# Inequality with Products of Integrals

Following from a previous post:

Let $h_i, h_j, f_i,f_j$ continuous functions such that $f$ are increasing and $f(0)=0$, Assume that: $\forall x$ $$h_i (x)>h_j (x)$$ $$f_i (x)>f_j (x)$$ and

$$\int_{0}^\infty h_i (x) f_i (x)dx>0$$ $$\int_{0}^\infty h_j (x) f_j (x)dx>0$$ Then, prove that $$\int_{0}^\infty h_i (x) f_i (x)dx>\int_{0}^\infty h_j (x) f_j (x) dx$$

I tried several things but haven't work to prove if the statement is right or wrong...

• I was thinking of adding the following conditions, but not sure if it adds anything... $\lim_{x\to \infty} h_i (x)=\lim_{x\to \infty} h_j (x)=0$ and for scale purposes,$\lim_{x\to \infty} f_i (x)=\lim_{x\to \infty} f_j (x)=1$ – Nick Aug 19 '17 at 14:45