Young's inequality

As Young's inequality - named after William Henry Young - various inequalities are called in mathematics. In this article three inequalities are described, which were named after Young and closely communicate with each other. The second and third inequality listed here, in each case is a special case of the preceding. All three versions make it possible to assess the product against a sum.

Statement

General form

Let be a continuous, strictly increasing and unbounded function with, and be their (so existing ) inverse function, which has the same properties.

Then for all:

The equality holds if and only if is.

Special case

With and, then:

With equality if and only if.

This is obtained from the general case, by setting. The inverse function is then.

On the other hand, we obtain this inequality as an application of the inequality of the weighted arithmetic and geometric means for the two summands and the weights and.

The special case can also be directly derived (see proof archive).

Scaled version of the special case

With note: For

This is obtained from the previous special case for and.