Lyapunov moment inequality ($\Vert Z{\Vert }_r\le \Vert Z{\Vert }_s(0<r<s)$)

For a random variable $Z$ and exponents $0<r<s$, the absolute moment norms satisfy $(\mathbb{E}|Z{|}^r{)}^{1/r}\le (\mathbb{E}|Z{|}^s{)}^{1/s}$, allowing infinity. For a finite $s$th moment, apply the Jensen inequality to the concave function $t^{r/s}$ and $|Z{|}^s$. The formula is for positive exponents; $r=0$ requires a separately defined limiting quantity.