# Limit property of a function: $\lim_{p \to 0} \frac{w(c p)}{w(p)} \in (0,\infty)$

I have a function that has (needs to have) the following property: $\lim_{p \to 0} \frac{w(c p)}{w(p)} = k \in (0,\infty)$ for all $c \in (0,\infty)$. Do you know how this property is called or whether it is equivalent to some well-known properties?

Background info: It is a part of a larger project and $w : [0,1] \to [0,1]$ is a "weighting function" that is twice differentiable, strictly increasing, $w(0)=0$, $w(1)=1$, and strictly concave in some interval $(0,\varepsilon)$.

Since $w$ is twice differentiable, a sufficient condition for finiteness is that $w'(0) \in (0,\infty)$, since by L'Hopital's rule, the limit is equal to $\lim_{p \to 0} \frac{w'(c p)}{w'(p)}$ (and similarly for higher derivatives), but I cannot use this approach, since in my applications I cannot put these restrictions on the derivatives.

• Are you looking for regularly varying functions? – Sangchul Lee Apr 20 '15 at 15:15
• Yes, that seems to be the answer (although not as useful for me as I hoped). Do you want to submit this as an answer, so that I can approve it? – TomH Apr 21 '15 at 10:35