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 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