Is there some general solution to the functional inequality:

$$ f(xy) \leq y f(x) + x f(y)$$

Where $x,y\in[0,1]$?

I can find many particular solutions but I just wonder if there is a more general description of functions f satisfying such functional inequality. I have the following conditions

1) $f : [0,1]\rightarrow \mathbb R$

2) $f$ is non-negative on $[0,1]$

3) $f(0) = 0$ and $f(1)$ is positive and finite, let say normalised to $1$

4) $f$ is one time continuously differentiable on $(0,1)$

Thanks for any suggestions