Young's Inequality

The AM-GM inequality is a very special cases of Young's inequality. We derive it in a general form, using the coupling functions introduced by Moreau. Suppose is a real-valued function on and is a real-valued function on , called the coupling function. Here and are arbitrary. Define the -conjugate of by Then and thus , which is the generalized Young's inequality. We can also write this in the form that directly suggests minorization

The classical coupling function is with both and in the positive reals. If we take , with , then The is attained for , from which we find with .

Thus if such that Then for all we have with equality if and only if .