description of a convex set of functions

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!

edited Jul 13 at 16:11

Igor Rivin
31.2k●3●37●71

asked Jul 13 at 15:46

Higgs88
43●4

	For the first two conditions, you can go as follows: $\phi(x)=e^{k-\int_0^x f(t)/t dt}$, for any $k$ and $f$ satisfying $0\leq f(t)\leq 1$. The second condition is harder from this perspective. – Will Sawin Jul 13 at 15:57
	@WillSawin: I think that should be $0 < f(t) < 1/t$. – Robert Israel Jul 13 at 16:21
	Cross-posted to MSE: math.stackexchange.com/questions/170387/… – Yemon Choi Jul 13 at 18:50