Littlewood interpolation inequality ($\Vert X{\Vert }_2^3\le \Vert X{\Vert }_1\Vert X{\Vert }_4^2$)

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 $\mathbb{E}|X{|}^2\le (\mathbb{E}|X|{)}^{2/3}(\mathbb{E}|X{|}^4{)}^{1/3}$. Raising to the power $3/2$ proves the displayed Lp norm interpolation inequality.