generalization
T. Ando proved a generalization of Young's inequality for complex matrices ordered
by
Loewner order In mathematics, Loewner order is the partial order defined by the convex cone of positive semi-definite matrices. This order is usually employed to generalize the definitions of monotone and concave/convex scalar functions to monotone and concave ...
ing. It states that for any pair
of complex matrices of order
there exists a
unitary matrix
In linear algebra, an invertible complex square matrix is unitary if its matrix inverse equals its conjugate transpose , that is, if
U^* U = UU^* = I,
where is the identity matrix.
In physics, especially in quantum mechanics, the conjugate ...
such that
where
denotes the
conjugate transpose
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \mathbf is an n \times m matrix obtained by transposing \mathbf and applying complex conjugation to each entry (the complex conjugate ...
of the matrix and
Standard version for increasing functions
For the standard version of the inequality,
let
denote a real-valued, continuous and strictly increasing function on