Hi everyone,

I have a question about the characterization of a set of functions.

Let $\Phi$ a set containing all the functions $\phi(x): \mathbb{R}_+\rightarrow \mathbb{R}_{+}$ that satisfy the following conditions:

$\phi'<0,\ \ (x\phi)'>0,\ \ x\phi(x)\leq\min(4, 2\sqrt{x})$

We can check easily that $\Phi$ is convex. How can we give a analytical characterization of $\Phi$? That is to say can we find a closed form for all the functions $\phi(x)$ in $\Phi$?

Thanks a lot for your help!