Young's Inequality

Young’s Inequality: Let $p,q>1$ satisfy $\frac1p+\frac1q=1$. Then for all $x,y\in\mathbb{R}$,

This inequality is a standard tool behind Hölder's inequality , Hölder's inequality for integrals , and many estimates in convex analysis . In the lecture notes it is obtained from the weighted AM–GM inequality applied to $a=|x|^p$ and $b=|y|^q$.

Examples:

• If $p=q=2$, then $|xy|\le \frac{x^2}{2}+\frac{y^2}{2}$.
• If $p=3$ and $q=\frac32$, then $|xy|\le \frac{|x|^3}{3}+\frac{2|y|^{3/2}}{3}$.