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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 b = 1 an then apply the resulting inequality to \tfrac . 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 a^\alpha b^\beta \leq \alpha a + \beta b, \qquad\, 0 \leq \alpha, \beta \leq 1, \quad\ \alpha + \beta = 1. 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 a = b 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 ...
2, a b \leq \frac + \frac, which also gives rise to the so-called Young's inequality with \varepsilon (valid for every \varepsilon > 0), 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" a b ~\leq~ \frac + \frac. Proof: Young's inequality with exponent 2 is the special case p = q = 2. 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 a and b we can write: 0 \leq (a-b)^2 Work out the square of the right hand side: 0 \leq a^2 - 2 a b + b^2 Add 2a b to both sides: 2 a b \leq a^2 + b^2 Divide both sides by 2 and we have Young's inequality with exponent 2: a b \leq \frac + \frac Young's inequality with \varepsilon follows by substituting a' and b' as below into Young's inequality with exponent 2: a' = a/\sqrt, \; b' = \sqrt b.


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 A, B of complex matrices of order n 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 ...
U such that U^* , A B^*, U \preceq \tfrac , A, ^p + \tfrac , B, ^q, 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 , A, = \sqrt.


Standard version for increasing functions

For the standard version of the inequality, let f denote a real-valued, continuous and strictly increasing function on
, c The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math> with c > 0 and f(0) = 0. Let f^ denote the
inverse function In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ . For a function f\colon ...
of f. Then, for all a \in
, c The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math> and b \in , f(c) a b ~\leq~ \int_0^a f(x)\,dx + \int_0^b f^(x)\,dx with equality if and only if b = f(a). With f(x) = x^ and f^(y) = y^, this reduces to standard version for conjugate Hölder exponents. For details and generalizations we refer to the paper of Mitroi & Niculescu.Mitroi, F. C., & Niculescu, C. P. (2011). An extension of Young's inequality. In Abstract and Applied Analysis (Vol. 2011). Hindawi.


Generalization using Fenchel–Legendre transforms

By denoting the
convex conjugate In mathematics and mathematical optimization, the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convex functions. It is also known as Legendre–Fenchel transformation, Fenchel transformati ...
of a real function f by g, we obtain a b ~\leq~ f(a) + g(b). This follows immediately from the definition of the convex conjugate. For a convex function f this also follows from the
Legendre transformation In mathematics, the Legendre transformation (or Legendre transform), first introduced by Adrien-Marie Legendre in 1787 when studying the minimal surface problem, is an involutive transformation on real-valued functions that are convex on a rea ...
. More generally, if f is defined on a real vector space X and its
convex conjugate In mathematics and mathematical optimization, the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convex functions. It is also known as Legendre–Fenchel transformation, Fenchel transformati ...
is denoted by f^\star (and is defined on the
dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V,'' together with the vector space structure of pointwise addition and scalar multiplication by cons ...
X^\star), then \langle u, v \rangle \leq f^\star(u) + f(v). where \langle \cdot , \cdot \rangle : X^\star \times X \to \Reals is the dual pairing.


Examples

The convex conjugate of f(a) = a^p / p is g(b) = b^q / q with q such that \tfrac + \tfrac = 1, and thus Young's inequality for conjugate Hölder exponents mentioned above is a special case. The Legendre transform of f(a) = e^a - 1 is g(b) = 1 - b + b \ln b, hence a b \leq e^a - b + b \ln b for all non-negative a and b. This estimate is useful in
large deviations theory In probability theory, the theory of large deviations concerns the asymptotic behaviour of remote tails of sequences of probability distributions. While some basic ideas of the theory can be traced to Laplace, the formalization started with insura ...
under exponential moment conditions, because b \ln b appears in the definition of
relative entropy Relative may refer to: General use *Kinship and family, the principle binding the most basic social units of society. If two people are connected by circumstances of birth, they are said to be ''relatives''. Philosophy *Relativism, the concept t ...
, which is the rate function in
Sanov's theorem In mathematics and information theory, Sanov's theorem gives a bound on the probability of observing an atypical sequence of samples from a given probability distribution. In the language of large deviations theory, Sanov's theorem identifies the ...
.


See also

* * * *


Notes


References

* *


External links


''Young's Inequality''
at
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