Prékopa-Leindler style inequality?

Does anyone know a simple proof of the following Prékopa-Leindler style inequality:

If we have $$f_1,f_2,g_1,g_2$$ strictly positive functions on $$\mathbb{R}$$ such that, for any $$x_1,x_2 \in \mathbb{R}$$, one has $$f_1(x_1)^2 f_2(x_2)^3 \leq g_1(x_1)^2g_2(x_2)^3$$ then $$\left(\int_\mathbb{R} f_1\right)^2\left(\int_\mathbb{R}f_2\right)^3 \leq \left(\int_\mathbb{R}g_1\right)^2\left(\int_\mathbb{R} g_2\right)^3.$$

We are given that for all $$x_1,x_2$$ we have $$(f_1/g_1)^2 (x_1)\leqslant (g_2/f_2)^3(x_2)$$, thus there exists $$c>0$$ such that $$(f_1/g_1)^2 (x_1)\leqslant c^6\leqslant (g_2/f_2)^3(x_2)$$, i.e. $$f_1(x)\leqslant c^3 g_1(x)$$, $$c^2 f_2(x)\leqslant g_2(x)$$ for all $$x$$, we integrate these pointwise inequalities, take appropriate powers of integrals and multiply.
• Hi ! Thx for your answer ! Then if you mix the right-hand side of the pointwise inequality as $g_1(x_1 + x_2)^2 g_2(x_1 - x_2)^3$, i guess it is not trivial anymore.. Apr 18 at 8:25