Cauchy-Schwartz inequality for 3 functions

Question

It is straightforward to see that the discrete Cauchy-Schwartz inequality can be generalized for 3 sequences: $$(\sum_{i=1}^{n}a_ib_ic_i)^2\le (\sum_{i=1}^na_i^2)(\sum_{i=1}^nb_i^2)(\sum_{i=1}^nc_i^2).$$ But is it true that the continuous version of the above inequality still hold? That is $$\left(\int_0^1 f(x)g(x)h(x)dx\right)^2\le \left(\int_0^1f(x)^2dx\right)\left(\int_0^1g(x)^2dx\right)\left(\int_0^1h(x)^2dx\right)?$$

Answer 1

$f(x)=g(x)=h(x)=x$ is a counter-example.