1 Lyapunov Inequality
Lyapunov’s inequality is a special case of the Hölder Inequality, replacing \(\lvert X \rvert\) by \(\lvert X \rvert^r\) with \(r>0\) and writing \(s=rp\), which states \[ (\mathbb{E}[\lvert X \rvert^r] )^{1/r}\leq (\mathbb{E}[\lvert X \rvert ^s])^{1/s};\quad 0 < r < s < \infty. \]
Proof: From Hölder we have that \[|\mathbb{E}[XY]|\leq \mathbb{E}|XY|\leq (\mathbb{E}|X|^p)^{1/p}\cdot (\mathbb{E}|Y|^q)^{1/q}.\] For \(Y=1\) and replacing \(\lvert X \rvert\) with \(\lvert X \rvert^r\) we have that \[ \mathbb{E}\lvert X \rvert ^r\leq (\mathbb{E}\lvert X \rvert ^{rp})^{1/p} = (\mathbb{E}\lvert X \rvert^s )^{r/s}, \] which we rearrange to obtain the result. \(\square\)