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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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    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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