Form for Hölder exponents
If are non-negative reals, and are positive reals that satisfy , then the following inequality holds for all possible values of and . with equality iff
Form for definite integrals
Suppose is a strictly increasing and continuous function on the interval where is a positive real number, and also . Then the following inequality holds for all and with equality iff .
The logarithm is concave and we know that , so by Jensen's Inequality, we have Young's Inequality then follows by exponentiation of both sides.