This tag is for questions relating to Young's inequality, a special case of the weighted AM-GM inequality. It is very useful in real analysis, including as a tool to prove Hölder's inequality. It is also a special case of a more general inequality known as Young's inequality for increasing functions.

Statement of the Inequality:Let $~p,~q~$ be positive real numbers satisfying $~\frac1{p} + \frac1{q} = 1~$. Then if $~a,~b~$ are non-negative real numbers, $$ab \le \frac{a^p}{p} + \frac{b^q}{q}~,$$ and equality holds if and only if $~a^p=b^q~$.

Young's inequality for products, bounding the product of two quantities, can be used to prove Hölder's inequality. It is also used widely 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.

Young's convolution inequality, bounding the convolution product of two functions. An example application is that Young's inequality can be used to show that the heat semigroup is a contracting semigroup using the $~L^2~$ norm (i.e. the Weierstrass transform does not enlarge the $~L^2~$ norm).

Young's inequality for integral operators, is a bound on the $~{\displaystyle L^{p}\to L^{q}}~$ operator norm of an integral operator in terms of $~{\displaystyle L^{r}}~$ norms of the kernel itself.

For more details see

http://mathworld.wolfram.com/YoungsInequality.html

https://brilliant.org/wiki/youngs-inequality

https://www.math.upenn.edu/~brweber/Courses/2011/Math361/Notes/YMandH.pdf