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: This form of Young's inequality can be proved by Jensen's inequality and can be used to prove Hölder'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 formula_2 or formula_3 Generalizations. Using formula_4 and replacing formula_5 (resp. formula_6) with formula_7(resp. formula_8) results in the inequality: which is useful for proving Hölder's inequality. for every formula_10 and then calculate its minimum. 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" (valid for every "ε" > 0), sometimes called the Peter–Paul inequality. 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: Work out the square of the right hand side: Add '2ab' to both sides: Divide both sides by 2 and we have Young 's inequality with exponent 2: Young's inequality with "ε" follows by substituting formula_17 and formula_18 as below into Young's inequality with exponent 2: 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 formula_21. Standard version for increasing functions. For the standard version of the inequality, let "f" denote a real-valued, continuous and strictly increasing function on [0 , "c"] with "c" > 0 and "f"(0) = 0. Let "f" denote the inverse function of "f". Then, for all "a" ∈ [0, "c"] and "b" ∈ [0, "f"("c")], with equality if and only if "b" = "f"("a"). With formula_23 and formula_24, 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 (convex conjugate) 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 formula_26 and its convex conjugate is denoted by formula_27 (and is defined on the dual space formula_28), then where formula_30 is the dual pairing.