I have a question about the characterization of a set of functions.
Let $\phi$ a set containing all the functions $\phi(x): \mathbb{R}_{+}\mapsto \mathbb{R}_{+}$ that satisfy the following conditions:
- $\phi'<0$,
- $(x\phi)'>0$,
- $x\phi(x)\le\min\{4,2\sqrt{x}\}$
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!