Littlewood interpolation inequality
= Littlewood interpolation inequality
{c}
{title2=$\|X\|_2^3\leq\|X\|_1\|X\|_4^2$}
For a <random variable> with finite fourth moment, apply <Holder inequality> with conjugate exponents $3/2$ and $3$ to $|X|^{2/3}|X|^{4/3}$. This gives $\mathbb E|X|^2\leq(\mathbb E|X|)^{2/3}(\mathbb E|X|^4)^{1/3}$. Raising to the power $3/2$ proves the displayed <Lp norm> interpolation inequality.