A continuous function $L :\mathbb{R}_+ \to \mathbb{R}_+$ is called regularly varying (at infinity) if for all $a > 0$,

$$ \lim_{x\to \infty} \frac{L(ax)}{L(x)}= g(a) $$

for some function $g(a)>0$. If $g(a)=1$ then $L$ is said to be slowly varying. This definition is due to Karamata, and a very good reference for this class of functions is the book:

*Bingham, N. H.; Goldie, C. M.; Teugels, J. L.*, Regular variation., Encyclopedia of Mathematics and its Applications, 27. Cambridge etc.: Cambridge University Press

I recently had to deal with a class of functions $S :\mathbb{R}_+ \to \mathbb{R}_+$ such that, for all $a>0$

$$ \lim_{x\to \infty} \frac{S(x^a)}{S(x)}= g(a), \tag{$\star$} $$

for some $g(a)>0$. It is clear that a function satisfyes $(\star)$ if and only if $S(x) = L(\log(x))$ for some regularly varying function $L$. Furthermore, one can show that $S$ is slowly varying.

Has the class of functions verying $(\star)$ been studied in the literature? Can someone point to a reference? If not, if you were to give a name to this class of functions, what would you suggest?