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

In its standard form, the inequality states that if a and b are nonnegative real numbers and p and q are real numbers greater than 1 such that 1/p + 1/q = 1, then
The equality holds if and only if. This form of Young's inequality can be proved by Jensen's inequality and can be used to prove Hölder's inequality.

Proof



Equivalently, it can be written as
This is the concavity of the logarithm function. The equality holds if and only if or.
Generally,
The equality holds if and only if all the s with non-zero s are equal.

Elementary case

An elementary case of Young's inequality is the inequality with exponent 2,
which also gives rise to the so-called Young's inequality with ε, 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


Matricial generalization

T. Ando proved a generalization of Young's inequality for complex matrices ordered
by Loewner ordering. It states that for any pair A, B of complex matrices of order n there exists a unitary matrix U such that
where * denotes the conjugate transpose of the matrix and.

Standard version for increasing functions

For the standard version of the inequality,
let f denote a real-valued, continuous and strictly increasing function on with c > 0 and f = 0. Let f−1 denote the inverse function of f. Then, for all a ∈ and b ∈ ,
with equality if and only if b = f.
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

If f is a convex function and its Legendre transformation is denoted by g, then
This follows immediately from the definition of the Legendre transform.
More generally, if f is a convex function defined on a real vector space and its convex conjugate is denoted by , then
where is the dual pairing.

Examples