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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!

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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 '12 at 15:57
    
@WillSawin: I think that should be $0 < f(t) < 1/t$. –  Robert Israel Jul 13 '12 at 16:21
    
Cross-posted to MSE: math.stackexchange.com/questions/170387/… –  Yemon Choi Jul 13 '12 at 18:50
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