proving $\left\lVert Z \right\rVert_{2} \leq \left\lVert Z \right\rVert_{1}^{\frac{1}{4}} \left\lVert Z \right\rVert_{3}^{\frac{3}{4}}$

Question

I'm trying to prove Khintchine’s inequality and the book suggested using this inequality to prove it. I'm assuming I should use Hölder's inequality but I can't seem to reach the correct result. My attempt is:

$\left\lVert z \right\rVert_{2} = \left( E|z|^2 \right)^{1/2}$

$E|z|^2 = E|z^{p} z^{q}|$

I assume I should factorize the power of 2 to some p and q such that when I use Hölder's inequality it works, but I can't seem to find the right combination.