0.1 Lyapunov’s Inequality
Lyapunov’s inequality is a special case of Hölder's Inequality, replacing \(\lvert X \rvert\) by \(\lvert X \rvert^r\) with \(r>0\) and writing \(s=rp\), which states \[ (\mathbb{E}[\lvert X \rvert^r] )^{1/r}\leq (\mathbb{E}[\lvert X \rvert ^s])^{1/s};\quad 0 < r < s < \infty. \]