Let f and g be functions, I am interested in finding (if there is any) the sufficient conditions on f and g such that we can satisfy an inequality of this type:
$$\frac{\int_{0}^{\infty}f(x)dx}{\int_{0}^{\infty}g(x)dx} \leq \int_{0}^{\infty}\frac{f(x)}{g(x)}dx$$
What I am interested in reality is to show that one limi its zero, but I want an upper bound that converges to 0 instead. I assumed that my functions f and g satisfies such conditions and I could get the limit. If anyone have experience with this kind of inequality, any help or insight will be appreciated.