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I am looking for a characterization of functions $f : \mathbb R \rightarrow \mathbb R$ such that

$$|f|^p \geq c |f'|$$

for some constants $p,c > 0$. A complete characterization would be ideal, but I would also be satisfied with a large class of functions which satisfies this property. I am also curious about which polynomials satisfy this condition for various $c$ and $p$.

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    What is the motivation?2012-05-20
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    One the (open) set where $f$ is positive, the function $g=f^{1-p}$ has $|g'|=|(1-p)f'/f^p|\le c(1-p)$. So you get a bijection with Lipschitz functions. For sign-changing functions try $g=f|f|^{-p}$.2012-05-20

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