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