Young's inequality for products

Young's inequality for products
In mathematics, Young's inequality for products is a mathematical inequality about the product of two numbers. The inequality is named after William Henry Young and should not be confused with Young's convolution inequality.
Young's inequality for products can be used to prove Hölder's inequality. 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:
It can be used to prove Hölder's inequality.
This form of Young's inequality can also be proved via Jensen's inequality.
Young's inequality may equivalently be written as
Where this is just the concavity of the logarithm 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
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 ordering. It states that for any pair of complex matrices of order there exists a unitary matrix such that
where denotes the conjugate transpose 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 with and Let denote the inverse function of Then, for all and
with equality if and only if
With and this reduces to standard version for conjugate Hölder exponents.
For details and generalizations we refer to the paper of Mitroi & Niculescu.
Generalization using Fenchel–Legendre transforms
By denoting the convex conjugate of a real function by we obtain
This follows immediately from the definition of the convex conjugate. For a convex function this also follows from the Legendre transformation.
More generally, if is defined on a real vector space and its convex conjugate is denoted by (and is defined on the dual space ), then
where is the dual pairing.
Examples
The convex conjugate of is with such that and thus Young's inequality for conjugate Hölder exponents mentioned above is a special case.
The Legendre transform of is , hence for all non-negative and This estimate is useful in large deviations theory under exponential moment conditions, because appears in the definition of relative entropy, which is the rate function in Sanov's theorem.
See also
Notes
References
External links
Young's Inequality at PlanetMath
Articles containing proofs
Inequalities