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 $p$.