Inequalities of integrals of periodic functions

Question

I have a function that has a shape similar to $\sin(x)^2$ (could be periodic extensions of $(x/(\pi/2))^2$ defined between $-\pi/2$ to $\pi/2$ for example). Let's call it $g(x)$. I want to show that the $$\int_0^\pi g(x)^2\cdot g(x+\pi/3)^2 g(x+2\pi/3)^2dx\leqslant \int_0^\pi g(x)^3 g(x+\pi/2)^3dx.$$ There is a generalized hypothesis that I want to prove but if one can prove this specific case then one can probably generalize. Thanks.