Young's inequality for products

In mathematics, Young's inequality for products is a mathematical inequality about the product of two numbers.

[1] The inequality is named after William Henry Young and should not be confused with Young's convolution 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.

From sketching a visual representation of the integrals of the area between this curve and the axes, and the area in the rectangle bounded by the lines

upper bounds the area of the rectangle below the curve (with equality when

upper bounds the area of the rectangle above the curve (with equality when

Young's inequality follows from evaluating the integrals.

Young's inequality follows by exponentiating.

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.

This can be shown by convexity arguments or by simply minimizing the single-variable function.

To prove full Young's inequality, clearly we assume that

Young's inequality may equivalently be written as

This also follows from the weighted AM-GM inequality.

which also gives rise to the so-called Young's inequality with

[5] 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

Start by observing that the square of every real number is zero or positive.

Work out the square of the right hand side:

Divide both sides by 2 and we have Young's inequality with exponent

T. Ando proved a generalization of Young's inequality for complex matrices ordered by Loewner ordering.

denotes the conjugate transpose of the matrix and

denote a real-valued, continuous and strictly increasing function on

this reduces to standard version for conjugate Hölder exponents.

For details and generalizations we refer to the paper of Mitroi & Niculescu.

[9] By denoting the convex conjugate of a real function

This follows immediately from the definition of the convex conjugate.

is defined on a real vector space

and thus Young's inequality for conjugate Hölder exponents mentioned above is a special case.

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.