# Functions whose derivative is bounded by a power of itself

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? – lhf May 20 '12 at 12:15
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 $g=f|f|^{-p}$. – user31373 May 20 '12 at 14:29

Suppose $p\ge 1$. Then the condition implies either $f\ne 0$ or $f\equiv 0$, by Gronwall's lemma. If $f\ne 0$, the matter reduces to the boundedness of $|f'|/|f|^p$ at infinity, since it is bounded on compact sets by continuity. In particular, a polynomial other than $0$ satisfies $|f|^p\ge c|f'|$ with some $c>0$ if and only if it has no real zeros.
Next, suppose $0<p<1$. Define $g(x)=f(x)|f(x)|^{-p}$ if $f(x)\ne 0$ and $g(x)=0$ otherwise. The computation $|g'|=|(1-p)f'/f^p|\le c(1-p)$ shows that $g$ is Lipschitz. Conversely, $g$ being Lipschitz implies that $|f|^p\ge c|f'|$ holds for some $c>0$.