Integral inequality for positive functions

Question

Is it true that $ \int_Sf(x)g(x)dx \leq \int_Sf(x)dx\int_Sg(x)dx \ \forall f(x),g(x)\geq0 $ ?

Answer 1

Take $S$ to be the interval $[0,\epsilon]$, Take $f,g$ to be the functions that send $x$ to $x$. Then $$\int^{\epsilon}_0f(x)dx=\int^{\epsilon}_0xdx={\epsilon}^2/2$$

$$\int^{\epsilon}_0g(x)dx=\int^{\epsilon}_0xdx={\epsilon}^2/2$$

$$\int^{\epsilon}_0f(x)g(x)dx=\int^{\epsilon}_0x^2dx=\frac{{\epsilon}^3}{3}$$

However $\frac{{\epsilon}^3}{3}\leq {\epsilon}^2/2\times {\epsilon}^2/2$ is false for small ${\epsilon}$. It is false if $\epsilon$ equals $10^{-6}$ for example.