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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:

  1. $\phi'<0$,
  2. $(x\phi)'>0$,
  3. $x\phi(x)\le\min\{4,2\sqrt{x}\}$

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!

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    Could you proofread your conditions on $\Phi$, and if possible convert them to LaTeX?2012-07-13
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    Your notations need to be changed, otherwise one would write $\phi \in \phi$!2012-07-13
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    It is not immediate to me that $\phi$ (the function) is convex.2012-07-13
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    cross-posted to MO: http://mathoverflow.net/questions/102151/description-of-a-convex-set-of-functions2012-07-13
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    There is confusion here introduced by edits. There was a set $\Phi$ of functions, each named $\phi$, but one of the edits lowercased all these. I won't bother to fix the problem; once you're aware of it, it is easy enough to figure out which is which.2012-07-13
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    I doubt that you will have a closed form that is more transparent than the definition. E.g., what is a closed form for decreasing functions $f$ such that $xf(x)$ is increasing?2012-07-13

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