Young's Inequality

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

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 $b=|y|^q$.

Replacing $x$ by $\varepsilon^{1/p}x$ and $y$ by $\varepsilon^{-1/p}y$ gives, for $\varepsilon>0$,

For quadratic estimates one often writes the equivalent form $|xy|\le\varepsilon x^2+y^2/(4\varepsilon)$, which also follows by expanding $(\sqrt\varepsilon |x|-|y|/(2\sqrt\varepsilon))^2\ge0$.