Generalization of regularly varying functions

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?

• How about "logarithmically regularly/slowly varying"? – Greg Martin Nov 20 '18 at 21:58
• I do not favour this name. For example, a function $f$ is log-concave if $\log f$ is concave. So A log-slowly varying function would be a function such that $\log f$ is slowly varying. Here it is quite the opposite. But I will add this to the list of names anyway. – Raziel Nov 21 '18 at 8:30